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space with a bounded evaluation functional and show that this implies the existence of a unique reproducing kernel by the Riesz
Representation theorem. Mercer's theorem can also be extended to address the vector-valued setting and we can therefore obtain a feature map view of the vvRKHS. Lastly, it can also be shown that the closure of the span of
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While this view of the vvRKHS can be useful in multi-task learning, this isometry does not reduce the study of the vector-valued case to that of the scalar-valued case. In fact, this isometry procedure can make both the scalar-valued kernel and the input space too difficult to work with in practice
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and is a mainstay in the architecture of neural networks where it is used as an activation function. One can construct a ReLU-like nonlinear function using the theory of reproducing kernel
Hilbert spaces. Below, we derive this construction and show how it implies the representation power of neural
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This second property parallels the reproducing property for the scalar-valued case. This definition can also be connected to integral operators, bounded evaluation functions, and feature maps as we saw for the scalar-valued RKHS. We can equivalently define the vvRKHS as a vector-valued
Hilbert
2738:{\displaystyle \sum _{i,j=1}^{n}c_{i}c_{j}K(x_{i},x_{j})=\sum _{i=1}^{n}c_{i}\left\langle K_{x_{i}},\sum _{j=1}^{n}c_{j}K_{x_{j}}\right\rangle _{H}=\left\langle \sum _{i=1}^{n}c_{i}K_{x_{i}},\sum _{j=1}^{n}c_{j}K_{x_{j}}\right\rangle _{H}=\left\|\sum _{i=1}^{n}c_{i}K_{x_{i}}\right\|_{H}^{2}\geq 0}
616:
is usually defined as a
Hilbert space whose elements are equivalence classes of functions it can be trivially redefined as a Hilbert space of functions by using choice to select a (total) function as a representative for each equivalence class. However, no choice of representatives can make this
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is a
Hilbert space which we will call the feature space. The first sections presented the connection between bounded/continuous evaluation functions, positive definite functions, and integral operators and in this section we provide another representation of the RKHS in terms of feature maps.
599:, but they do not converge uniformly (i.e., they do not converge with respect to the supremum norm). This illustrates that pointwise convergence does not imply convergence in norm. It is important to note that the supremum norm does not arise from any inner product, as it does not satisfy the
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at every point in the domain, it does not lend itself to easy application in practice. A more intuitive definition of the RKHS can be obtained by observing that this property guarantees that the evaluation functional can be represented by taking the inner product of
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For ease of understanding, we provide the framework for real-valued
Hilbert spaces. The theory can be easily extended to spaces of complex-valued functions and hence include the many important examples of reproducing kernel Hilbert spaces that are spaces of
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is small). This follows from the fact that the inner product induces pointwise evaluation control. Roughly speaking, this means that closeness in the RKHS norm implies pointwise closeness, but the converse does not necessarily hold.
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4060:{\displaystyle |f(x)|\leq {\frac {1}{2\pi }}{\sqrt {2a\int _{-a}^{a}|F(\omega )|^{2}\,d\omega }}={\frac {\sqrt {2a}}{2\pi }}{\sqrt {\int _{-\infty }^{\infty }|F(\omega )|^{2}\,d\omega }}={\sqrt {\frac {a}{\pi }}}\|f\|_{L^{2}}.}
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2007:
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4856:. The Moore–Aronszajn theorem goes in the other direction; it states that every symmetric, positive definite kernel defines a unique reproducing kernel Hilbert space. The theorem first appeared in Aronszajn's
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9195:. This representation implies that the elements of the RKHS are inner products of elements in the feature space and can accordingly be seen as hyperplanes. This view of the RKHS is related to the
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8292:{\displaystyle \left\langle f,g\right\rangle _{H}=\sum _{i=1}^{\infty }{\frac {\left\langle f,\varphi _{i}\right\rangle _{L_{2}}\left\langle g,\varphi _{i}\right\rangle _{L_{2}}}{\sigma _{i}}}.}
6507:{\displaystyle f(x)=\langle K_{x},f\rangle _{G}=\langle K_{x},f_{H}\rangle _{G}+\langle K_{x},f_{H^{\bot }}\rangle _{G}=\langle K_{x},f_{H}\rangle _{G}=\langle K_{x},f_{H}\rangle _{H}=f_{H}(x),}
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4710:{\displaystyle \langle f,K_{x}\rangle _{L^{2}}=\int _{-\infty }^{\infty }f(y)\cdot {\overline {K_{x}(y)}}\,dy={\frac {1}{2\pi }}\int _{-a}^{a}F(\omega )\cdot e^{i\omega x}\,d\omega =f(x).}
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We lastly remark that the above theory can be further extended to spaces of functions with values in function spaces but obtaining kernels for these spaces is a more difficult task.
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13198:. As the scalar-valued kernel encodes dependencies between the inputs, we can observe that the matrix-valued kernel encodes dependencies among both the inputs and the outputs.
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5521:{\displaystyle f(x)=\sum _{i=1}^{\infty }a_{i}K_{x_{i}}(x)\quad {\text{where}}\quad \lim _{n\to \infty }\sup _{p\geq 0}\left\|\sum _{i=n}^{n+p}a_{i}K_{x_{i}}\right\|_{H_{0}}=0.}
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It is not entirely straightforward to construct natural examples of a
Hilbert space which are not an RKHS in a non-trivial fashion. Some examples, however, have been found.
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8100:{\displaystyle H=\left\{f\in L_{2}(X)\,{\Bigg \vert }\,\sum _{i=1}^{\infty }{\frac {\left\langle f,\varphi _{i}\right\rangle _{L_{2}}^{2}}{\sigma _{i}}}<\infty \right\}}
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This connection between kernels and feature maps provides us with a new way to understand positive definite functions and hence reproducing kernels as inner products in
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5188:{\displaystyle \left\langle \sum _{j=1}^{n}b_{j}K_{y_{j}},\sum _{i=1}^{m}a_{i}K_{x_{i}}\right\rangle _{H_{0}}=\sum _{i=1}^{m}\sum _{j=1}^{n}{a_{i}}b_{j}K(y_{j},x_{i}),}
4480:{\displaystyle \int _{-\infty }^{\infty }K_{x}(y)e^{-i\omega y}\,dy={\begin{cases}e^{-i\omega x}&{\text{if }}\omega \in ,\\0&{\textrm {otherwise}},\end{cases}}}
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5744:{\displaystyle \langle f,K_{x}\rangle _{H}=\sum _{i=1}^{\infty }a_{i}\left\langle K_{x_{i}},K_{x}\right\rangle _{H_{0}}=\sum _{i=1}^{\infty }a_{i}K(x_{i},x)=f(x).}
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8629:) is satisfied by the reproducing property. Another classical example of a feature map relates to the previous section regarding integral operators by taking
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8520:. Conversely, every positive definite function and corresponding reproducing kernel Hilbert space has infinitely many associated feature maps such that (
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We can gain intuition for the vvRKHS by taking a component-wise perspective on these spaces. In particular, we find that every vvRKHS is isometrically
12541:
The connection to the scalar-valued RKHS can then be made by the fact that every matrix-valued kernel can be identified with a kernel of the form of (
12238:
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is a reproducing kernel so that the corresponding RKHS can be defined in terms of these eigenvalues and eigenfunctions. We provide the details below.
1643:
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1190:
11457:
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14407:
Alpay, D., and T. M. Mills. "A family of
Hilbert spaces which are not reproducing kernel Hilbert spaces." J. Anal. Appl. 1.2 (2003): 107–111.
12713:
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14108:{\displaystyle K(x,y)=K_{y}(x)=\int _{0}^{x}G(z,y)\,dz={\begin{cases}x,&0\leq x<y\\y,&{\text{otherwise.}}\end{cases}}=\min(x,y)}
9958:
5201:
11712:
In this section we extend the definition of the RKHS to spaces of vector-valued functions as this extension is particularly important in
8902:{\displaystyle H_{\varphi }=\{f:X\to \mathbb {R} \mid \exists w\in F,f(x)=\langle w,\varphi (x)\rangle _{F},\forall {\text{ }}x\in X\}.}
8386:
9120:
14725:
Steinwart, Ingo; Scovel, Clint (2012). "Mercer's theorem on general domains: On the interaction between measures, kernels, and RKHSs".
7224:
4268:{\displaystyle K_{x}(y)={\frac {a}{\pi }}\operatorname {sinc} \left({\frac {a}{\pi }}(y-x)\right)={\frac {\sin(a(y-x))}{\pi (y-x)}}.}
14502:
Thomas-Agnan C . Computing a family of reproducing kernels for statistical applications. Numerical
Algorithms, 13, pp. 21-32 (1996)
911:
12067:
8766:
Lastly, feature maps allow us to construct function spaces that reveal another perspective on the RKHS. Consider the linear space
14761:
Rosasco, Lorenzo and Poggio, Thomas. "A Regularization Tour of
Machine Learning – MIT 9.520 Lecture Notes" Manuscript, Dec. 2014.
7878:{\displaystyle \lim _{n\to \infty }\sup _{u,v}\left|K(u,v)-\sum _{j=1}^{n}\sigma _{j}\,\varphi _{j}(u)\,\varphi _{j}(v)\right|=0.}
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2183:
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of the kernel function evaluated at the training points. This is a practically useful result as it effectively simplifies the
12309:
11616:
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De Vito, Ernest, Umanita, Veronica, and Villa, Silvia. "An extension of Mercer theorem to vector-valued measurable kernels,"
14192:
11069:{\displaystyle \|f\|_{H}^{2}=\int _{\mathbb {R} }{\Big (}{\frac {1}{\sigma }}f(x)^{2}+\sigma f'(x)^{2}{\Big )}\mathrm {d} x.}
13270:
13001:
9080:{\displaystyle \|f\|_{\varphi }=\inf\{\|w\|_{F}:w\in F,f(x)=\langle w,\varphi (x)\rangle _{F},\forall {\text{ }}x\in X\}.}
726:. The idea of the reproducing kernel remained untouched for nearly twenty years until it appeared in the dissertations of
699:
12995:-dimensional symmetric positive semi-definite matrix. In light of our previous discussion these kernels are of the form
9561:
1432:) is the weakest condition that ensures both the existence of an inner product and the evaluation of every function in
17:
8763:. Moreover, every feature map can naturally define a RKHS by means of the definition of a positive definite function.
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12478:
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We have seen how a reproducing kernel Hilbert space defines a reproducing kernel function that is both symmetric and
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https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=440218056738e05b5ab43679f932a9f33fccee87
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which states that every function in an RKHS that minimises an empirical risk functional can be written as a
651:
An RKHS is associated with a kernel that reproduces every function in the space in the sense that for every
13507:
11834:
10608:
6094:
2855:{\displaystyle n\in \mathbb {N} ,x_{1},\dots ,x_{n}\in X,{\text{ and }}c_{1},\dots ,c_{n}\in \mathbb {R} .}
1545:
715:
10597:{\displaystyle K(x,y)=(\alpha \langle x,y\rangle +1)^{d},\qquad \alpha \in \mathbb {R} ,d\in \mathbb {N} }
10236:
9547:{\displaystyle K((x_{1},\ldots ,x_{p}),(y_{1},\ldots ,y_{p}))=K_{1}(x_{1},y_{1})\cdots K_{p}(x_{p},y_{p})}
7496:
644:
would need to be the non-existent Dirac delta function). However, there are RKHSs in which the norm is an
14551:
Alvarez, Mauricio, Rosasco, Lorenzo and Lawrence, Neil, “Kernels for Vector-Valued Functions: a Review,”
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8558:
3533:{\displaystyle \langle f,g\rangle _{L^{2}}=\int _{-\infty }^{\infty }f(x)\cdot {\overline {g(x)}}\,dx.}
3144:
610:
13739:{\displaystyle f(y)=\int _{0}^{y}f'(x)\,dx=\int _{0}^{\infty }G(x,y)f'(x)\,dx=\langle K_{y},f\rangle }
13216:
12864:
12306:
As noted above, the RKHS associated to this reproducing kernel is given by the closure of the span of
12207:
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1324:
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to the RKHS, letting one prove the optimality of using ReLU activations in neural network settings.
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13785:
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11196:
4395:
3076:
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This representation of the RKHS has application in probability and statistics, for example to the
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Okutmustur, Baver. “Reproducing Kernel Hilbert Spaces,” M.S. dissertation, Bilkent University,
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2015:
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This above series representation is referred to as a Mercer kernel or Mercer representation of
3624:
703:
691:" can be performed by taking an inner product with a function determined by the kernel. Such a
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145:
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11790:. More formally, we define a vector-valued RKHS (vvRKHS) as a Hilbert space of functions
7467:
7126:
for self-adjoint operators implies that there is an at most countable decreasing sequence
6958:
6681:
2975:
8:
14388:
14373:
14356:
13832:{\displaystyle G(x,y)={\begin{cases}1,&x<y\\0,&{\text{otherwise}}\end{cases}}}
13155:
13109:
11747:
11713:
8532:
7683:{\displaystyle K(x,y)=\sum _{j=1}^{\infty }\sigma _{j}\,\varphi _{j}(x)\,\varphi _{j}(y)}
6652:
3116:
762:
711:
39:
11590:
7538:
7417:
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is symmetric and positive definiteness follows from the properties of inner product in
8483:
8357:
8113:
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7891:
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7028:
7008:
6658:
6634:
6062:
5892:{\displaystyle \langle K_{x},K_{y}\rangle _{H}=K(x,y)=\langle K_{x},K_{y}\rangle _{G}.}
4827:
4073:
3804:
3783:{\displaystyle f(x)={\frac {1}{2\pi }}\int _{-a}^{a}F(\omega )e^{ix\omega }\,d\omega .}
3630:
3584:
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3121:
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2999:
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81:
61:
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a continuous, symmetric, and positive definite function. Define the integral operator
14754:
14717:
Paulsen, Vern. “An introduction to the theory of reproducing kernel Hilbert spaces,”
14694:
14559:
14416:
Z. Pasternak-Winiarski, "On weights which admit reproducing kernel of Bergman type",
13504:
To construct the reproducing kernel it suffices to consider a dense subspace, so let
13494:{\displaystyle \langle f,g\rangle _{\mathcal {H}}=\int _{0}^{\infty }f'(x)g'(x)\,dx.}
3401:
3109:
2862:
The Moore–Aronszajn theorem (see below) is a sort of converse to this: if a function
2326:
810:
754:
750:
739:
723:
707:
14345:{\displaystyle \min(x,y)=x-\operatorname {ReLU} (x-y)=y-\operatorname {ReLU} (y-x).}
9938:{\displaystyle d_{K}(x,y)=\|K_{x}-K_{y}\|_{H}^{2}=2(1-K(x,y))\qquad \forall x\in X.}
14734:
14689:
14629:
14577:
10788:{\displaystyle K(x,y)=e^{-{\frac {\|x-y\|^{2}}{2\sigma ^{2}}}},\qquad \sigma >0}
7123:
5959:{\displaystyle \langle \cdot ,\cdot \rangle _{H}=\langle \cdot ,\cdot \rangle _{G}}
3264:{\displaystyle H=\{f\in L^{2}(\mathbb {R} )\mid \operatorname {supp} (F)\subset \}}
3139:
2993:
1126:
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problem from an infinite dimensional to a finite dimensional optimization problem.
746:
727:
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12661:) defines a matrix-valued kernel with the above expression. Now letting the map
6978:
Mercer's theorem states that the spectral decomposition of the integral operator
735:
12283:{\displaystyle \gamma :X\times \Lambda \times X\times \Lambda \to \mathbb {R} .}
1735:{\displaystyle f(x)=L_{x}(f)=\langle f,\ K_{x}\rangle _{H}\quad \forall f\in H.}
757:. Reproducing kernel Hilbert spaces are particularly important in the field of
14711:
11091:
11084:
7493:
and therefore we may choose continuous functions as the eigenvectors, that is,
1298:{\displaystyle |L_{x}(f)|:=|f(x)|\leq M_{x}\,\|f\|_{H}\qquad \forall f\in H.\,}
731:
14738:
11539:{\displaystyle K(x,y)={\frac {1}{\pi }}{\frac {1}{(1-x{\overline {y}})^{2}}}.}
4070:
This inequality shows that the evaluation functional is bounded, proving that
14802:
11342:
11111:
6676:
861:, equipped with pointwise addition and pointwise scalar multiplication. The
834:
738:. The subject was eventually systematically developed in the early 1950s by
51:
12975:
kernels which can factorized as the product of a scalar valued kernel and a
7578:
may be written in terms of the eigenvalues and continuous eigenfunctions as
648:-norm, such as the space of band-limited functions (see the example below).
14661:
Durrett, Greg. 9.520 Course Notes, Massachusetts Institute of Technology,
14378:
9196:
3113:
10886:{\displaystyle K(x,y)=e^{-{\frac {\|x-y\|}{\sigma }}},\qquad \sigma >0}
30:
14764:
14670:
11241:{\displaystyle K(x,y)={\begin{cases}1&x=y\\0&x\neq y\end{cases}}}
4861:
2882:
satisfies these conditions then there is a Hilbert space of functions on
14429:
T. Ł. Żynda, "On weights which admit reproducing kernel of Szegő type",
10388:
corresponding to this kernel is the dual space, consisting of functions
3393:{\textstyle F(\omega )=\int _{-\infty }^{\infty }f(t)e^{-i\omega t}\,dt}
2910:
The simplest example of a reproducing kernel Hilbert space is the space
2002:{\displaystyle K_{x}(y)=L_{y}(K_{x})=\langle K_{x},\ K_{y}\rangle _{H},}
14591:
12797:{\displaystyle (Df)(x,t)=\langle f(x),e_{t}\rangle _{\mathbb {R} ^{T}}}
12157:
8307:
2325:
in the complex case) is both symmetric (resp. conjugate symmetric) and
12053:{\displaystyle \langle f,\Gamma _{x}c\rangle _{H}=f(x)^{\intercal }c.}
11970:{\displaystyle \Gamma _{x}c(y)=\Gamma (x,y)c\in H{\text{ for }}y\in X}
8733:{\displaystyle \varphi (x)=({\sqrt {\sigma _{i}}}\varphi _{i}(x))_{i}}
11338:
10057:{\displaystyle K(x,y)^{2}\leq K(x,x)K(y,y)=1\qquad \forall x,y\in X.}
5276:{\displaystyle K(x,y)=\left\langle K_{x},K_{y}\right\rangle _{H_{0}}}
7122:
is a compact, continuous, self-adjoint, and positive operator. The
358:
An immediate consequence of this property is that if two functions
11345:
8457:{\displaystyle K(x,y)=\langle \varphi (x),\varphi (y)\rangle _{F}.}
14655:
14552:
9188:{\displaystyle K(x,y)=\langle \varphi (x),\varphi (y)\rangle _{F}}
5283:. The symmetry of this inner product follows from the symmetry of
3112:, as we now illustrate by example. Consider the Hilbert space of
315:
is called the reproducing kernel, and it reproduces the value of
13204:
695:
exists if and only if every evaluation functional is continuous.
34:
Figure illustrates related but varying approaches to viewing RKHS
11110:
can be represented as an array of complex numbers. If the usual
9313:
be a collection of corresponding positive definite functions on
698:
The reproducing kernel was first introduced in the 1907 work of
14608:
Reproducing kernel Hilbert spaces in Probability and Statistics
14490:
Reproducing kernel Hilbert spaces in Probability and Statistics
7292:{\displaystyle T_{K}\varphi _{i}(x)=\sigma _{i}\varphi _{i}(x)}
885:
is a linear functional that evaluates each function at a point
14418:
International Journal of Mathematics and Mathematical Sciences
969:{\displaystyle L_{x}:f\mapsto f(x){\text{ }}\forall f\in H.}
671:
in the set on which the functions are defined, "evaluation at
54:
of functions in which point evaluation is a continuous linear
14772:
12126:{\displaystyle \{\Gamma _{x}c:x\in X,c\in \mathbb {R} ^{T}\}}
11740:
is a symmetric function that is now a positive semi-definite
12971:
An important class of matrix-valued reproducing kernels are
6955:
is the space of square integrable functions with respect to
6675:
be a compact space equipped with a strictly positive finite
14663:
https://www.mit.edu/~9.520/scribe-notes/class03_gdurett.pdf
14251:
has the following representations with the ReLu function:
14080:
13825:
11234:
6909:{\displaystyle (\cdot )=\int _{X}K(\cdot ,t)f(t)\,d\mu (t)}
4473:
3108:
Nontrivial reproducing kernel Hilbert spaces often involve
12645:{\displaystyle \Gamma (x,y)_{(t,s)}=\gamma ((x,t),(y,s)).}
6626:
12467:{\displaystyle \gamma _{(x,t)}(y,s)=\gamma ((x,t),(y,s))}
12160:
to a scalar-valued RKHS on a particular input space. Let
7464:
maps continuously into the space of continuous functions
6631:
We may characterize a symmetric positive definite kernel
2253:{\displaystyle K(x,y)=\langle K_{x},\ K_{y}\rangle _{H}.}
12368:{\displaystyle \{\gamma _{(x,t)}:x\in X,t\in \Lambda \}}
11697:{\displaystyle K(x,y)={\frac {\sin a(x-y)}{\pi (x-y)}}.}
10614:
These are another common class of kernels which satisfy
1544:
from which the RKHS takes its name. More formally, the
722:
which satisfy the reproducing property in the theory of
14244:{\displaystyle X\times X=[0,\infty )\times [0,\infty )}
11707:
4883:. Then there is a unique Hilbert space of functions on
4720:
Thus we obtain the reproducing property of the kernel.
14779:
Zhang, Haizhang; Xu, Yuesheng; Zhang, Qinghui (2012).
13326:{\displaystyle {\mathcal {H}}=L_{2}^{1}(0)[0,\infty )}
12968:
as properties of the original kernels are often lost.
12153:, another property similar to the scalar-valued case.
11720:. The main difference is that the reproducing kernel
7178:
3318:
1386:
14260:
14195:
14172:
14124:
13942:
13848:
13758:
13596:
13558:
13510:
13404:
13374:
13339:
13273:
13219:
13184:
13158:
13138:
13112:
13096:{\displaystyle \gamma ((x,t),(y,s))=K(x,y)K_{T}(t,s)}
13004:
12981:
12943:
12916:
12896:
12867:
12840:
12813:
12716:
12667:
12556:
12481:
12381:
12312:
12241:
12210:
12166:
12139:
12070:
11989:
11900:
11872:
11837:
11796:
11776:
11750:
11726:
11619:
11593:
11555:
11460:
11419:
11397:
11351:
11321:
11293:
11261:
11169:
11131:
10945:
10923:
10903:
10814:
10697:
10620:
10506:
10441:
10394:
10374:
10318:
10284:
10239:
10197:
10165:
10130:
10098:
10074:
9961:
9813:
9783:
9742:
9722:
9679:
9659:
9623:
9564:
9377:
9319:
9268:
9217:
9123:
9096:
8948:
8918:
8775:
8749:
8668:
8635:
8603:
8561:
8535:
8506:
8486:
8389:
8360:
8328:
8139:
8116:
7960:
7937:
7917:
7894:
7734:
7699:
7587:
7564:
7541:
7499:
7470:
7443:
7420:
7374:
7338:
7305:
7227:
7132:
7101:
7078:
7051:
7031:
7011:
6984:
6961:
6925:
6818:
6744:
6704:
6684:
6661:
6637:
6592:
6554:
6523:
6271:
6230:
6183:
6150:
6097:
6065:
6017:
5972:
5908:
5789:
5546:
5322:
5204:
4959:
4830:
4795:
4759:
4728:
4507:
4323:
4284:
4129:
4099:
4076:
3830:
3807:
3693:
3653:
3633:
3607:
3587:
3549:
3433:
3410:
3280:
3182:
3147:
3124:
3079:
3048:
3022:
3002:
2978:
2958:
2916:
2888:
2868:
2754:
2338:
2309:
2269:
2186:
2161:
2121:
2101:
2071:
2051:
2018:
1905:
1882:
1855:
1833:
1811:
1791:
1764:
1646:
1621:
1594:
1574:
1554:
1530:
1506:
1479:
1459:
1438:
1360:
1327:
1193:
1155:
1135:
1104:
1084:
1064:
1033:
1013:
993:
914:
891:
871:
847:
819:
795:
677:
657:
623:
579:
537:
476:
456:
436:
404:
384:
364:
341:
321:
294:
235:
207:
174:
148:
126:
104:
84:
64:
3581:, it is a HIlbert space. Moreover, the elements of
14781:"Refinement of Operator-valued Reproducing Kernels"
13587:. The Fundamental Theorem of Calculus then gives
2095:This allows us to define the reproducing kernel of
14344:
14243:
14178:
14158:
14107:
13925:
13831:
13738:
13579:
13544:
13493:
13387:
13360:
13325:
13255:
13190:
13170:
13144:
13124:
13095:
12987:
12956:
12929:
12902:
12882:
12853:
12826:
12796:
12699:
12644:
12529:
12466:
12367:
12282:
12222:
12196:
12145:
12125:
12052:
11969:
11884:
11858:
11823:
11782:
11762:
11732:
11696:
11602:
11579:
11538:
11443:
11413:. It can be shown that the reproducing kernel for
11405:
11375:
11329:
11307:
11276:
11240:
11152:
11068:
10929:
10909:
10885:
10787:
10668:
10596:
10484:
10427:
10380:
10357:
10290:
10270:
10218:
10183:
10151:
10116:
10080:
10056:
9937:
9795:
9769:
9728:
9705:
9665:
9645:
9605:
9546:
9359:
9305:
9254:
9187:
9109:
9079:
8931:
8901:
8755:
8732:
8654:
8615:
8589:
8547:
8512:
8492:
8456:
8366:
8346:
8291:
8122:
8099:
7943:
7923:
7900:
7877:
7717:
7682:
7570:
7550:
7527:
7485:
7456:
7429:
7406:
7360:
7324:
7291:
7213:
7164:
7114:
7084:
7064:
7045:in terms of the eigenvalues and eigenfunctions of
7037:
7017:
6997:
6967:
6947:
6908:
6801:
6730:
6690:
6667:
6643:
6611:
6574:
6536:
6506:
6242:
6216:
6169:
6136:
6071:
6029:
6003:
5958:
5891:
5743:
5520:
5287:and the non-degeneracy follows from the fact that
5275:
5187:
4836:
4816:
4781:
4741:
4709:
4479:
4306:
4267:
4112:
4082:
4059:
3813:
3782:
3661:
3639:
3623:that tend to zero at infinity, essentially by the
3615:
3593:
3573:
3532:
3416:
3392:
3304:
3263:
3165:
3130:
3097:
3061:
3034:
3008:
2984:
2964:
2944:
2894:
2874:
2854:
2737:
2317:
2295:
2252:
2169:
2147:
2107:
2084:
2057:
2037:
2001:
1888:
1868:
1841:
1819:
1797:
1777:
1734:
1627:
1607:
1580:
1560:
1536:
1512:
1492:
1465:
1444:
1415:
1372:
1346:
1297:
1174:
1141:
1117:
1090:
1070:
1046:
1019:
999:
968:
897:
877:
853:
825:
801:
683:
663:
636:
591:
565:
519:
462:
442:
422:
390:
370:
347:
327:
307:
278:
219:
193:
160:
134:
112:
90:
70:
14682:Journal of Mathematical Analysis and Applications
14675:"Some results on Tchebycheffian Spline Functions"
14569:Transactions of the American Mathematical Society
11050:
10984:
9606:{\displaystyle X=X_{1}\times \dots \times X_{p}.}
8000:
4749:in this case is the "bandlimited version" of the
531:For example, consider the sequence of functions
14800:
14261:
14087:
13235:
10607:
8968:
7752:
7736:
7180:
5758:be another Hilbert space of functions for which
5419:
5403:
1388:
573:. These functions converge pointwise to 0 as
477:
14724:
12530:{\displaystyle (x,t),(y,s)\in X\times \Lambda }
11549:Lastly, the space of band limited functions in
6655:and obtain an additional view of the RKHS. Let
4492:time-shifting property of the Fourier transform
3312:is the set of square integrable functions, and
865:functional over the Hilbert space of functions
745:These spaces have wide applications, including
14668:
13205:Connection between RKHSs and the ReLU function
11160:can be thought of as an identity matrix since
7214:{\textstyle \lim _{i\to \infty }\sigma _{i}=0}
1849:in the case of complex Hilbert spaces) and as
14621:Bulletin of the American Mathematical Society
14616:"On the Mathematical Foundations of Learning"
14431:Journal of Contemporary Mathematical Analysis
13926:{\displaystyle K_{y}'(x)=G(x,y),\ K_{y}(0)=0}
10428:{\displaystyle f(x)=\langle x,\beta \rangle }
8347:{\displaystyle \varphi \colon X\rightarrow F}
279:{\displaystyle \langle f,K_{x}\rangle =f(x).}
14778:
14712:http://www.thesis.bilkent.edu.tr/0002953.pdf
13733:
13714:
13418:
13405:
13250:
13238:
12776:
12747:
12700:{\displaystyle D:H_{\Gamma }\to H_{\gamma }}
12362:
12313:
12191:
12173:
12120:
12071:
12013:
11990:
10953:
10946:
10859:
10847:
10743:
10730:
10660:
10648:
10546:
10534:
10485:{\displaystyle \|f\|_{H}^{2}=\|\beta \|^{2}}
10473:
10466:
10449:
10442:
10422:
10410:
10352:
10340:
10265:
10240:
9869:
9842:
9176:
9145:
9071:
9042:
9020:
8981:
8974:
8971:
8956:
8949:
8893:
8864:
8842:
8789:
8442:
8411:
7319:
7306:
6470:
6443:
6431:
6404:
6392:
6358:
6346:
6319:
6307:
6287:
5998:
5973:
5947:
5934:
5922:
5909:
5877:
5850:
5817:
5790:
5567:
5547:
4528:
4508:
4038:
4031:
3447:
3434:
3258:
3189:
3086:
3080:
2263:From this definition it is easy to see that
2238:
2208:
1987:
1957:
1707:
1684:
1269:
1262:
417:
405:
255:
236:
14613:
11102:consist of all complex-valued functions on
7911:Furthermore, it can be shown that the RKHS
6731:{\displaystyle K:X\times X\to \mathbb {R} }
6051:Now we need to prove that every element of
5531:Now we can check the reproducing property (
2296:{\displaystyle K:X\times X\to \mathbb {R} }
2148:{\displaystyle K:X\times X\to \mathbb {R} }
14433:(Armenian Academy of Sciences), 55, 2020.
10358:{\displaystyle K(x,y)=\langle x,y\rangle }
6802:{\displaystyle T_{K}:L_{2}(X)\to L_{2}(X)}
6217:{\displaystyle f_{H^{\bot }}\in H^{\bot }}
4847:
4824:in the weak sense as the cutoff frequency
14693:
14633:
14581:
14558:
14355:Using this formulation, we can apply the
14019:
13704:
13644:
13481:
12870:
12782:
12655:Moreover, every kernel with the form of (
12273:
12230:and the corresponding reproducing kernel
12110:
11846:
11811:
11570:
11434:
11399:
11366:
11323:
11301:
11264:
10976:
10590:
10576:
8805:
8005:
7997:
7844:
7824:
7660:
7640:
6890:
6724:
5309:with respect to this inner product. Then
4682:
4608:
4380:
4007:
3926:
3770:
3655:
3609:
3564:
3520:
3383:
3295:
3212:
2845:
2762:
2311:
2289:
2163:
2141:
1835:
1813:
1294:
1261:
128:
106:
13395:) derivative. It has the inner product
13333:of absolutely continuous functions with
29:
14606:Berlinet, Alain and Thomas, Christine.
14487:Berlinet, Alain and Thomas, Christine.
12197:{\displaystyle \Lambda =\{1,\dots ,T\}}
11824:{\displaystyle f:X\to \mathbb {R} ^{T}}
8378:Every feature map defines a kernel via
7165:{\displaystyle (\sigma _{i})_{i}\geq 0}
6627:Integral operators and Mercer's theorem
27:In functional analysis, a Hilbert space
14:
14801:
7407:{\displaystyle T_{K},\sigma _{i}>0}
2902:for which it is a reproducing kernel.
398:in the RKHS are close in norm (i.e.,
14614:Cucker, Felipe; Smale, Steve (2002).
13545:{\displaystyle f\in C^{1}[0,\infty )}
12910:is bijective and an isometry between
12861:component of the canonical basis for
11859:{\displaystyle c\in \mathbb {R} ^{T}}
10495:
6137:{\displaystyle f=f_{H}+f_{H^{\bot }}}
14788:Journal of Machine Learning Research
14769:Spline Models for Observational Data
13267:We will work with the Hilbert space
12232:
11708:Extension to vector-valued functions
11121:is the function whose value is 1 at
10271:{\displaystyle \{K_{x}\mid x\in X\}}
8380:
7528:{\displaystyle \varphi _{i}\in C(X)}
1637:
1184:
14610:, Kluwer Academic Publishers, 2004.
11580:{\displaystyle L^{2}(\mathbb {R} )}
11444:{\displaystyle H^{2}(\mathbb {D} )}
11376:{\displaystyle H^{2}(\mathbb {D} )}
10307:
8529:For example, we can trivially take
6048:and hence contains its completion.
5762:is a reproducing kernel. For every
3574:{\displaystyle L^{2}(\mathbb {R} )}
3543:Since this is a closed subspace of
3305:{\displaystyle L^{2}(\mathbb {R} )}
1416:{\textstyle \sup _{x}M_{x}=\infty }
24:
14493:, Kluwer Academic Publishers, 2004
14235:
14217:
13664:
13536:
13442:
13423:
13317:
13276:
12922:
12679:
12557:
12524:
12359:
12266:
12254:
12217:
12167:
12075:
12000:
11926:
11902:
11727:
11610:is a RKHS with reproducing kernel
11341:) is more sophisticated. Here the
11083:
11056:
10302:
10068:This inequality allows us to view
10033:
9920:
9360:{\displaystyle (X_{i})_{i=1}^{p}.}
9054:
8876:
8812:
8185:
8089:
8022:
7746:
7625:
7190:
7025:yields a series representation of
6565:
6384:
6209:
6194:
6127:
5683:
5595:
5413:
5354:
5313:consists of functions of the form
4560:
4555:
4337:
4332:
3973:
3968:
3479:
3474:
3347:
3342:
3160:
1717:
1520:. This function is the so-called
1410:
1380:, it might still be the case that
1341:
1279:
951:
586:
25:
14820:
14583:10.1090/S0002-9947-1950-0051437-7
14384:Kernel embedding of distributions
14189:Moreover the minimum function on
14159:{\displaystyle K_{y}=K(\cdot ,y)}
13264:networks with ReLU activations.
10669:{\displaystyle K(x,y)=K(\|x-y\|)}
9803:. Define a pseudo-metric on X as
9706:{\displaystyle X_{0}\times X_{0}}
9306:{\displaystyle (K_{i})_{i=1}^{p}}
9255:{\displaystyle (X_{i})_{i=1}^{p}}
9117:is a RKHS with kernel defined by
8590:{\displaystyle \varphi (x)=K_{x}}
6517:where we have used the fact that
3166:{\displaystyle 0<a<\infty }
470:are also pointwise close (i.e.,
58:. Specifically, a Hilbert space
13256:{\displaystyle f(x)=\max\{0,x\}}
12883:{\displaystyle \mathbb {R} ^{T}}
12223:{\displaystyle X\times \Lambda }
11277:{\displaystyle \mathbb {C} ^{n}}
7325:{\displaystyle \{\varphi _{i}\}}
6651:via the integral operator using
6004:{\displaystyle \{K_{x}:x\in X\}}
1785:is itself a function defined on
1347:{\displaystyle M_{x}<\infty }
985:reproducing kernel Hilbert space
520:{\displaystyle \sup |f(x)-g(x)|}
44:reproducing kernel Hilbert space
14564:"Theory of Reproducing Kernels"
14553:https://arxiv.org/abs/1106.6251
14532:
14523:
14514:
14505:
14496:
10897:The squared norm of a function
10873:
10775:
10568:
10032:
9919:
9646:{\displaystyle X_{0}\subset X,}
8313:
6548:so that its inner product with
5401:
5395:
4860:, although he attributes it to
3424:. As the inner product, we use
1716:
1635:with the reproducing property,
1278:
14481:
14472:
14463:
14454:
14445:
14436:
14423:
14410:
14401:
14336:
14324:
14306:
14294:
14276:
14264:
14238:
14226:
14220:
14208:
14153:
14141:
14102:
14090:
14016:
14004:
13980:
13974:
13958:
13946:
13914:
13908:
13889:
13877:
13868:
13862:
13774:
13762:
13701:
13695:
13684:
13672:
13641:
13635:
13606:
13600:
13568:
13562:
13539:
13527:
13478:
13472:
13461:
13455:
13349:
13343:
13320:
13308:
13305:
13299:
13229:
13223:
13090:
13078:
13065:
13053:
13044:
13041:
13029:
13023:
13011:
13008:
12759:
12753:
12741:
12729:
12726:
12717:
12684:
12636:
12633:
12621:
12615:
12603:
12600:
12589:
12577:
12573:
12560:
12512:
12500:
12494:
12482:
12461:
12458:
12446:
12440:
12428:
12425:
12416:
12404:
12399:
12387:
12333:
12321:
12269:
12035:
12028:
11941:
11929:
11920:
11914:
11806:
11685:
11673:
11665:
11653:
11635:
11623:
11574:
11566:
11521:
11498:
11476:
11464:
11438:
11430:
11370:
11362:
11308:{\displaystyle X=\mathbb {D} }
11185:
11173:
11147:
11135:
11039:
11032:
11009:
11002:
10830:
10818:
10713:
10701:
10663:
10645:
10636:
10624:
10556:
10528:
10522:
10510:
10404:
10398:
10334:
10322:
10213:
10201:
10146:
10134:
10023:
10011:
10005:
9993:
9978:
9965:
9916:
9913:
9901:
9889:
9836:
9824:
9758:
9746:
9541:
9515:
9499:
9473:
9457:
9454:
9422:
9416:
9384:
9381:
9334:
9320:
9283:
9269:
9232:
9218:
9172:
9166:
9157:
9151:
9139:
9127:
9038:
9032:
9014:
9008:
8860:
8854:
8836:
8830:
8801:
8721:
8717:
8711:
8684:
8678:
8672:
8571:
8565:
8438:
8432:
8423:
8417:
8405:
8393:
8338:
7994:
7988:
7861:
7855:
7841:
7835:
7787:
7775:
7743:
7677:
7671:
7657:
7651:
7603:
7591:
7522:
7516:
7480:
7474:
7355:
7349:
7286:
7280:
7254:
7248:
7187:
7147:
7133:
6942:
6936:
6903:
6897:
6887:
6881:
6875:
6863:
6844:
6838:
6835:
6819:
6796:
6790:
6777:
6774:
6768:
6720:
6498:
6492:
6281:
6275:
5844:
5832:
5735:
5729:
5720:
5701:
5495:
5436:
5410:
5392:
5386:
5332:
5326:
5220:
5208:
5179:
5153:
4943:}. Define an inner product on
4811:
4799:
4776:
4770:
4701:
4695:
4660:
4654:
4599:
4593:
4574:
4568:
4490:which is a consequence of the
4445:
4430:
4358:
4352:
4301:
4295:
4256:
4244:
4236:
4233:
4221:
4215:
4195:
4183:
4146:
4140:
3997:
3992:
3986:
3979:
3916:
3911:
3905:
3898:
3849:
3845:
3839:
3832:
3751:
3745:
3703:
3697:
3568:
3560:
3511:
3505:
3493:
3487:
3361:
3355:
3328:
3322:
3299:
3291:
3255:
3240:
3234:
3228:
3216:
3208:
3098:{\displaystyle \{x\}\subset X}
2939:
2927:
2714:
2661:
2415:
2389:
2285:
2202:
2190:
2137:
1951:
1938:
1922:
1916:
1678:
1672:
1656:
1650:
1588:there exists a unique element
1244:
1240:
1234:
1227:
1219:
1215:
1209:
1195:
943:
937:
931:
583:
560:
554:
513:
509:
503:
494:
488:
481:
270:
264:
13:
1:
14635:10.1090/S0273-0979-01-00923-5
14545:
12854:{\displaystyle t^{\text{th}}}
10609:Radial basis function kernels
10159:will be closer to 1 while if
9716:Consider a normalized kernel
9713:is also a reproducing kernel.
9202:
8306:for stochastic processes and
8304:Karhunen-Loève representation
7332:form an orthonormal basis of
6575:{\displaystyle f_{H^{\bot }}}
4858:Theory of Reproducing Kernels
3173:and define the Hilbert space
2945:{\displaystyle L^{2}(X,\mu )}
784:
566:{\displaystyle \sin ^{2n}(x)}
14695:10.1016/0022-247X(71)90184-3
13368:and square integrable (i.e.
11515:
11406:{\displaystyle \mathbb {D} }
11330:{\displaystyle \mathbb {D} }
11090:We also provide examples of
9207:Useful properties of RKHSs:
9110:{\displaystyle H_{\varphi }}
8932:{\displaystyle H_{\varphi }}
4817:{\displaystyle \delta (y-x)}
4603:
3662:{\displaystyle \mathbb {R} }
3627:. In fact, the elements of
3616:{\displaystyle \mathbb {R} }
3515:
2318:{\displaystyle \mathbb {C} }
2170:{\displaystyle \mathbb {C} }
1842:{\displaystyle \mathbb {C} }
1820:{\displaystyle \mathbb {R} }
1546:Riesz representation theorem
592:{\displaystyle n\to \infty }
135:{\displaystyle \mathbb {C} }
113:{\displaystyle \mathbb {R} }
7:
14362:
12957:{\displaystyle H_{\gamma }}
12930:{\displaystyle H_{\Gamma }}
12657:
12543:
12296:
11125:and 0 everywhere else, and
10233:The closure of the span of
8655:{\displaystyle F=\ell ^{2}}
8625:
8522:
8470:
8110:where the inner product of
6254:is a reproducing kernel of
5776:
5533:
2905:
1748:
1428:
1311:
771:empirical risk minimization
759:statistical learning theory
168:, there exists a function
142:) is an RKHS if, for each
10:
14825:
14420:, vol. 15, Issue 1, 1992.
10685:squared exponential kernel
9262:be a sequence of sets and
7558:Then by Mercer's theorem
6170:{\displaystyle f_{H}\in H}
6030:{\displaystyle H\subset G}
4314:defined above is given by
2038:{\displaystyle K_{y}\in H}
1175:{\displaystyle M_{x}>0}
761:because of the celebrated
194:{\displaystyle K_{x}\in H}
14739:10.1007/s00365-012-9153-3
10676:. Some examples include:
9951:Cauchy–Schwarz inequality
7072:. This then implies that
6623:and concludes the proof.
6586:is zero. This shows that
6041:is complete and contains
5754:To prove uniqueness, let
4278:The Fourier transform of
4120:in this case is given by
3795:Cauchy–Schwarz inequality
3682:Fourier inversion theorem
3042:, the reproducing kernel
1805:with values in the field
1149:, i.e. there exists some
78:of functions from a set
14394:
14369:Positive definite kernel
10184:{\displaystyle x,y\in X}
10117:{\displaystyle x,y\in X}
9770:{\displaystyle K(x,x)=1}
9653:then the restriction of
8912:We can define a norm on
7718:{\displaystyle x,y\in X}
7361:{\displaystyle L_{2}(X)}
7095:Under these assumptions
6948:{\displaystyle L_{2}(X)}
6087:is a closed subspace of
4891:is a reproducing kernel.
4877:positive definite kernel
4782:{\displaystyle K_{x}(y)}
4307:{\displaystyle K_{x}(y)}
3647:are the restrictions to
3601:are smooth functions on
2177:in the complex case) by
718:simultaneously examined
13213:is commonly defined as
12474:for every set of pairs
11733:{\displaystyle \Gamma }
11718:manifold regularization
7437:One can also show that
7368:. By the positivity of
6612:{\displaystyle f=f_{H}}
4928:be the linear span of {
4848:Moore–Aronszajn theorem
3793:It then follows by the
704:boundary value problems
423:{\displaystyle \|f-g\|}
355:via the inner product.
14346:
14245:
14180:
14160:
14109:
13927:
13833:
13740:
13581:
13580:{\displaystyle f(0)=0}
13546:
13495:
13389:
13362:
13361:{\displaystyle f(0)=0}
13327:
13257:
13192:
13172:
13146:
13126:
13097:
12989:
12958:
12931:
12904:
12884:
12855:
12828:
12798:
12701:
12646:
12531:
12468:
12369:
12284:
12224:
12204:. Consider the space
12198:
12147:
12127:
12054:
11971:
11886:
11885:{\displaystyle x\in X}
11860:
11825:
11784:
11764:
11734:
11698:
11604:
11581:
11540:
11445:
11407:
11377:
11331:
11309:
11278:
11242:
11154:
11153:{\displaystyle K(x,y)}
11106:. Then an element of
11070:
10931:
10911:
10887:
10789:
10670:
10598:
10486:
10429:
10382:
10359:
10292:
10272:
10220:
10219:{\displaystyle K(x,y)}
10185:
10153:
10152:{\displaystyle K(x,y)}
10118:
10082:
10058:
9939:
9797:
9796:{\displaystyle x\in X}
9771:
9730:
9707:
9667:
9647:
9607:
9548:
9361:
9307:
9256:
9189:
9111:
9081:
8933:
8903:
8757:
8734:
8656:
8617:
8616:{\displaystyle x\in X}
8591:
8549:
8514:
8494:
8458:
8368:
8348:
8293:
8189:
8124:
8101:
8026:
7945:
7925:
7902:
7879:
7813:
7719:
7684:
7629:
7572:
7552:
7529:
7487:
7458:
7431:
7408:
7362:
7326:
7293:
7215:
7166:
7116:
7086:
7066:
7039:
7019:
6999:
6969:
6949:
6910:
6803:
6732:
6692:
6669:
6645:
6613:
6576:
6538:
6508:
6244:
6243:{\displaystyle x\in X}
6218:
6171:
6138:
6073:
6031:
6005:
5960:
5893:
5745:
5687:
5599:
5522:
5466:
5358:
5291:is positive definite.
5277:
5189:
5127:
5106:
5037:
4986:
4838:
4818:
4783:
4743:
4711:
4494:. Consequently, using
4481:
4308:
4269:
4114:
4084:
4061:
3815:
3784:
3663:
3641:
3625:Riemann-Lebesgue lemma
3617:
3595:
3575:
3534:
3418:
3394:
3306:
3265:
3167:
3132:
3099:
3063:
3036:
3035:{\displaystyle x\in X}
3010:
2986:
2966:
2946:
2896:
2876:
2856:
2739:
2685:
2617:
2566:
2498:
2441:
2365:
2319:
2297:
2254:
2171:
2149:
2109:
2086:
2059:
2039:
2003:
1890:
1870:
1843:
1821:
1799:
1779:
1736:
1629:
1609:
1582:
1562:
1538:
1524:for the Hilbert space
1514:
1494:
1467:
1446:
1417:
1374:
1373:{\displaystyle x\in X}
1348:
1299:
1176:
1143:
1119:
1092:
1072:
1048:
1021:
1001:
970:
899:
879:
855:
827:
803:
685:
665:
638:
593:
567:
521:
464:
444:
424:
392:
372:
349:
329:
309:
280:
221:
220:{\displaystyle f\in H}
195:
162:
161:{\displaystyle x\in X}
136:
114:
92:
72:
35:
14347:
14246:
14181:
14161:
14110:
13928:
13834:
13741:
13582:
13547:
13496:
13390:
13388:{\displaystyle L_{2}}
13363:
13328:
13258:
13193:
13173:
13147:
13127:
13098:
12990:
12959:
12932:
12905:
12885:
12856:
12829:
12827:{\displaystyle e_{t}}
12799:
12702:
12647:
12532:
12469:
12370:
12285:
12225:
12199:
12148:
12128:
12055:
11972:
11887:
11861:
11826:
11785:
11765:
11735:
11699:
11605:
11582:
11541:
11446:
11408:
11389:holomorphic functions
11378:
11332:
11310:
11279:
11243:
11155:
11071:
10937:with this kernel is:
10932:
10912:
10888:
10790:
10671:
10599:
10487:
10430:
10383:
10360:
10293:
10273:
10221:
10186:
10154:
10119:
10090:measure of similarity
10083:
10059:
9940:
9798:
9772:
9731:
9708:
9668:
9648:
9608:
9549:
9367:It then follows that
9362:
9308:
9257:
9199:in machine learning.
9190:
9112:
9090:It can be shown that
9082:
8934:
8904:
8758:
8735:
8657:
8618:
8592:
8550:
8515:
8495:
8459:
8369:
8349:
8294:
8169:
8125:
8102:
8006:
7946:
7926:
7903:
7880:
7793:
7720:
7685:
7609:
7573:
7553:
7530:
7488:
7459:
7457:{\displaystyle T_{K}}
7432:
7409:
7363:
7327:
7294:
7216:
7167:
7117:
7115:{\displaystyle T_{K}}
7087:
7067:
7065:{\displaystyle T_{K}}
7040:
7020:
7000:
6998:{\displaystyle T_{K}}
6970:
6950:
6911:
6804:
6733:
6693:
6670:
6646:
6614:
6577:
6539:
6537:{\displaystyle K_{x}}
6509:
6245:
6219:
6172:
6139:
6074:
6032:
6006:
5961:
5894:
5746:
5667:
5579:
5523:
5440:
5338:
5278:
5190:
5107:
5086:
5017:
4966:
4839:
4819:
4784:
4744:
4742:{\displaystyle K_{x}}
4712:
4482:
4309:
4270:
4115:
4113:{\displaystyle K_{x}}
4085:
4062:
3816:
3785:
3671:holomorphic functions
3664:
3642:
3618:
3596:
3576:
3535:
3419:
3395:
3307:
3266:
3168:
3133:
3100:
3073:of the one point set
3064:
3062:{\displaystyle K_{x}}
3037:
3011:
2987:
2967:
2947:
2897:
2877:
2857:
2740:
2665:
2597:
2546:
2478:
2421:
2339:
2320:
2298:
2255:
2172:
2150:
2110:
2087:
2085:{\displaystyle L_{y}}
2060:
2040:
2004:
1891:
1871:
1869:{\displaystyle K_{x}}
1844:
1822:
1800:
1780:
1778:{\displaystyle K_{x}}
1737:
1630:
1610:
1608:{\displaystyle K_{x}}
1583:
1563:
1548:implies that for all
1539:
1515:
1495:
1493:{\displaystyle K_{x}}
1468:
1447:
1418:
1375:
1349:
1300:
1177:
1144:
1120:
1118:{\displaystyle L_{x}}
1098:or, equivalently, if
1093:
1073:
1049:
1047:{\displaystyle L_{x}}
1022:
1002:
971:
900:
880:
856:
839:real-valued functions
828:
804:
686:
666:
639:
637:{\displaystyle K_{0}}
594:
568:
522:
465:
445:
425:
393:
373:
350:
330:
310:
308:{\displaystyle K_{x}}
281:
222:
196:
163:
137:
115:
93:
73:
33:
14258:
14193:
14170:
14122:
13940:
13846:
13756:
13594:
13556:
13508:
13402:
13372:
13337:
13271:
13217:
13182:
13156:
13136:
13110:
13002:
12979:
12941:
12914:
12894:
12890:, one can show that
12865:
12838:
12811:
12714:
12665:
12554:
12479:
12379:
12310:
12239:
12208:
12164:
12137:
12068:
11987:
11898:
11870:
11835:
11794:
11774:
11748:
11724:
11617:
11591:
11553:
11458:
11417:
11395:
11349:
11319:
11291:
11259:
11167:
11129:
10943:
10921:
10901:
10812:
10695:
10618:
10504:
10439:
10392:
10372:
10316:
10282:
10237:
10226:will be closer to 0.
10195:
10191:are dissimilar then
10163:
10128:
10096:
10072:
9959:
9811:
9781:
9740:
9720:
9677:
9657:
9621:
9562:
9375:
9317:
9266:
9215:
9121:
9094:
8946:
8916:
8773:
8747:
8666:
8633:
8601:
8559:
8533:
8504:
8484:
8387:
8358:
8326:
8137:
8114:
7958:
7935:
7915:
7892:
7732:
7697:
7585:
7562:
7539:
7497:
7486:{\displaystyle C(X)}
7468:
7441:
7418:
7372:
7336:
7303:
7225:
7176:
7130:
7099:
7076:
7049:
7029:
7009:
6982:
6968:{\displaystyle \mu }
6959:
6923:
6816:
6742:
6702:
6691:{\displaystyle \mu }
6682:
6659:
6635:
6590:
6552:
6521:
6269:
6228:
6181:
6148:
6095:
6063:
6015:
5970:
5906:
5787:
5544:
5320:
5202:
4957:
4828:
4793:
4757:
4751:Dirac delta function
4726:
4505:
4496:Plancherel's theorem
4321:
4282:
4127:
4097:
4093:The kernel function
4074:
3828:
3805:
3799:Plancherel's theorem
3691:
3675:Paley–Wiener theorem
3651:
3631:
3605:
3585:
3547:
3431:
3408:
3316:
3278:
3180:
3145:
3122:
3117:continuous functions
3077:
3046:
3020:
3000:
2985:{\displaystyle \mu }
2976:
2956:
2914:
2886:
2866:
2752:
2336:
2307:
2267:
2184:
2159:
2119:
2099:
2069:
2049:
2016:
1903:
1880:
1853:
1831:
1809:
1789:
1762:
1644:
1619:
1592:
1572:
1552:
1528:
1504:
1477:
1457:
1436:
1384:
1358:
1325:
1191:
1153:
1133:
1102:
1082:
1062:
1031:
1011:
991:
912:
889:
869:
845:
817:
793:
742:and Stefan Bergman.
712:biharmonic functions
675:
655:
621:
577:
535:
474:
454:
434:
402:
382:
362:
339:
319:
292:
233:
205:
172:
146:
124:
102:
82:
62:
14669:Kimeldorf, George;
14389:Representer theorem
14357:representer theorem
14000:
13861:
13668:
13626:
13446:
13298:
13171:{\displaystyle t,s}
13125:{\displaystyle x,y}
11763:{\displaystyle x,y}
11714:multi-task learning
10966:
10462:
10092:between inputs. If
9882:
9353:
9302:
9251:
8548:{\displaystyle F=H}
8073:
4844:tends to infinity.
4650:
4564:
4341:
3977:
3896:
3741:
3483:
3351:
2728:
1354:is assumed for all
763:representer theorem
201:such that for all
40:functional analysis
14560:Aronszajn, Nachman
14342:
14241:
14176:
14156:
14105:
14079:
13986:
13923:
13849:
13829:
13824:
13736:
13654:
13612:
13577:
13542:
13491:
13432:
13385:
13358:
13323:
13284:
13253:
13188:
13168:
13142:
13122:
13093:
12985:
12954:
12927:
12900:
12880:
12851:
12824:
12794:
12697:
12642:
12527:
12464:
12365:
12280:
12220:
12194:
12143:
12123:
12050:
11967:
11882:
11856:
11831:such that for all
11821:
11780:
11760:
11730:
11694:
11603:{\displaystyle 2a}
11600:
11577:
11536:
11441:
11403:
11373:
11327:
11305:
11274:
11238:
11233:
11150:
11098:be finite and let
11066:
10952:
10927:
10907:
10883:
10785:
10666:
10594:
10496:Polynomial kernels
10482:
10448:
10425:
10378:
10355:
10288:
10268:
10216:
10181:
10149:
10114:
10078:
10054:
9935:
9868:
9793:
9767:
9726:
9703:
9663:
9643:
9603:
9544:
9357:
9333:
9303:
9282:
9252:
9231:
9185:
9107:
9077:
8929:
8899:
8753:
8730:
8652:
8613:
8587:
8545:
8510:
8490:
8454:
8364:
8344:
8289:
8120:
8097:
8029:
7941:
7921:
7898:
7875:
7766:
7750:
7715:
7680:
7568:
7551:{\displaystyle i.}
7548:
7525:
7483:
7454:
7430:{\displaystyle i.}
7427:
7404:
7358:
7322:
7289:
7211:
7194:
7162:
7112:
7082:
7062:
7035:
7015:
6995:
6965:
6945:
6906:
6799:
6728:
6688:
6665:
6641:
6609:
6572:
6534:
6504:
6240:
6214:
6167:
6134:
6069:
6027:
6001:
5956:
5889:
5741:
5518:
5433:
5417:
5273:
5185:
4834:
4814:
4779:
4739:
4707:
4633:
4547:
4477:
4472:
4324:
4304:
4265:
4110:
4090:is indeed a RKHS.
4080:
4057:
3960:
3879:
3811:
3780:
3724:
3659:
3637:
3613:
3591:
3571:
3530:
3466:
3414:
3390:
3334:
3302:
3261:
3163:
3128:
3110:analytic functions
3095:
3071:indicator function
3059:
3032:
3006:
2982:
2962:
2942:
2892:
2872:
2852:
2735:
2659:
2315:
2293:
2250:
2167:
2145:
2105:
2082:
2055:
2045:is the element in
2035:
1999:
1886:
1866:
1839:
1817:
1795:
1775:
1732:
1625:
1605:
1578:
1558:
1534:
1522:reproducing kernel
1510:
1490:
1463:
1442:
1413:
1396:
1370:
1344:
1295:
1172:
1139:
1115:
1088:
1068:
1044:
1017:
997:
966:
895:
875:
851:
823:
799:
779:analytic functions
767:linear combination
724:integral equations
693:reproducing kernel
681:
661:
634:
589:
563:
517:
460:
440:
420:
388:
368:
345:
325:
305:
276:
217:
191:
158:
132:
110:
88:
68:
36:
18:Reproducing kernel
14179:{\displaystyle f}
14075:
13897:
13820:
13191:{\displaystyle T}
13145:{\displaystyle X}
12988:{\displaystyle T}
12903:{\displaystyle D}
12848:
12304:
12303:
12146:{\displaystyle H}
11956:
11783:{\displaystyle X}
11689:
11531:
11518:
11490:
11386:square-integrable
11255:is isomorphic to
10997:
10930:{\displaystyle H}
10910:{\displaystyle f}
10866:
10768:
10381:{\displaystyle H}
10291:{\displaystyle H}
10124:are similar then
10081:{\displaystyle K}
9729:{\displaystyle K}
9666:{\displaystyle K}
9060:
8882:
8756:{\displaystyle H}
8699:
8513:{\displaystyle F}
8493:{\displaystyle K}
8478:
8477:
8367:{\displaystyle F}
8284:
8123:{\displaystyle H}
8084:
7944:{\displaystyle K}
7924:{\displaystyle H}
7901:{\displaystyle K}
7751:
7735:
7571:{\displaystyle K}
7179:
7085:{\displaystyle K}
7038:{\displaystyle K}
7018:{\displaystyle K}
6668:{\displaystyle X}
6644:{\displaystyle K}
6079:be an element of
6072:{\displaystyle f}
5418:
5402:
5399:
4854:positive definite
4837:{\displaystyle a}
4631:
4606:
4464:
4422:
4260:
4181:
4160:
4083:{\displaystyle H}
4029:
4028:
4014:
3956:
3947:
3933:
3869:
3814:{\displaystyle x}
3722:
3640:{\displaystyle H}
3594:{\displaystyle H}
3518:
3417:{\displaystyle f}
3402:Fourier transform
3131:{\displaystyle H}
3009:{\displaystyle X}
2965:{\displaystyle X}
2895:{\displaystyle X}
2875:{\displaystyle K}
2810:
2327:positive definite
2226:
2108:{\displaystyle H}
2058:{\displaystyle H}
1975:
1889:{\displaystyle H}
1798:{\displaystyle X}
1756:
1755:
1695:
1628:{\displaystyle H}
1581:{\displaystyle X}
1561:{\displaystyle x}
1537:{\displaystyle H}
1513:{\displaystyle H}
1466:{\displaystyle f}
1445:{\displaystyle H}
1387:
1319:
1318:
1142:{\displaystyle H}
1091:{\displaystyle H}
1071:{\displaystyle f}
1020:{\displaystyle X}
1000:{\displaystyle x}
949:
898:{\displaystyle x}
878:{\displaystyle H}
854:{\displaystyle X}
826:{\displaystyle H}
802:{\displaystyle X}
755:quantum mechanics
751:harmonic analysis
740:Nachman Aronszajn
700:Stanisław Zaremba
684:{\displaystyle x}
664:{\displaystyle x}
601:parallelogram law
463:{\displaystyle g}
443:{\displaystyle f}
430:is small), then
391:{\displaystyle g}
371:{\displaystyle f}
348:{\displaystyle x}
328:{\displaystyle f}
91:{\displaystyle X}
71:{\displaystyle H}
16:(Redirected from
14816:
14795:
14785:
14758:
14707:
14697:
14679:
14665:, February 2010.
14647:
14637:
14603:
14585:
14539:
14536:
14530:
14527:
14521:
14518:
14512:
14509:
14503:
14500:
14494:
14485:
14479:
14476:
14470:
14467:
14461:
14458:
14452:
14449:
14443:
14440:
14434:
14427:
14421:
14414:
14408:
14405:
14374:Mercer's theorem
14351:
14349:
14348:
14343:
14250:
14248:
14247:
14242:
14185:
14183:
14182:
14177:
14165:
14163:
14162:
14157:
14134:
14133:
14114:
14112:
14111:
14106:
14083:
14082:
14076:
14073:
13999:
13994:
13973:
13972:
13932:
13930:
13929:
13924:
13907:
13906:
13895:
13857:
13838:
13836:
13835:
13830:
13828:
13827:
13821:
13818:
13745:
13743:
13742:
13737:
13726:
13725:
13694:
13667:
13662:
13634:
13625:
13620:
13586:
13584:
13583:
13578:
13551:
13549:
13548:
13543:
13526:
13525:
13500:
13498:
13497:
13492:
13471:
13454:
13445:
13440:
13428:
13427:
13426:
13394:
13392:
13391:
13386:
13384:
13383:
13367:
13365:
13364:
13359:
13332:
13330:
13329:
13324:
13297:
13292:
13280:
13279:
13262:
13260:
13259:
13254:
13197:
13195:
13194:
13189:
13177:
13175:
13174:
13169:
13151:
13149:
13148:
13143:
13131:
13129:
13128:
13123:
13102:
13100:
13099:
13094:
13077:
13076:
12994:
12992:
12991:
12986:
12963:
12961:
12960:
12955:
12953:
12952:
12936:
12934:
12933:
12928:
12926:
12925:
12909:
12907:
12906:
12901:
12889:
12887:
12886:
12881:
12879:
12878:
12873:
12860:
12858:
12857:
12852:
12850:
12849:
12846:
12833:
12831:
12830:
12825:
12823:
12822:
12803:
12801:
12800:
12795:
12793:
12792:
12791:
12790:
12785:
12774:
12773:
12706:
12704:
12703:
12698:
12696:
12695:
12683:
12682:
12651:
12649:
12648:
12643:
12593:
12592:
12538:
12536:
12534:
12533:
12528:
12473:
12471:
12470:
12465:
12403:
12402:
12374:
12372:
12371:
12366:
12337:
12336:
12298:
12289:
12287:
12286:
12281:
12276:
12233:
12229:
12227:
12226:
12221:
12203:
12201:
12200:
12195:
12152:
12150:
12149:
12144:
12132:
12130:
12129:
12124:
12119:
12118:
12113:
12083:
12082:
12059:
12057:
12056:
12051:
12043:
12042:
12021:
12020:
12008:
12007:
11976:
11974:
11973:
11968:
11957:
11954:
11910:
11909:
11891:
11889:
11888:
11883:
11865:
11863:
11862:
11857:
11855:
11854:
11849:
11830:
11828:
11827:
11822:
11820:
11819:
11814:
11789:
11787:
11786:
11781:
11769:
11767:
11766:
11761:
11739:
11737:
11736:
11731:
11703:
11701:
11700:
11695:
11690:
11688:
11668:
11642:
11609:
11607:
11606:
11601:
11586:
11584:
11583:
11578:
11573:
11565:
11564:
11545:
11543:
11542:
11537:
11532:
11530:
11529:
11528:
11519:
11511:
11493:
11491:
11483:
11450:
11448:
11447:
11442:
11437:
11429:
11428:
11412:
11410:
11409:
11404:
11402:
11384:is the space of
11382:
11380:
11379:
11374:
11369:
11361:
11360:
11336:
11334:
11333:
11328:
11326:
11314:
11312:
11311:
11306:
11304:
11283:
11281:
11280:
11275:
11273:
11272:
11267:
11247:
11245:
11244:
11239:
11237:
11236:
11159:
11157:
11156:
11151:
11075:
11073:
11072:
11067:
11059:
11054:
11053:
11047:
11046:
11031:
11017:
11016:
10998:
10990:
10988:
10987:
10981:
10980:
10979:
10965:
10960:
10936:
10934:
10933:
10928:
10916:
10914:
10913:
10908:
10892:
10890:
10889:
10884:
10869:
10868:
10867:
10862:
10845:
10802:Laplacian kernel
10794:
10792:
10791:
10786:
10771:
10770:
10769:
10767:
10766:
10765:
10752:
10751:
10750:
10728:
10675:
10673:
10672:
10667:
10603:
10601:
10600:
10595:
10593:
10579:
10564:
10563:
10491:
10489:
10488:
10483:
10481:
10480:
10461:
10456:
10434:
10432:
10431:
10426:
10387:
10385:
10384:
10379:
10364:
10362:
10361:
10356:
10308:Bilinear kernels
10297:
10295:
10294:
10289:
10277:
10275:
10274:
10269:
10252:
10251:
10225:
10223:
10222:
10217:
10190:
10188:
10187:
10182:
10158:
10156:
10155:
10150:
10123:
10121:
10120:
10115:
10087:
10085:
10084:
10079:
10063:
10061:
10060:
10055:
9986:
9985:
9944:
9942:
9941:
9936:
9881:
9876:
9867:
9866:
9854:
9853:
9823:
9822:
9802:
9800:
9799:
9794:
9776:
9774:
9773:
9768:
9735:
9733:
9732:
9727:
9712:
9710:
9709:
9704:
9702:
9701:
9689:
9688:
9672:
9670:
9669:
9664:
9652:
9650:
9649:
9644:
9633:
9632:
9612:
9610:
9609:
9604:
9599:
9598:
9580:
9579:
9553:
9551:
9550:
9545:
9540:
9539:
9527:
9526:
9514:
9513:
9498:
9497:
9485:
9484:
9472:
9471:
9453:
9452:
9434:
9433:
9415:
9414:
9396:
9395:
9366:
9364:
9363:
9358:
9352:
9347:
9332:
9331:
9312:
9310:
9309:
9304:
9301:
9296:
9281:
9280:
9261:
9259:
9258:
9253:
9250:
9245:
9230:
9229:
9194:
9192:
9191:
9186:
9184:
9183:
9116:
9114:
9113:
9108:
9106:
9105:
9086:
9084:
9083:
9078:
9061:
9058:
9050:
9049:
8989:
8988:
8964:
8963:
8938:
8936:
8935:
8930:
8928:
8927:
8908:
8906:
8905:
8900:
8883:
8880:
8872:
8871:
8808:
8785:
8784:
8762:
8760:
8759:
8754:
8739:
8737:
8736:
8731:
8729:
8728:
8710:
8709:
8700:
8698:
8697:
8688:
8661:
8659:
8658:
8653:
8651:
8650:
8622:
8620:
8619:
8614:
8596:
8594:
8593:
8588:
8586:
8585:
8554:
8552:
8551:
8546:
8519:
8517:
8516:
8511:
8499:
8497:
8496:
8491:
8472:
8463:
8461:
8460:
8455:
8450:
8449:
8381:
8373:
8371:
8370:
8365:
8353:
8351:
8350:
8345:
8298:
8296:
8295:
8290:
8285:
8283:
8282:
8273:
8272:
8271:
8270:
8269:
8259:
8255:
8254:
8253:
8232:
8231:
8230:
8229:
8219:
8215:
8214:
8213:
8191:
8188:
8183:
8165:
8164:
8159:
8155:
8129:
8127:
8126:
8121:
8106:
8104:
8103:
8098:
8096:
8092:
8085:
8083:
8082:
8072:
8067:
8066:
8065:
8055:
8051:
8050:
8049:
8028:
8025:
8020:
8004:
8003:
7987:
7986:
7950:
7948:
7947:
7942:
7930:
7928:
7927:
7922:
7907:
7905:
7904:
7899:
7884:
7882:
7881:
7876:
7868:
7864:
7854:
7853:
7834:
7833:
7823:
7822:
7812:
7807:
7765:
7749:
7724:
7722:
7721:
7716:
7689:
7687:
7686:
7681:
7670:
7669:
7650:
7649:
7639:
7638:
7628:
7623:
7577:
7575:
7574:
7569:
7557:
7555:
7554:
7549:
7534:
7532:
7531:
7526:
7509:
7508:
7492:
7490:
7489:
7484:
7463:
7461:
7460:
7455:
7453:
7452:
7436:
7434:
7433:
7428:
7413:
7411:
7410:
7405:
7397:
7396:
7384:
7383:
7367:
7365:
7364:
7359:
7348:
7347:
7331:
7329:
7328:
7323:
7318:
7317:
7298:
7296:
7295:
7290:
7279:
7278:
7269:
7268:
7247:
7246:
7237:
7236:
7220:
7218:
7217:
7212:
7204:
7203:
7193:
7171:
7169:
7168:
7163:
7155:
7154:
7145:
7144:
7124:spectral theorem
7121:
7119:
7118:
7113:
7111:
7110:
7091:
7089:
7088:
7083:
7071:
7069:
7068:
7063:
7061:
7060:
7044:
7042:
7041:
7036:
7024:
7022:
7021:
7016:
7004:
7002:
7001:
6996:
6994:
6993:
6974:
6972:
6971:
6966:
6954:
6952:
6951:
6946:
6935:
6934:
6915:
6913:
6912:
6907:
6859:
6858:
6831:
6830:
6808:
6806:
6805:
6800:
6789:
6788:
6767:
6766:
6754:
6753:
6737:
6735:
6734:
6729:
6727:
6697:
6695:
6694:
6689:
6674:
6672:
6671:
6666:
6653:Mercer's theorem
6650:
6648:
6647:
6642:
6618:
6616:
6615:
6610:
6608:
6607:
6581:
6579:
6578:
6573:
6571:
6570:
6569:
6568:
6543:
6541:
6540:
6535:
6533:
6532:
6513:
6511:
6510:
6505:
6491:
6490:
6478:
6477:
6468:
6467:
6455:
6454:
6439:
6438:
6429:
6428:
6416:
6415:
6400:
6399:
6390:
6389:
6388:
6387:
6370:
6369:
6354:
6353:
6344:
6343:
6331:
6330:
6315:
6314:
6299:
6298:
6249:
6247:
6246:
6241:
6223:
6221:
6220:
6215:
6213:
6212:
6200:
6199:
6198:
6197:
6176:
6174:
6173:
6168:
6160:
6159:
6143:
6141:
6140:
6135:
6133:
6132:
6131:
6130:
6113:
6112:
6078:
6076:
6075:
6070:
6036:
6034:
6033:
6028:
6010:
6008:
6007:
6002:
5985:
5984:
5965:
5963:
5962:
5957:
5955:
5954:
5930:
5929:
5898:
5896:
5895:
5890:
5885:
5884:
5875:
5874:
5862:
5861:
5825:
5824:
5815:
5814:
5802:
5801:
5750:
5748:
5747:
5742:
5713:
5712:
5697:
5696:
5686:
5681:
5663:
5662:
5661:
5660:
5650:
5646:
5645:
5644:
5632:
5631:
5630:
5629:
5609:
5608:
5598:
5593:
5575:
5574:
5565:
5564:
5527:
5525:
5524:
5519:
5511:
5510:
5509:
5508:
5498:
5494:
5493:
5492:
5491:
5490:
5476:
5475:
5465:
5454:
5432:
5416:
5400:
5397:
5385:
5384:
5383:
5382:
5368:
5367:
5357:
5352:
5282:
5280:
5279:
5274:
5272:
5271:
5270:
5269:
5259:
5255:
5254:
5253:
5241:
5240:
5194:
5192:
5191:
5186:
5178:
5177:
5165:
5164:
5149:
5148:
5139:
5138:
5137:
5126:
5121:
5105:
5100:
5082:
5081:
5080:
5079:
5069:
5065:
5064:
5063:
5062:
5061:
5047:
5046:
5036:
5031:
5013:
5012:
5011:
5010:
4996:
4995:
4985:
4980:
4875:is a symmetric,
4843:
4841:
4840:
4835:
4823:
4821:
4820:
4815:
4788:
4786:
4785:
4780:
4769:
4768:
4748:
4746:
4745:
4740:
4738:
4737:
4716:
4714:
4713:
4708:
4681:
4680:
4649:
4644:
4632:
4630:
4619:
4607:
4602:
4592:
4591:
4581:
4563:
4558:
4543:
4542:
4541:
4540:
4526:
4525:
4486:
4484:
4483:
4478:
4476:
4475:
4466:
4465:
4462:
4423:
4420:
4416:
4415:
4379:
4378:
4351:
4350:
4340:
4335:
4313:
4311:
4310:
4305:
4294:
4293:
4274:
4272:
4271:
4266:
4261:
4259:
4239:
4207:
4202:
4198:
4182:
4174:
4161:
4153:
4139:
4138:
4119:
4117:
4116:
4111:
4109:
4108:
4089:
4087:
4086:
4081:
4066:
4064:
4063:
4058:
4053:
4052:
4051:
4050:
4030:
4021:
4020:
4015:
4006:
4005:
4000:
3982:
3976:
3971:
3959:
3957:
3955:
3940:
3939:
3934:
3925:
3924:
3919:
3901:
3895:
3890:
3872:
3870:
3868:
3857:
3852:
3835:
3820:
3818:
3817:
3812:
3789:
3787:
3786:
3781:
3769:
3768:
3740:
3735:
3723:
3721:
3710:
3668:
3666:
3665:
3660:
3658:
3646:
3644:
3643:
3638:
3622:
3620:
3619:
3614:
3612:
3600:
3598:
3597:
3592:
3580:
3578:
3577:
3572:
3567:
3559:
3558:
3539:
3537:
3536:
3531:
3519:
3514:
3500:
3482:
3477:
3462:
3461:
3460:
3459:
3423:
3421:
3420:
3415:
3399:
3397:
3396:
3391:
3382:
3381:
3350:
3345:
3311:
3309:
3308:
3303:
3298:
3290:
3289:
3270:
3268:
3267:
3262:
3215:
3207:
3206:
3172:
3170:
3169:
3164:
3140:cutoff frequency
3137:
3135:
3134:
3129:
3104:
3102:
3101:
3096:
3068:
3066:
3065:
3060:
3058:
3057:
3041:
3039:
3038:
3033:
3015:
3013:
3012:
3007:
2994:counting measure
2991:
2989:
2988:
2983:
2971:
2969:
2968:
2963:
2951:
2949:
2948:
2943:
2926:
2925:
2901:
2899:
2898:
2893:
2881:
2879:
2878:
2873:
2861:
2859:
2858:
2853:
2848:
2840:
2839:
2821:
2820:
2811:
2808:
2797:
2796:
2778:
2777:
2765:
2744:
2742:
2741:
2736:
2727:
2722:
2717:
2713:
2712:
2711:
2710:
2709:
2695:
2694:
2684:
2679:
2655:
2654:
2649:
2645:
2644:
2643:
2642:
2641:
2627:
2626:
2616:
2611:
2593:
2592:
2591:
2590:
2576:
2575:
2565:
2560:
2536:
2535:
2530:
2526:
2525:
2524:
2523:
2522:
2508:
2507:
2497:
2492:
2474:
2473:
2472:
2471:
2451:
2450:
2440:
2435:
2414:
2413:
2401:
2400:
2385:
2384:
2375:
2374:
2364:
2359:
2324:
2322:
2321:
2316:
2314:
2302:
2300:
2299:
2294:
2292:
2259:
2257:
2256:
2251:
2246:
2245:
2236:
2235:
2224:
2220:
2219:
2176:
2174:
2173:
2168:
2166:
2154:
2152:
2151:
2146:
2144:
2114:
2112:
2111:
2106:
2091:
2089:
2088:
2083:
2081:
2080:
2064:
2062:
2061:
2056:
2044:
2042:
2041:
2036:
2028:
2027:
2008:
2006:
2005:
2000:
1995:
1994:
1985:
1984:
1973:
1969:
1968:
1950:
1949:
1937:
1936:
1915:
1914:
1895:
1893:
1892:
1887:
1875:
1873:
1872:
1867:
1865:
1864:
1848:
1846:
1845:
1840:
1838:
1826:
1824:
1823:
1818:
1816:
1804:
1802:
1801:
1796:
1784:
1782:
1781:
1776:
1774:
1773:
1750:
1741:
1739:
1738:
1733:
1715:
1714:
1705:
1704:
1693:
1671:
1670:
1638:
1634:
1632:
1631:
1626:
1614:
1612:
1611:
1606:
1604:
1603:
1587:
1585:
1584:
1579:
1567:
1565:
1564:
1559:
1543:
1541:
1540:
1535:
1519:
1517:
1516:
1511:
1499:
1497:
1496:
1491:
1489:
1488:
1473:with a function
1472:
1470:
1469:
1464:
1451:
1449:
1448:
1443:
1426:While property (
1422:
1420:
1419:
1414:
1406:
1405:
1395:
1379:
1377:
1376:
1371:
1353:
1351:
1350:
1345:
1337:
1336:
1313:
1304:
1302:
1301:
1296:
1277:
1276:
1260:
1259:
1247:
1230:
1222:
1208:
1207:
1198:
1185:
1181:
1179:
1178:
1173:
1165:
1164:
1148:
1146:
1145:
1140:
1127:bounded operator
1124:
1122:
1121:
1116:
1114:
1113:
1097:
1095:
1094:
1089:
1077:
1075:
1074:
1069:
1053:
1051:
1050:
1045:
1043:
1042:
1026:
1024:
1023:
1018:
1006:
1004:
1003:
998:
975:
973:
972:
967:
950:
947:
924:
923:
904:
902:
901:
896:
884:
882:
881:
876:
860:
858:
857:
852:
832:
830:
829:
824:
809:be an arbitrary
808:
806:
805:
800:
747:complex analysis
690:
688:
687:
682:
670:
668:
667:
662:
643:
641:
640:
635:
633:
632:
598:
596:
595:
590:
572:
570:
569:
564:
550:
549:
526:
524:
523:
518:
516:
484:
469:
467:
466:
461:
449:
447:
446:
441:
429:
427:
426:
421:
397:
395:
394:
389:
377:
375:
374:
369:
354:
352:
351:
346:
334:
332:
331:
326:
314:
312:
311:
306:
304:
303:
285:
283:
282:
277:
254:
253:
226:
224:
223:
218:
200:
198:
197:
192:
184:
183:
167:
165:
164:
159:
141:
139:
138:
133:
131:
119:
117:
116:
111:
109:
97:
95:
94:
89:
77:
75:
74:
69:
21:
14824:
14823:
14819:
14818:
14817:
14815:
14814:
14813:
14799:
14798:
14783:
14677:
14548:
14543:
14542:
14537:
14533:
14528:
14524:
14519:
14515:
14510:
14506:
14501:
14497:
14486:
14482:
14477:
14473:
14468:
14464:
14459:
14455:
14450:
14446:
14441:
14437:
14428:
14424:
14415:
14411:
14406:
14402:
14397:
14365:
14259:
14256:
14255:
14194:
14191:
14190:
14171:
14168:
14167:
14129:
14125:
14123:
14120:
14119:
14078:
14077:
14072:
14070:
14061:
14060:
14043:
14030:
14029:
13995:
13990:
13968:
13964:
13941:
13938:
13937:
13902:
13898:
13853:
13847:
13844:
13843:
13823:
13822:
13817:
13815:
13806:
13805:
13794:
13781:
13780:
13757:
13754:
13753:
13721:
13717:
13687:
13663:
13658:
13627:
13621:
13616:
13595:
13592:
13591:
13557:
13554:
13553:
13521:
13517:
13509:
13506:
13505:
13464:
13447:
13441:
13436:
13422:
13421:
13417:
13403:
13400:
13399:
13379:
13375:
13373:
13370:
13369:
13338:
13335:
13334:
13293:
13288:
13275:
13274:
13272:
13269:
13268:
13218:
13215:
13214:
13207:
13183:
13180:
13179:
13157:
13154:
13153:
13137:
13134:
13133:
13111:
13108:
13107:
13072:
13068:
13003:
13000:
12999:
12980:
12977:
12976:
12948:
12944:
12942:
12939:
12938:
12921:
12917:
12915:
12912:
12911:
12895:
12892:
12891:
12874:
12869:
12868:
12866:
12863:
12862:
12845:
12841:
12839:
12836:
12835:
12818:
12814:
12812:
12809:
12808:
12786:
12781:
12780:
12779:
12775:
12769:
12765:
12715:
12712:
12711:
12691:
12687:
12678:
12674:
12666:
12663:
12662:
12576:
12572:
12555:
12552:
12551:
12480:
12477:
12476:
12475:
12386:
12382:
12380:
12377:
12376:
12320:
12316:
12311:
12308:
12307:
12272:
12240:
12237:
12236:
12209:
12206:
12205:
12165:
12162:
12161:
12138:
12135:
12134:
12133:coincides with
12114:
12109:
12108:
12078:
12074:
12069:
12066:
12065:
12038:
12034:
12016:
12012:
12003:
11999:
11988:
11985:
11984:
11955: for
11953:
11905:
11901:
11899:
11896:
11895:
11871:
11868:
11867:
11850:
11845:
11844:
11836:
11833:
11832:
11815:
11810:
11809:
11795:
11792:
11791:
11775:
11772:
11771:
11749:
11746:
11745:
11725:
11722:
11721:
11710:
11669:
11643:
11641:
11618:
11615:
11614:
11592:
11589:
11588:
11587:with bandwidth
11569:
11560:
11556:
11554:
11551:
11550:
11524:
11520:
11510:
11497:
11492:
11482:
11459:
11456:
11455:
11433:
11424:
11420:
11418:
11415:
11414:
11398:
11396:
11393:
11392:
11365:
11356:
11352:
11350:
11347:
11346:
11322:
11320:
11317:
11316:
11300:
11292:
11289:
11288:
11268:
11263:
11262:
11260:
11257:
11256:
11232:
11231:
11220:
11214:
11213:
11202:
11192:
11191:
11168:
11165:
11164:
11130:
11127:
11126:
11119:
11092:Bergman kernels
11088:
11085:Bergman kernels
11055:
11049:
11048:
11042:
11038:
11024:
11012:
11008:
10989:
10983:
10982:
10975:
10974:
10970:
10961:
10956:
10944:
10941:
10940:
10922:
10919:
10918:
10902:
10899:
10898:
10846:
10844:
10840:
10836:
10813:
10810:
10809:
10761:
10757:
10753:
10746:
10742:
10729:
10727:
10723:
10719:
10696:
10693:
10692:
10619:
10616:
10615:
10612:
10589:
10575:
10559:
10555:
10505:
10502:
10501:
10498:
10476:
10472:
10457:
10452:
10440:
10437:
10436:
10393:
10390:
10389:
10373:
10370:
10369:
10317:
10314:
10313:
10310:
10305:
10303:Common examples
10283:
10280:
10279:
10278:coincides with
10247:
10243:
10238:
10235:
10234:
10196:
10193:
10192:
10164:
10161:
10160:
10129:
10126:
10125:
10097:
10094:
10093:
10073:
10070:
10069:
9981:
9977:
9960:
9957:
9956:
9877:
9872:
9862:
9858:
9849:
9845:
9818:
9814:
9812:
9809:
9808:
9782:
9779:
9778:
9741:
9738:
9737:
9721:
9718:
9717:
9697:
9693:
9684:
9680:
9678:
9675:
9674:
9658:
9655:
9654:
9628:
9624:
9622:
9619:
9618:
9594:
9590:
9575:
9571:
9563:
9560:
9559:
9558:is a kernel on
9535:
9531:
9522:
9518:
9509:
9505:
9493:
9489:
9480:
9476:
9467:
9463:
9448:
9444:
9429:
9425:
9410:
9406:
9391:
9387:
9376:
9373:
9372:
9348:
9337:
9327:
9323:
9318:
9315:
9314:
9297:
9286:
9276:
9272:
9267:
9264:
9263:
9246:
9235:
9225:
9221:
9216:
9213:
9212:
9205:
9179:
9175:
9122:
9119:
9118:
9101:
9097:
9095:
9092:
9091:
9057:
9045:
9041:
8984:
8980:
8959:
8955:
8947:
8944:
8943:
8923:
8919:
8917:
8914:
8913:
8879:
8867:
8863:
8804:
8780:
8776:
8774:
8771:
8770:
8748:
8745:
8744:
8724:
8720:
8705:
8701:
8693:
8689:
8687:
8667:
8664:
8663:
8646:
8642:
8634:
8631:
8630:
8602:
8599:
8598:
8581:
8577:
8560:
8557:
8556:
8534:
8531:
8530:
8505:
8502:
8501:
8485:
8482:
8481:
8445:
8441:
8388:
8385:
8384:
8359:
8356:
8355:
8327:
8324:
8323:
8316:
8278:
8274:
8265:
8261:
8260:
8249:
8245:
8238:
8234:
8233:
8225:
8221:
8220:
8209:
8205:
8198:
8194:
8193:
8192:
8190:
8184:
8173:
8160:
8145:
8141:
8140:
8138:
8135:
8134:
8115:
8112:
8111:
8078:
8074:
8068:
8061:
8057:
8056:
8045:
8041:
8034:
8030:
8027:
8021:
8010:
7999:
7998:
7982:
7978:
7971:
7967:
7959:
7956:
7955:
7936:
7933:
7932:
7916:
7913:
7912:
7893:
7890:
7889:
7849:
7845:
7829:
7825:
7818:
7814:
7808:
7797:
7771:
7767:
7755:
7739:
7733:
7730:
7729:
7698:
7695:
7694:
7665:
7661:
7645:
7641:
7634:
7630:
7624:
7613:
7586:
7583:
7582:
7563:
7560:
7559:
7540:
7537:
7536:
7504:
7500:
7498:
7495:
7494:
7469:
7466:
7465:
7448:
7444:
7442:
7439:
7438:
7419:
7416:
7415:
7392:
7388:
7379:
7375:
7373:
7370:
7369:
7343:
7339:
7337:
7334:
7333:
7313:
7309:
7304:
7301:
7300:
7274:
7270:
7264:
7260:
7242:
7238:
7232:
7228:
7226:
7223:
7222:
7199:
7195:
7183:
7177:
7174:
7173:
7150:
7146:
7140:
7136:
7131:
7128:
7127:
7106:
7102:
7100:
7097:
7096:
7077:
7074:
7073:
7056:
7052:
7050:
7047:
7046:
7030:
7027:
7026:
7010:
7007:
7006:
6989:
6985:
6983:
6980:
6979:
6960:
6957:
6956:
6930:
6926:
6924:
6921:
6920:
6854:
6850:
6826:
6822:
6817:
6814:
6813:
6784:
6780:
6762:
6758:
6749:
6745:
6743:
6740:
6739:
6723:
6703:
6700:
6699:
6683:
6680:
6679:
6660:
6657:
6656:
6636:
6633:
6632:
6629:
6603:
6599:
6591:
6588:
6587:
6564:
6560:
6559:
6555:
6553:
6550:
6549:
6528:
6524:
6522:
6519:
6518:
6486:
6482:
6473:
6469:
6463:
6459:
6450:
6446:
6434:
6430:
6424:
6420:
6411:
6407:
6395:
6391:
6383:
6379:
6378:
6374:
6365:
6361:
6349:
6345:
6339:
6335:
6326:
6322:
6310:
6306:
6294:
6290:
6270:
6267:
6266:
6229:
6226:
6225:
6208:
6204:
6193:
6189:
6188:
6184:
6182:
6179:
6178:
6155:
6151:
6149:
6146:
6145:
6126:
6122:
6121:
6117:
6108:
6104:
6096:
6093:
6092:
6091:, we can write
6064:
6061:
6060:
6047:
6016:
6013:
6012:
5980:
5976:
5971:
5968:
5967:
5966:on the span of
5950:
5946:
5925:
5921:
5907:
5904:
5903:
5880:
5876:
5870:
5866:
5857:
5853:
5820:
5816:
5810:
5806:
5797:
5793:
5788:
5785:
5784:
5780:) implies that
5708:
5704:
5692:
5688:
5682:
5671:
5656:
5652:
5651:
5640:
5636:
5625:
5621:
5620:
5616:
5615:
5611:
5610:
5604:
5600:
5594:
5583:
5570:
5566:
5560:
5556:
5545:
5542:
5541:
5504:
5500:
5499:
5486:
5482:
5481:
5477:
5471:
5467:
5455:
5444:
5439:
5435:
5434:
5422:
5406:
5396:
5378:
5374:
5373:
5369:
5363:
5359:
5353:
5342:
5321:
5318:
5317:
5308:
5265:
5261:
5260:
5249:
5245:
5236:
5232:
5231:
5227:
5226:
5203:
5200:
5199:
5173:
5169:
5160:
5156:
5144:
5140:
5133:
5129:
5128:
5122:
5111:
5101:
5090:
5075:
5071:
5070:
5057:
5053:
5052:
5048:
5042:
5038:
5032:
5021:
5006:
5002:
5001:
4997:
4991:
4987:
4981:
4970:
4965:
4961:
4960:
4958:
4955:
4954:
4949:
4933:
4927:
4911:
4850:
4829:
4826:
4825:
4794:
4791:
4790:
4764:
4760:
4758:
4755:
4754:
4733:
4729:
4727:
4724:
4723:
4670:
4666:
4645:
4637:
4623:
4618:
4587:
4583:
4582:
4580:
4559:
4551:
4536:
4532:
4531:
4527:
4521:
4517:
4506:
4503:
4502:
4471:
4470:
4461:
4460:
4458:
4452:
4451:
4419:
4417:
4402:
4398:
4391:
4390:
4365:
4361:
4346:
4342:
4336:
4328:
4322:
4319:
4318:
4289:
4285:
4283:
4280:
4279:
4240:
4208:
4206:
4173:
4172:
4168:
4152:
4134:
4130:
4128:
4125:
4124:
4104:
4100:
4098:
4095:
4094:
4075:
4072:
4071:
4046:
4042:
4041:
4037:
4019:
4001:
3996:
3995:
3978:
3972:
3964:
3958:
3948:
3938:
3920:
3915:
3914:
3897:
3891:
3883:
3871:
3861:
3856:
3848:
3831:
3829:
3826:
3825:
3806:
3803:
3802:
3758:
3754:
3736:
3728:
3714:
3709:
3692:
3689:
3688:
3654:
3652:
3649:
3648:
3632:
3629:
3628:
3608:
3606:
3603:
3602:
3586:
3583:
3582:
3563:
3554:
3550:
3548:
3545:
3544:
3501:
3499:
3478:
3470:
3455:
3451:
3450:
3446:
3432:
3429:
3428:
3409:
3406:
3405:
3368:
3364:
3346:
3338:
3317:
3314:
3313:
3294:
3285:
3281:
3279:
3276:
3275:
3211:
3202:
3198:
3181:
3178:
3177:
3146:
3143:
3142:
3123:
3120:
3119:
3078:
3075:
3074:
3053:
3049:
3047:
3044:
3043:
3021:
3018:
3017:
3001:
2998:
2997:
2977:
2974:
2973:
2957:
2954:
2953:
2921:
2917:
2915:
2912:
2911:
2908:
2887:
2884:
2883:
2867:
2864:
2863:
2844:
2835:
2831:
2816:
2812:
2809: and
2807:
2792:
2788:
2773:
2769:
2761:
2753:
2750:
2749:
2723:
2718:
2705:
2701:
2700:
2696:
2690:
2686:
2680:
2669:
2664:
2660:
2650:
2637:
2633:
2632:
2628:
2622:
2618:
2612:
2601:
2586:
2582:
2581:
2577:
2571:
2567:
2561:
2550:
2545:
2541:
2540:
2531:
2518:
2514:
2513:
2509:
2503:
2499:
2493:
2482:
2467:
2463:
2462:
2458:
2457:
2453:
2452:
2446:
2442:
2436:
2425:
2409:
2405:
2396:
2392:
2380:
2376:
2370:
2366:
2360:
2343:
2337:
2334:
2333:
2310:
2308:
2305:
2304:
2288:
2268:
2265:
2264:
2241:
2237:
2231:
2227:
2215:
2211:
2185:
2182:
2181:
2162:
2160:
2157:
2156:
2140:
2120:
2117:
2116:
2100:
2097:
2096:
2076:
2072:
2070:
2067:
2066:
2050:
2047:
2046:
2023:
2019:
2017:
2014:
2013:
1990:
1986:
1980:
1976:
1964:
1960:
1945:
1941:
1932:
1928:
1910:
1906:
1904:
1901:
1900:
1881:
1878:
1877:
1860:
1856:
1854:
1851:
1850:
1834:
1832:
1829:
1828:
1812:
1810:
1807:
1806:
1790:
1787:
1786:
1769:
1765:
1763:
1760:
1759:
1710:
1706:
1700:
1696:
1666:
1662:
1645:
1642:
1641:
1620:
1617:
1616:
1599:
1595:
1593:
1590:
1589:
1573:
1570:
1569:
1553:
1550:
1549:
1529:
1526:
1525:
1505:
1502:
1501:
1484:
1480:
1478:
1475:
1474:
1458:
1455:
1454:
1437:
1434:
1433:
1401:
1397:
1391:
1385:
1382:
1381:
1359:
1356:
1355:
1332:
1328:
1326:
1323:
1322:
1272:
1268:
1255:
1251:
1243:
1226:
1218:
1203:
1199:
1194:
1192:
1189:
1188:
1160:
1156:
1154:
1151:
1150:
1134:
1131:
1130:
1109:
1105:
1103:
1100:
1099:
1083:
1080:
1079:
1063:
1060:
1059:
1038:
1034:
1032:
1029:
1028:
1012:
1009:
1008:
992:
989:
988:
946:
919:
915:
913:
910:
909:
890:
887:
886:
870:
867:
866:
846:
843:
842:
818:
815:
814:
794:
791:
790:
787:
736:Salomon Bochner
676:
673:
672:
656:
653:
652:
628:
624:
622:
619:
618:
617:space an RKHS (
578:
575:
574:
542:
538:
536:
533:
532:
512:
480:
475:
472:
471:
455:
452:
451:
435:
432:
431:
403:
400:
399:
383:
380:
379:
363:
360:
359:
340:
337:
336:
320:
317:
316:
299:
295:
293:
290:
289:
249:
245:
234:
231:
230:
206:
203:
202:
179:
175:
173:
170:
169:
147:
144:
143:
127:
125:
122:
121:
105:
103:
100:
99:
83:
80:
79:
63:
60:
59:
28:
23:
22:
15:
12:
11:
5:
14822:
14812:
14811:
14809:Hilbert spaces
14797:
14796:
14776:
14762:
14759:
14733:(3): 363–417.
14727:Constr. Approx
14722:
14715:
14714:, August 2005.
14708:
14666:
14659:
14648:
14611:
14604:
14576:(3): 337–404.
14556:
14547:
14544:
14541:
14540:
14531:
14522:
14513:
14504:
14495:
14480:
14471:
14462:
14453:
14444:
14435:
14422:
14409:
14399:
14398:
14396:
14393:
14392:
14391:
14386:
14381:
14376:
14371:
14364:
14361:
14353:
14352:
14341:
14338:
14335:
14332:
14329:
14326:
14323:
14320:
14317:
14314:
14311:
14308:
14305:
14302:
14299:
14296:
14293:
14290:
14287:
14284:
14281:
14278:
14275:
14272:
14269:
14266:
14263:
14240:
14237:
14234:
14231:
14228:
14225:
14222:
14219:
14216:
14213:
14210:
14207:
14204:
14201:
14198:
14175:
14155:
14152:
14149:
14146:
14143:
14140:
14137:
14132:
14128:
14116:
14115:
14104:
14101:
14098:
14095:
14092:
14089:
14086:
14081:
14071:
14069:
14066:
14063:
14062:
14059:
14056:
14053:
14050:
14047:
14044:
14042:
14039:
14036:
14035:
14033:
14028:
14025:
14022:
14018:
14015:
14012:
14009:
14006:
14003:
13998:
13993:
13989:
13985:
13982:
13979:
13976:
13971:
13967:
13963:
13960:
13957:
13954:
13951:
13948:
13945:
13922:
13919:
13916:
13913:
13910:
13905:
13901:
13894:
13891:
13888:
13885:
13882:
13879:
13876:
13873:
13870:
13867:
13864:
13860:
13856:
13852:
13840:
13839:
13826:
13816:
13814:
13811:
13808:
13807:
13804:
13801:
13798:
13795:
13793:
13790:
13787:
13786:
13784:
13779:
13776:
13773:
13770:
13767:
13764:
13761:
13747:
13746:
13735:
13732:
13729:
13724:
13720:
13716:
13713:
13710:
13707:
13703:
13700:
13697:
13693:
13690:
13686:
13683:
13680:
13677:
13674:
13671:
13666:
13661:
13657:
13653:
13650:
13647:
13643:
13640:
13637:
13633:
13630:
13624:
13619:
13615:
13611:
13608:
13605:
13602:
13599:
13576:
13573:
13570:
13567:
13564:
13561:
13541:
13538:
13535:
13532:
13529:
13524:
13520:
13516:
13513:
13502:
13501:
13490:
13487:
13484:
13480:
13477:
13474:
13470:
13467:
13463:
13460:
13457:
13453:
13450:
13444:
13439:
13435:
13431:
13425:
13420:
13416:
13413:
13410:
13407:
13382:
13378:
13357:
13354:
13351:
13348:
13345:
13342:
13322:
13319:
13316:
13313:
13310:
13307:
13304:
13301:
13296:
13291:
13287:
13283:
13278:
13252:
13249:
13246:
13243:
13240:
13237:
13234:
13231:
13228:
13225:
13222:
13206:
13203:
13187:
13167:
13164:
13161:
13141:
13121:
13118:
13115:
13104:
13103:
13092:
13089:
13086:
13083:
13080:
13075:
13071:
13067:
13064:
13061:
13058:
13055:
13052:
13049:
13046:
13043:
13040:
13037:
13034:
13031:
13028:
13025:
13022:
13019:
13016:
13013:
13010:
13007:
12984:
12951:
12947:
12924:
12920:
12899:
12877:
12872:
12844:
12821:
12817:
12805:
12804:
12789:
12784:
12778:
12772:
12768:
12764:
12761:
12758:
12755:
12752:
12749:
12746:
12743:
12740:
12737:
12734:
12731:
12728:
12725:
12722:
12719:
12707:be defined as
12694:
12690:
12686:
12681:
12677:
12673:
12670:
12653:
12652:
12641:
12638:
12635:
12632:
12629:
12626:
12623:
12620:
12617:
12614:
12611:
12608:
12605:
12602:
12599:
12596:
12591:
12588:
12585:
12582:
12579:
12575:
12571:
12568:
12565:
12562:
12559:
12526:
12523:
12520:
12517:
12514:
12511:
12508:
12505:
12502:
12499:
12496:
12493:
12490:
12487:
12484:
12463:
12460:
12457:
12454:
12451:
12448:
12445:
12442:
12439:
12436:
12433:
12430:
12427:
12424:
12421:
12418:
12415:
12412:
12409:
12406:
12401:
12398:
12395:
12392:
12389:
12385:
12364:
12361:
12358:
12355:
12352:
12349:
12346:
12343:
12340:
12335:
12332:
12329:
12326:
12323:
12319:
12315:
12302:
12301:
12292:
12290:
12279:
12275:
12271:
12268:
12265:
12262:
12259:
12256:
12253:
12250:
12247:
12244:
12219:
12216:
12213:
12193:
12190:
12187:
12184:
12181:
12178:
12175:
12172:
12169:
12142:
12122:
12117:
12112:
12107:
12104:
12101:
12098:
12095:
12092:
12089:
12086:
12081:
12077:
12073:
12061:
12060:
12049:
12046:
12041:
12037:
12033:
12030:
12027:
12024:
12019:
12015:
12011:
12006:
12002:
11998:
11995:
11992:
11978:
11977:
11966:
11963:
11960:
11952:
11949:
11946:
11943:
11940:
11937:
11934:
11931:
11928:
11925:
11922:
11919:
11916:
11913:
11908:
11904:
11881:
11878:
11875:
11853:
11848:
11843:
11840:
11818:
11813:
11808:
11805:
11802:
11799:
11779:
11759:
11756:
11753:
11729:
11709:
11706:
11705:
11704:
11693:
11687:
11684:
11681:
11678:
11675:
11672:
11667:
11664:
11661:
11658:
11655:
11652:
11649:
11646:
11640:
11637:
11634:
11631:
11628:
11625:
11622:
11599:
11596:
11576:
11572:
11568:
11563:
11559:
11547:
11546:
11535:
11527:
11523:
11517:
11514:
11509:
11506:
11503:
11500:
11496:
11489:
11486:
11481:
11478:
11475:
11472:
11469:
11466:
11463:
11440:
11436:
11432:
11427:
11423:
11401:
11372:
11368:
11364:
11359:
11355:
11325:
11303:
11299:
11296:
11271:
11266:
11251:In this case,
11249:
11248:
11235:
11230:
11227:
11224:
11221:
11219:
11216:
11215:
11212:
11209:
11206:
11203:
11201:
11198:
11197:
11195:
11190:
11187:
11184:
11181:
11178:
11175:
11172:
11149:
11146:
11143:
11140:
11137:
11134:
11117:
11114:is used, then
11087:
11082:
11081:
11080:
11079:
11078:
11077:
11076:
11065:
11062:
11058:
11052:
11045:
11041:
11037:
11034:
11030:
11027:
11023:
11020:
11015:
11011:
11007:
11004:
11001:
10996:
10993:
10986:
10978:
10973:
10969:
10964:
10959:
10955:
10951:
10948:
10926:
10906:
10895:
10894:
10893:
10882:
10879:
10876:
10872:
10865:
10861:
10858:
10855:
10852:
10849:
10843:
10839:
10835:
10832:
10829:
10826:
10823:
10820:
10817:
10799:
10798:
10797:
10796:
10795:
10784:
10781:
10778:
10774:
10764:
10760:
10756:
10749:
10745:
10741:
10738:
10735:
10732:
10726:
10722:
10718:
10715:
10712:
10709:
10706:
10703:
10700:
10665:
10662:
10659:
10656:
10653:
10650:
10647:
10644:
10641:
10638:
10635:
10632:
10629:
10626:
10623:
10611:
10606:
10605:
10604:
10592:
10588:
10585:
10582:
10578:
10574:
10571:
10567:
10562:
10558:
10554:
10551:
10548:
10545:
10542:
10539:
10536:
10533:
10530:
10527:
10524:
10521:
10518:
10515:
10512:
10509:
10497:
10494:
10479:
10475:
10471:
10468:
10465:
10460:
10455:
10451:
10447:
10444:
10424:
10421:
10418:
10415:
10412:
10409:
10406:
10403:
10400:
10397:
10377:
10366:
10365:
10354:
10351:
10348:
10345:
10342:
10339:
10336:
10333:
10330:
10327:
10324:
10321:
10309:
10306:
10304:
10301:
10300:
10299:
10287:
10267:
10264:
10261:
10258:
10255:
10250:
10246:
10242:
10230:
10229:
10228:
10227:
10215:
10212:
10209:
10206:
10203:
10200:
10180:
10177:
10174:
10171:
10168:
10148:
10145:
10142:
10139:
10136:
10133:
10113:
10110:
10107:
10104:
10101:
10077:
10066:
10065:
10064:
10053:
10050:
10047:
10044:
10041:
10038:
10035:
10031:
10028:
10025:
10022:
10019:
10016:
10013:
10010:
10007:
10004:
10001:
9998:
9995:
9992:
9989:
9984:
9980:
9976:
9973:
9970:
9967:
9964:
9947:
9946:
9945:
9934:
9931:
9928:
9925:
9922:
9918:
9915:
9912:
9909:
9906:
9903:
9900:
9897:
9894:
9891:
9888:
9885:
9880:
9875:
9871:
9865:
9861:
9857:
9852:
9848:
9844:
9841:
9838:
9835:
9832:
9829:
9826:
9821:
9817:
9792:
9789:
9786:
9766:
9763:
9760:
9757:
9754:
9751:
9748:
9745:
9725:
9714:
9700:
9696:
9692:
9687:
9683:
9662:
9642:
9639:
9636:
9631:
9627:
9615:
9614:
9613:
9602:
9597:
9593:
9589:
9586:
9583:
9578:
9574:
9570:
9567:
9556:
9555:
9554:
9543:
9538:
9534:
9530:
9525:
9521:
9517:
9512:
9508:
9504:
9501:
9496:
9492:
9488:
9483:
9479:
9475:
9470:
9466:
9462:
9459:
9456:
9451:
9447:
9443:
9440:
9437:
9432:
9428:
9424:
9421:
9418:
9413:
9409:
9405:
9402:
9399:
9394:
9390:
9386:
9383:
9380:
9356:
9351:
9346:
9343:
9340:
9336:
9330:
9326:
9322:
9300:
9295:
9292:
9289:
9285:
9279:
9275:
9271:
9249:
9244:
9241:
9238:
9234:
9228:
9224:
9220:
9204:
9201:
9182:
9178:
9174:
9171:
9168:
9165:
9162:
9159:
9156:
9153:
9150:
9147:
9144:
9141:
9138:
9135:
9132:
9129:
9126:
9104:
9100:
9088:
9087:
9076:
9073:
9070:
9067:
9064:
9056:
9053:
9048:
9044:
9040:
9037:
9034:
9031:
9028:
9025:
9022:
9019:
9016:
9013:
9010:
9007:
9004:
9001:
8998:
8995:
8992:
8987:
8983:
8979:
8976:
8973:
8970:
8967:
8962:
8958:
8954:
8951:
8926:
8922:
8910:
8909:
8898:
8895:
8892:
8889:
8886:
8878:
8875:
8870:
8866:
8862:
8859:
8856:
8853:
8850:
8847:
8844:
8841:
8838:
8835:
8832:
8829:
8826:
8823:
8820:
8817:
8814:
8811:
8807:
8803:
8800:
8797:
8794:
8791:
8788:
8783:
8779:
8752:
8727:
8723:
8719:
8716:
8713:
8708:
8704:
8696:
8692:
8686:
8683:
8680:
8677:
8674:
8671:
8649:
8645:
8641:
8638:
8612:
8609:
8606:
8584:
8580:
8576:
8573:
8570:
8567:
8564:
8544:
8541:
8538:
8509:
8489:
8476:
8475:
8466:
8464:
8453:
8448:
8444:
8440:
8437:
8434:
8431:
8428:
8425:
8422:
8419:
8416:
8413:
8410:
8407:
8404:
8401:
8398:
8395:
8392:
8363:
8343:
8340:
8337:
8334:
8331:
8315:
8312:
8300:
8299:
8288:
8281:
8277:
8268:
8264:
8258:
8252:
8248:
8244:
8241:
8237:
8228:
8224:
8218:
8212:
8208:
8204:
8201:
8197:
8187:
8182:
8179:
8176:
8172:
8168:
8163:
8158:
8154:
8151:
8148:
8144:
8119:
8108:
8107:
8095:
8091:
8088:
8081:
8077:
8071:
8064:
8060:
8054:
8048:
8044:
8040:
8037:
8033:
8024:
8019:
8016:
8013:
8009:
8002:
7996:
7993:
7990:
7985:
7981:
7977:
7974:
7970:
7966:
7963:
7940:
7920:
7897:
7886:
7885:
7874:
7871:
7867:
7863:
7860:
7857:
7852:
7848:
7843:
7840:
7837:
7832:
7828:
7821:
7817:
7811:
7806:
7803:
7800:
7796:
7792:
7789:
7786:
7783:
7780:
7777:
7774:
7770:
7764:
7761:
7758:
7754:
7748:
7745:
7742:
7738:
7714:
7711:
7708:
7705:
7702:
7691:
7690:
7679:
7676:
7673:
7668:
7664:
7659:
7656:
7653:
7648:
7644:
7637:
7633:
7627:
7622:
7619:
7616:
7612:
7608:
7605:
7602:
7599:
7596:
7593:
7590:
7567:
7547:
7544:
7524:
7521:
7518:
7515:
7512:
7507:
7503:
7482:
7479:
7476:
7473:
7451:
7447:
7426:
7423:
7403:
7400:
7395:
7391:
7387:
7382:
7378:
7357:
7354:
7351:
7346:
7342:
7321:
7316:
7312:
7308:
7288:
7285:
7282:
7277:
7273:
7267:
7263:
7259:
7256:
7253:
7250:
7245:
7241:
7235:
7231:
7210:
7207:
7202:
7198:
7192:
7189:
7186:
7182:
7161:
7158:
7153:
7149:
7143:
7139:
7135:
7109:
7105:
7081:
7059:
7055:
7034:
7014:
6992:
6988:
6964:
6944:
6941:
6938:
6933:
6929:
6917:
6916:
6905:
6902:
6899:
6896:
6893:
6889:
6886:
6883:
6880:
6877:
6874:
6871:
6868:
6865:
6862:
6857:
6853:
6849:
6846:
6843:
6840:
6837:
6834:
6829:
6825:
6821:
6798:
6795:
6792:
6787:
6783:
6779:
6776:
6773:
6770:
6765:
6761:
6757:
6752:
6748:
6726:
6722:
6719:
6716:
6713:
6710:
6707:
6687:
6664:
6640:
6628:
6625:
6606:
6602:
6598:
6595:
6567:
6563:
6558:
6531:
6527:
6515:
6514:
6503:
6500:
6497:
6494:
6489:
6485:
6481:
6476:
6472:
6466:
6462:
6458:
6453:
6449:
6445:
6442:
6437:
6433:
6427:
6423:
6419:
6414:
6410:
6406:
6403:
6398:
6394:
6386:
6382:
6377:
6373:
6368:
6364:
6360:
6357:
6352:
6348:
6342:
6338:
6334:
6329:
6325:
6321:
6318:
6313:
6309:
6305:
6302:
6297:
6293:
6289:
6286:
6283:
6280:
6277:
6274:
6239:
6236:
6233:
6211:
6207:
6203:
6196:
6192:
6187:
6166:
6163:
6158:
6154:
6129:
6125:
6120:
6116:
6111:
6107:
6103:
6100:
6068:
6045:
6026:
6023:
6020:
6000:
5997:
5994:
5991:
5988:
5983:
5979:
5975:
5953:
5949:
5945:
5942:
5939:
5936:
5933:
5928:
5924:
5920:
5917:
5914:
5911:
5902:By linearity,
5900:
5899:
5888:
5883:
5879:
5873:
5869:
5865:
5860:
5856:
5852:
5849:
5846:
5843:
5840:
5837:
5834:
5831:
5828:
5823:
5819:
5813:
5809:
5805:
5800:
5796:
5792:
5752:
5751:
5740:
5737:
5734:
5731:
5728:
5725:
5722:
5719:
5716:
5711:
5707:
5703:
5700:
5695:
5691:
5685:
5680:
5677:
5674:
5670:
5666:
5659:
5655:
5649:
5643:
5639:
5635:
5628:
5624:
5619:
5614:
5607:
5603:
5597:
5592:
5589:
5586:
5582:
5578:
5573:
5569:
5563:
5559:
5555:
5552:
5549:
5529:
5528:
5517:
5514:
5507:
5503:
5497:
5489:
5485:
5480:
5474:
5470:
5464:
5461:
5458:
5453:
5450:
5447:
5443:
5438:
5431:
5428:
5425:
5421:
5415:
5412:
5409:
5405:
5394:
5391:
5388:
5381:
5377:
5372:
5366:
5362:
5356:
5351:
5348:
5345:
5341:
5337:
5334:
5331:
5328:
5325:
5306:
5268:
5264:
5258:
5252:
5248:
5244:
5239:
5235:
5230:
5225:
5222:
5219:
5216:
5213:
5210:
5207:
5198:which implies
5196:
5195:
5184:
5181:
5176:
5172:
5168:
5163:
5159:
5155:
5152:
5147:
5143:
5136:
5132:
5125:
5120:
5117:
5114:
5110:
5104:
5099:
5096:
5093:
5089:
5085:
5078:
5074:
5068:
5060:
5056:
5051:
5045:
5041:
5035:
5030:
5027:
5024:
5020:
5016:
5009:
5005:
5000:
4994:
4990:
4984:
4979:
4976:
4973:
4969:
4964:
4947:
4931:
4925:
4909:
4893:
4892:
4849:
4846:
4833:
4813:
4810:
4807:
4804:
4801:
4798:
4778:
4775:
4772:
4767:
4763:
4736:
4732:
4718:
4717:
4706:
4703:
4700:
4697:
4694:
4691:
4688:
4685:
4679:
4676:
4673:
4669:
4665:
4662:
4659:
4656:
4653:
4648:
4643:
4640:
4636:
4629:
4626:
4622:
4617:
4614:
4611:
4605:
4601:
4598:
4595:
4590:
4586:
4579:
4576:
4573:
4570:
4567:
4562:
4557:
4554:
4550:
4546:
4539:
4535:
4530:
4524:
4520:
4516:
4513:
4510:
4488:
4487:
4474:
4469:
4459:
4457:
4454:
4453:
4450:
4447:
4444:
4441:
4438:
4435:
4432:
4429:
4426:
4418:
4414:
4411:
4408:
4405:
4401:
4397:
4396:
4394:
4389:
4386:
4383:
4377:
4374:
4371:
4368:
4364:
4360:
4357:
4354:
4349:
4345:
4339:
4334:
4331:
4327:
4303:
4300:
4297:
4292:
4288:
4276:
4275:
4264:
4258:
4255:
4252:
4249:
4246:
4243:
4238:
4235:
4232:
4229:
4226:
4223:
4220:
4217:
4214:
4211:
4205:
4201:
4197:
4194:
4191:
4188:
4185:
4180:
4177:
4171:
4167:
4164:
4159:
4156:
4151:
4148:
4145:
4142:
4137:
4133:
4107:
4103:
4079:
4068:
4067:
4056:
4049:
4045:
4040:
4036:
4033:
4027:
4024:
4018:
4013:
4010:
4004:
3999:
3994:
3991:
3988:
3985:
3981:
3975:
3970:
3967:
3963:
3954:
3951:
3946:
3943:
3937:
3932:
3929:
3923:
3918:
3913:
3910:
3907:
3904:
3900:
3894:
3889:
3886:
3882:
3878:
3875:
3867:
3864:
3860:
3855:
3851:
3847:
3844:
3841:
3838:
3834:
3810:
3801:that, for all
3791:
3790:
3779:
3776:
3773:
3767:
3764:
3761:
3757:
3753:
3750:
3747:
3744:
3739:
3734:
3731:
3727:
3720:
3717:
3713:
3708:
3705:
3702:
3699:
3696:
3657:
3636:
3611:
3590:
3570:
3566:
3562:
3557:
3553:
3541:
3540:
3529:
3526:
3523:
3517:
3513:
3510:
3507:
3504:
3498:
3495:
3492:
3489:
3486:
3481:
3476:
3473:
3469:
3465:
3458:
3454:
3449:
3445:
3442:
3439:
3436:
3413:
3389:
3386:
3380:
3377:
3374:
3371:
3367:
3363:
3360:
3357:
3354:
3349:
3344:
3341:
3337:
3333:
3330:
3327:
3324:
3321:
3301:
3297:
3293:
3288:
3284:
3272:
3271:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3236:
3233:
3230:
3227:
3224:
3221:
3218:
3214:
3210:
3205:
3201:
3197:
3194:
3191:
3188:
3185:
3162:
3159:
3156:
3153:
3150:
3127:
3094:
3091:
3088:
3085:
3082:
3056:
3052:
3031:
3028:
3025:
3005:
2981:
2961:
2941:
2938:
2935:
2932:
2929:
2924:
2920:
2907:
2904:
2891:
2871:
2851:
2847:
2843:
2838:
2834:
2830:
2827:
2824:
2819:
2815:
2806:
2803:
2800:
2795:
2791:
2787:
2784:
2781:
2776:
2772:
2768:
2764:
2760:
2757:
2746:
2745:
2734:
2731:
2726:
2721:
2716:
2708:
2704:
2699:
2693:
2689:
2683:
2678:
2675:
2672:
2668:
2663:
2658:
2653:
2648:
2640:
2636:
2631:
2625:
2621:
2615:
2610:
2607:
2604:
2600:
2596:
2589:
2585:
2580:
2574:
2570:
2564:
2559:
2556:
2553:
2549:
2544:
2539:
2534:
2529:
2521:
2517:
2512:
2506:
2502:
2496:
2491:
2488:
2485:
2481:
2477:
2470:
2466:
2461:
2456:
2449:
2445:
2439:
2434:
2431:
2428:
2424:
2420:
2417:
2412:
2408:
2404:
2399:
2395:
2391:
2388:
2383:
2379:
2373:
2369:
2363:
2358:
2355:
2352:
2349:
2346:
2342:
2313:
2291:
2287:
2284:
2281:
2278:
2275:
2272:
2261:
2260:
2249:
2244:
2240:
2234:
2230:
2223:
2218:
2214:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2165:
2143:
2139:
2136:
2133:
2130:
2127:
2124:
2115:as a function
2104:
2079:
2075:
2065:associated to
2054:
2034:
2031:
2026:
2022:
2010:
2009:
1998:
1993:
1989:
1983:
1979:
1972:
1967:
1963:
1959:
1956:
1953:
1948:
1944:
1940:
1935:
1931:
1927:
1924:
1921:
1918:
1913:
1909:
1885:
1863:
1859:
1837:
1815:
1794:
1772:
1768:
1754:
1753:
1744:
1742:
1731:
1728:
1725:
1722:
1719:
1713:
1709:
1703:
1699:
1692:
1689:
1686:
1683:
1680:
1677:
1674:
1669:
1665:
1661:
1658:
1655:
1652:
1649:
1624:
1602:
1598:
1577:
1557:
1533:
1509:
1487:
1483:
1462:
1441:
1412:
1409:
1404:
1400:
1394:
1390:
1369:
1366:
1363:
1343:
1340:
1335:
1331:
1317:
1316:
1307:
1305:
1293:
1290:
1287:
1284:
1281:
1275:
1271:
1267:
1264:
1258:
1254:
1250:
1246:
1242:
1239:
1236:
1233:
1229:
1225:
1221:
1217:
1214:
1211:
1206:
1202:
1197:
1171:
1168:
1163:
1159:
1138:
1112:
1108:
1087:
1067:
1041:
1037:
1016:
996:
977:
976:
965:
962:
959:
956:
953:
945:
942:
939:
936:
933:
930:
927:
922:
918:
894:
874:
850:
822:
798:
786:
783:
732:Stefan Bergman
680:
660:
631:
627:
588:
585:
582:
562:
559:
556:
553:
548:
545:
541:
515:
511:
508:
505:
502:
499:
496:
493:
490:
487:
483:
479:
459:
439:
419:
416:
413:
410:
407:
387:
367:
344:
324:
302:
298:
288:The function
275:
272:
269:
266:
263:
260:
257:
252:
248:
244:
241:
238:
216:
213:
210:
190:
187:
182:
178:
157:
154:
151:
130:
108:
87:
67:
26:
9:
6:
4:
3:
2:
14821:
14810:
14807:
14806:
14804:
14793:
14789:
14782:
14777:
14774:
14770:
14766:
14763:
14760:
14756:
14752:
14748:
14744:
14740:
14736:
14732:
14728:
14723:
14720:
14716:
14713:
14709:
14705:
14701:
14696:
14691:
14687:
14683:
14676:
14672:
14667:
14664:
14660:
14657:
14653:
14649:
14645:
14641:
14636:
14631:
14627:
14623:
14622:
14617:
14612:
14609:
14605:
14601:
14597:
14593:
14589:
14584:
14579:
14575:
14571:
14570:
14565:
14561:
14557:
14554:
14550:
14549:
14535:
14526:
14517:
14508:
14499:
14492:
14491:
14484:
14475:
14466:
14457:
14448:
14439:
14432:
14426:
14419:
14413:
14404:
14400:
14390:
14387:
14385:
14382:
14380:
14377:
14375:
14372:
14370:
14367:
14366:
14360:
14358:
14339:
14333:
14330:
14327:
14321:
14318:
14315:
14312:
14309:
14303:
14300:
14297:
14291:
14288:
14285:
14282:
14279:
14273:
14270:
14267:
14254:
14253:
14252:
14232:
14229:
14223:
14214:
14211:
14205:
14202:
14199:
14196:
14187:
14173:
14150:
14147:
14144:
14138:
14135:
14130:
14126:
14118:This implies
14099:
14096:
14093:
14084:
14067:
14064:
14057:
14054:
14051:
14048:
14045:
14040:
14037:
14031:
14026:
14023:
14020:
14013:
14010:
14007:
14001:
13996:
13991:
13987:
13983:
13977:
13969:
13965:
13961:
13955:
13952:
13949:
13943:
13936:
13935:
13934:
13920:
13917:
13911:
13903:
13899:
13892:
13886:
13883:
13880:
13874:
13871:
13865:
13858:
13854:
13850:
13812:
13809:
13802:
13799:
13796:
13791:
13788:
13782:
13777:
13771:
13768:
13765:
13759:
13752:
13751:
13750:
13730:
13727:
13722:
13718:
13711:
13708:
13705:
13698:
13691:
13688:
13681:
13678:
13675:
13669:
13659:
13655:
13651:
13648:
13645:
13638:
13631:
13628:
13622:
13617:
13613:
13609:
13603:
13597:
13590:
13589:
13588:
13574:
13571:
13565:
13559:
13533:
13530:
13522:
13518:
13514:
13511:
13488:
13485:
13482:
13475:
13468:
13465:
13458:
13451:
13448:
13437:
13433:
13429:
13414:
13411:
13408:
13398:
13397:
13396:
13380:
13376:
13355:
13352:
13346:
13340:
13314:
13311:
13302:
13294:
13289:
13285:
13281:
13265:
13247:
13244:
13241:
13232:
13226:
13220:
13212:
13211:ReLU function
13202:
13199:
13185:
13165:
13162:
13159:
13139:
13119:
13116:
13113:
13087:
13084:
13081:
13073:
13069:
13062:
13059:
13056:
13050:
13047:
13038:
13035:
13032:
13026:
13020:
13017:
13014:
13005:
12998:
12997:
12996:
12982:
12974:
12969:
12965:
12949:
12945:
12918:
12897:
12875:
12842:
12819:
12815:
12787:
12770:
12766:
12762:
12756:
12750:
12744:
12738:
12735:
12732:
12723:
12720:
12710:
12709:
12708:
12692:
12688:
12675:
12671:
12668:
12660:
12659:
12639:
12630:
12627:
12624:
12618:
12612:
12609:
12606:
12597:
12594:
12586:
12583:
12580:
12569:
12566:
12563:
12550:
12549:
12548:
12546:
12545:
12539:
12521:
12518:
12515:
12509:
12506:
12503:
12497:
12491:
12488:
12485:
12455:
12452:
12449:
12443:
12437:
12434:
12431:
12422:
12419:
12413:
12410:
12407:
12396:
12393:
12390:
12383:
12356:
12353:
12350:
12347:
12344:
12341:
12338:
12330:
12327:
12324:
12317:
12300:
12293:
12291:
12277:
12263:
12260:
12257:
12251:
12248:
12245:
12242:
12235:
12234:
12231:
12214:
12211:
12188:
12185:
12182:
12179:
12176:
12170:
12159:
12154:
12140:
12115:
12105:
12102:
12099:
12096:
12093:
12090:
12087:
12084:
12079:
12047:
12044:
12039:
12031:
12025:
12022:
12017:
12009:
12004:
11996:
11993:
11983:
11982:
11981:
11964:
11961:
11958:
11950:
11947:
11944:
11938:
11935:
11932:
11923:
11917:
11911:
11906:
11894:
11893:
11892:
11879:
11876:
11873:
11851:
11841:
11838:
11816:
11803:
11800:
11797:
11777:
11757:
11754:
11751:
11743:
11719:
11715:
11691:
11682:
11679:
11676:
11670:
11662:
11659:
11656:
11650:
11647:
11644:
11638:
11632:
11629:
11626:
11620:
11613:
11612:
11611:
11597:
11594:
11561:
11557:
11533:
11525:
11512:
11507:
11504:
11501:
11494:
11487:
11484:
11479:
11473:
11470:
11467:
11461:
11454:
11453:
11452:
11425:
11421:
11390:
11387:
11383:
11357:
11353:
11344:
11343:Bergman space
11340:
11297:
11294:
11285:
11269:
11254:
11228:
11225:
11222:
11217:
11210:
11207:
11204:
11199:
11193:
11188:
11182:
11179:
11176:
11170:
11163:
11162:
11161:
11144:
11141:
11138:
11132:
11124:
11120:
11113:
11112:inner product
11109:
11105:
11101:
11097:
11093:
11086:
11063:
11060:
11043:
11035:
11028:
11025:
11021:
11018:
11013:
11005:
10999:
10994:
10991:
10971:
10967:
10962:
10957:
10949:
10939:
10938:
10924:
10904:
10896:
10880:
10877:
10874:
10870:
10863:
10856:
10853:
10850:
10841:
10837:
10833:
10827:
10824:
10821:
10815:
10808:
10807:
10806:
10805:
10803:
10800:
10782:
10779:
10776:
10772:
10762:
10758:
10754:
10747:
10739:
10736:
10733:
10724:
10720:
10716:
10710:
10707:
10704:
10698:
10691:
10690:
10689:
10688:
10686:
10682:
10679:
10678:
10677:
10657:
10654:
10651:
10642:
10639:
10633:
10630:
10627:
10621:
10610:
10586:
10583:
10580:
10572:
10569:
10565:
10560:
10552:
10549:
10543:
10540:
10537:
10531:
10525:
10519:
10516:
10513:
10507:
10500:
10499:
10493:
10477:
10469:
10463:
10458:
10453:
10445:
10419:
10416:
10413:
10407:
10401:
10395:
10375:
10349:
10346:
10343:
10337:
10331:
10328:
10325:
10319:
10312:
10311:
10285:
10262:
10259:
10256:
10253:
10248:
10244:
10232:
10231:
10210:
10207:
10204:
10198:
10178:
10175:
10172:
10169:
10166:
10143:
10140:
10137:
10131:
10111:
10108:
10105:
10102:
10099:
10091:
10075:
10067:
10051:
10048:
10045:
10042:
10039:
10036:
10029:
10026:
10020:
10017:
10014:
10008:
10002:
9999:
9996:
9990:
9987:
9982:
9974:
9971:
9968:
9962:
9955:
9954:
9952:
9948:
9932:
9929:
9926:
9923:
9910:
9907:
9904:
9898:
9895:
9892:
9886:
9883:
9878:
9873:
9863:
9859:
9855:
9850:
9846:
9839:
9833:
9830:
9827:
9819:
9815:
9807:
9806:
9805:
9804:
9790:
9787:
9784:
9764:
9761:
9755:
9752:
9749:
9743:
9723:
9715:
9698:
9694:
9690:
9685:
9681:
9660:
9640:
9637:
9634:
9629:
9625:
9616:
9600:
9595:
9591:
9587:
9584:
9581:
9576:
9572:
9568:
9565:
9557:
9536:
9532:
9528:
9523:
9519:
9510:
9506:
9502:
9494:
9490:
9486:
9481:
9477:
9468:
9464:
9460:
9449:
9445:
9441:
9438:
9435:
9430:
9426:
9419:
9411:
9407:
9403:
9400:
9397:
9392:
9388:
9378:
9371:
9370:
9369:
9368:
9354:
9349:
9344:
9341:
9338:
9328:
9324:
9298:
9293:
9290:
9287:
9277:
9273:
9247:
9242:
9239:
9236:
9226:
9222:
9210:
9209:
9208:
9200:
9198:
9180:
9169:
9163:
9160:
9154:
9148:
9142:
9136:
9133:
9130:
9124:
9102:
9098:
9074:
9068:
9065:
9062:
9051:
9046:
9035:
9029:
9026:
9023:
9017:
9011:
9005:
9002:
8999:
8996:
8993:
8990:
8985:
8977:
8965:
8960:
8952:
8942:
8941:
8940:
8924:
8920:
8896:
8890:
8887:
8884:
8873:
8868:
8857:
8851:
8848:
8845:
8839:
8833:
8827:
8824:
8821:
8818:
8815:
8809:
8798:
8795:
8792:
8786:
8781:
8777:
8769:
8768:
8767:
8764:
8750:
8741:
8725:
8714:
8706:
8702:
8694:
8690:
8681:
8675:
8669:
8647:
8643:
8639:
8636:
8628:
8627:
8610:
8607:
8604:
8582:
8578:
8574:
8568:
8562:
8542:
8539:
8536:
8527:
8525:
8524:
8507:
8487:
8474:
8467:
8465:
8451:
8446:
8435:
8429:
8426:
8420:
8414:
8408:
8402:
8399:
8396:
8390:
8383:
8382:
8379:
8376:
8361:
8341:
8335:
8332:
8329:
8321:
8311:
8309:
8305:
8286:
8279:
8275:
8266:
8262:
8256:
8250:
8246:
8242:
8239:
8235:
8226:
8222:
8216:
8210:
8206:
8202:
8199:
8195:
8180:
8177:
8174:
8170:
8166:
8161:
8156:
8152:
8149:
8146:
8142:
8133:
8132:
8131:
8117:
8093:
8086:
8079:
8075:
8069:
8062:
8058:
8052:
8046:
8042:
8038:
8035:
8031:
8017:
8014:
8011:
8007:
7991:
7983:
7979:
7975:
7972:
7968:
7964:
7961:
7954:
7953:
7952:
7938:
7918:
7909:
7895:
7872:
7869:
7865:
7858:
7850:
7846:
7838:
7830:
7826:
7819:
7815:
7809:
7804:
7801:
7798:
7794:
7790:
7784:
7781:
7778:
7772:
7768:
7762:
7759:
7756:
7740:
7728:
7727:
7726:
7712:
7709:
7706:
7703:
7700:
7674:
7666:
7662:
7654:
7646:
7642:
7635:
7631:
7620:
7617:
7614:
7610:
7606:
7600:
7597:
7594:
7588:
7581:
7580:
7579:
7565:
7545:
7542:
7519:
7513:
7510:
7505:
7501:
7477:
7471:
7449:
7445:
7424:
7421:
7401:
7398:
7393:
7389:
7385:
7380:
7376:
7352:
7344:
7340:
7314:
7310:
7283:
7275:
7271:
7265:
7261:
7257:
7251:
7243:
7239:
7233:
7229:
7208:
7205:
7200:
7196:
7184:
7159:
7156:
7151:
7141:
7137:
7125:
7107:
7103:
7093:
7079:
7057:
7053:
7032:
7012:
6990:
6986:
6976:
6962:
6939:
6931:
6927:
6900:
6894:
6891:
6884:
6878:
6872:
6869:
6866:
6860:
6855:
6851:
6847:
6841:
6832:
6827:
6823:
6812:
6811:
6810:
6793:
6785:
6781:
6771:
6763:
6759:
6755:
6750:
6746:
6717:
6714:
6711:
6708:
6705:
6685:
6678:
6677:Borel measure
6662:
6654:
6638:
6624:
6622:
6604:
6600:
6596:
6593:
6585:
6561:
6556:
6547:
6529:
6525:
6501:
6495:
6487:
6483:
6479:
6474:
6464:
6460:
6456:
6451:
6447:
6440:
6435:
6425:
6421:
6417:
6412:
6408:
6401:
6396:
6380:
6375:
6371:
6366:
6362:
6355:
6350:
6340:
6336:
6332:
6327:
6323:
6316:
6311:
6303:
6300:
6295:
6291:
6284:
6278:
6272:
6265:
6264:
6263:
6261:
6257:
6253:
6237:
6234:
6231:
6205:
6201:
6190:
6185:
6164:
6161:
6156:
6152:
6123:
6118:
6114:
6109:
6105:
6101:
6098:
6090:
6086:
6082:
6066:
6058:
6054:
6049:
6044:
6040:
6024:
6021:
6018:
5995:
5992:
5989:
5986:
5981:
5977:
5951:
5943:
5940:
5937:
5931:
5926:
5918:
5915:
5912:
5886:
5881:
5871:
5867:
5863:
5858:
5854:
5847:
5841:
5838:
5835:
5829:
5826:
5821:
5811:
5807:
5803:
5798:
5794:
5783:
5782:
5781:
5779:
5778:
5773:
5769:
5765:
5761:
5757:
5738:
5732:
5726:
5723:
5717:
5714:
5709:
5705:
5698:
5693:
5689:
5678:
5675:
5672:
5668:
5664:
5657:
5653:
5647:
5641:
5637:
5633:
5626:
5622:
5617:
5612:
5605:
5601:
5590:
5587:
5584:
5580:
5576:
5571:
5561:
5557:
5553:
5550:
5540:
5539:
5538:
5536:
5535:
5515:
5512:
5505:
5501:
5487:
5483:
5478:
5472:
5468:
5462:
5459:
5456:
5451:
5448:
5445:
5441:
5429:
5426:
5423:
5407:
5389:
5379:
5375:
5370:
5364:
5360:
5349:
5346:
5343:
5339:
5335:
5329:
5323:
5316:
5315:
5314:
5312:
5305:
5301:
5297:
5292:
5290:
5286:
5266:
5262:
5256:
5250:
5246:
5242:
5237:
5233:
5228:
5223:
5217:
5214:
5211:
5205:
5182:
5174:
5170:
5166:
5161:
5157:
5150:
5145:
5141:
5134:
5130:
5123:
5118:
5115:
5112:
5108:
5102:
5097:
5094:
5091:
5087:
5083:
5076:
5072:
5066:
5058:
5054:
5049:
5043:
5039:
5033:
5028:
5025:
5022:
5018:
5014:
5007:
5003:
4998:
4992:
4988:
4982:
4977:
4974:
4971:
4967:
4962:
4953:
4952:
4951:
4946:
4942:
4938:
4934:
4924:
4920:
4916:
4912:
4905:
4901:
4897:
4890:
4886:
4882:
4878:
4874:
4870:
4867:
4866:
4865:
4863:
4859:
4855:
4845:
4831:
4808:
4805:
4802:
4796:
4789:converges to
4773:
4765:
4761:
4752:
4734:
4730:
4721:
4704:
4698:
4692:
4689:
4686:
4683:
4677:
4674:
4671:
4667:
4663:
4657:
4651:
4646:
4641:
4638:
4634:
4627:
4624:
4620:
4615:
4612:
4609:
4596:
4588:
4584:
4577:
4571:
4565:
4552:
4548:
4544:
4537:
4533:
4522:
4518:
4514:
4511:
4501:
4500:
4499:
4497:
4493:
4467:
4455:
4448:
4442:
4439:
4436:
4433:
4427:
4424:
4412:
4409:
4406:
4403:
4399:
4392:
4387:
4384:
4381:
4375:
4372:
4369:
4366:
4362:
4355:
4347:
4343:
4329:
4325:
4317:
4316:
4315:
4298:
4290:
4286:
4262:
4253:
4250:
4247:
4241:
4230:
4227:
4224:
4218:
4212:
4209:
4203:
4199:
4192:
4189:
4186:
4178:
4175:
4169:
4165:
4162:
4157:
4154:
4149:
4143:
4135:
4131:
4123:
4122:
4121:
4105:
4101:
4091:
4077:
4054:
4047:
4043:
4034:
4025:
4022:
4016:
4011:
4008:
4002:
3989:
3983:
3965:
3961:
3952:
3949:
3944:
3941:
3935:
3930:
3927:
3921:
3908:
3902:
3892:
3887:
3884:
3880:
3876:
3873:
3865:
3862:
3858:
3853:
3842:
3836:
3824:
3823:
3822:
3808:
3800:
3796:
3777:
3774:
3771:
3765:
3762:
3759:
3755:
3748:
3742:
3737:
3732:
3729:
3725:
3718:
3715:
3711:
3706:
3700:
3694:
3687:
3686:
3685:
3683:
3678:
3676:
3672:
3634:
3626:
3588:
3555:
3551:
3527:
3524:
3521:
3508:
3502:
3496:
3490:
3484:
3471:
3467:
3463:
3456:
3452:
3443:
3440:
3437:
3427:
3426:
3425:
3411:
3403:
3387:
3384:
3378:
3375:
3372:
3369:
3365:
3358:
3352:
3339:
3335:
3331:
3325:
3319:
3286:
3282:
3252:
3249:
3246:
3243:
3237:
3231:
3225:
3222:
3219:
3203:
3199:
3195:
3192:
3186:
3183:
3176:
3175:
3174:
3157:
3154:
3151:
3148:
3141:
3125:
3118:
3115:
3111:
3106:
3092:
3089:
3083:
3072:
3054:
3050:
3029:
3026:
3023:
3003:
2995:
2979:
2972:is a set and
2959:
2936:
2933:
2930:
2922:
2918:
2903:
2889:
2869:
2849:
2841:
2836:
2832:
2828:
2825:
2822:
2817:
2813:
2804:
2801:
2798:
2793:
2789:
2785:
2782:
2779:
2774:
2770:
2766:
2758:
2755:
2732:
2729:
2724:
2719:
2706:
2702:
2697:
2691:
2687:
2681:
2676:
2673:
2670:
2666:
2656:
2651:
2646:
2638:
2634:
2629:
2623:
2619:
2613:
2608:
2605:
2602:
2598:
2594:
2587:
2583:
2578:
2572:
2568:
2562:
2557:
2554:
2551:
2547:
2542:
2537:
2532:
2527:
2519:
2515:
2510:
2504:
2500:
2494:
2489:
2486:
2483:
2479:
2475:
2468:
2464:
2459:
2454:
2447:
2443:
2437:
2432:
2429:
2426:
2422:
2418:
2410:
2406:
2402:
2397:
2393:
2386:
2381:
2377:
2371:
2367:
2361:
2356:
2353:
2350:
2347:
2344:
2340:
2332:
2331:
2330:
2328:
2282:
2279:
2276:
2273:
2270:
2247:
2242:
2232:
2228:
2221:
2216:
2212:
2205:
2199:
2196:
2193:
2187:
2180:
2179:
2178:
2134:
2131:
2128:
2125:
2122:
2102:
2093:
2077:
2073:
2052:
2032:
2029:
2024:
2020:
1996:
1991:
1981:
1977:
1970:
1965:
1961:
1954:
1946:
1942:
1933:
1929:
1925:
1919:
1911:
1907:
1899:
1898:
1897:
1896:we have that
1883:
1861:
1857:
1792:
1770:
1766:
1752:
1745:
1743:
1729:
1726:
1723:
1720:
1711:
1701:
1697:
1690:
1687:
1681:
1675:
1667:
1663:
1659:
1653:
1647:
1640:
1639:
1636:
1622:
1600:
1596:
1575:
1555:
1547:
1531:
1523:
1507:
1485:
1481:
1460:
1439:
1431:
1430:
1424:
1407:
1402:
1398:
1392:
1367:
1364:
1361:
1338:
1333:
1329:
1315:
1308:
1306:
1291:
1288:
1285:
1282:
1273:
1265:
1256:
1252:
1248:
1237:
1231:
1223:
1212:
1204:
1200:
1187:
1186:
1183:
1169:
1166:
1161:
1157:
1136:
1128:
1110:
1106:
1085:
1065:
1057:
1039:
1035:
1014:
994:
986:
982:
963:
960:
957:
954:
940:
934:
928:
925:
920:
916:
908:
907:
906:
892:
872:
864:
848:
840:
836:
835:Hilbert space
820:
812:
796:
782:
780:
774:
772:
768:
764:
760:
756:
752:
748:
743:
741:
737:
733:
729:
725:
721:
717:
713:
709:
705:
701:
696:
694:
678:
658:
649:
647:
629:
625:
615:
613:
607:
604:
602:
580:
557:
551:
546:
543:
539:
529:
506:
500:
497:
491:
485:
457:
437:
414:
411:
408:
385:
365:
356:
342:
322:
300:
296:
286:
273:
267:
261:
258:
250:
246:
242:
239:
228:
214:
211:
208:
188:
185:
180:
176:
155:
152:
149:
85:
65:
57:
53:
52:Hilbert space
49:
45:
41:
32:
19:
14791:
14787:
14768:
14765:Wahba, Grace
14730:
14726:
14688:(1): 82–95.
14685:
14681:
14671:Wahba, Grace
14658:, June 2013.
14625:
14619:
14607:
14573:
14567:
14555:, June 2011.
14534:
14525:
14516:
14507:
14498:
14488:
14483:
14474:
14465:
14456:
14447:
14438:
14430:
14425:
14417:
14412:
14403:
14379:Kernel trick
14354:
14188:
14117:
13841:
13748:
13503:
13266:
13208:
13200:
13105:
12972:
12970:
12966:
12806:
12656:
12654:
12542:
12540:
12305:
12294:
12155:
12062:
11979:
11741:
11711:
11548:
11337:denotes the
11287:The case of
11286:
11252:
11250:
11122:
11115:
11107:
11103:
11099:
11095:
11089:
10917:in the RKHS
10801:
10684:
10680:
10613:
10367:
9206:
9197:kernel trick
9089:
8911:
8765:
8742:
8624:
8528:
8521:
8479:
8468:
8377:
8319:
8317:
8314:Feature maps
8301:
8109:
7951:is given by
7910:
7887:
7692:
7299:, where the
7094:
6977:
6918:
6630:
6620:
6583:
6545:
6516:
6259:
6255:
6251:
6250:then, since
6088:
6084:
6080:
6056:
6052:
6050:
6042:
6038:
5901:
5775:
5771:
5767:
5763:
5759:
5755:
5753:
5532:
5530:
5310:
5303:
5295:
5293:
5288:
5284:
5197:
4944:
4940:
4936:
4929:
4922:
4918:
4914:
4907:
4903:
4899:
4895:
4894:
4888:
4884:
4880:
4872:
4868:
4857:
4851:
4722:
4719:
4489:
4277:
4092:
4069:
3792:
3679:
3542:
3273:
3138:. Fix some
3107:
2909:
2747:
2262:
2094:
2011:
1757:
1746:
1521:
1427:
1425:
1320:
1309:
987:if, for all
984:
980:
979:We say that
978:
788:
775:
744:
716:James Mercer
697:
692:
650:
645:
611:
608:
605:
530:
357:
287:
229:
47:
43:
37:
14628:(1): 1–49.
14166:reproduces
10435:satisfying
8320:feature map
6544:belongs to
4921:, ⋅ ). Let
4862:E. H. Moore
4753:, and that
3114:bandlimited
728:Gábor Szegő
702:concerning
14546:References
14442:Okutmustur
14074:otherwise.
12158:isomorphic
11744:for every
9736:such that
9203:Properties
8308:kernel PCA
7725:such that
7172:such that
5300:completion
4898:. For all
4887:for which
4871:. Suppose
4498:, we have
3684:, we have
3669:of entire
2748:for every
1182:such that
1056:continuous
863:evaluation
785:Definition
56:functional
14794:: 91–136.
14755:253885172
14656:1110.4017
14331:−
14322:
14316:−
14301:−
14292:
14286:−
14236:∞
14224:×
14218:∞
14200:×
14145:⋅
14049:≤
13988:∫
13819:otherwise
13734:⟩
13715:⟨
13665:∞
13656:∫
13614:∫
13537:∞
13515:∈
13443:∞
13434:∫
13419:⟩
13406:⟨
13318:∞
13006:γ
12973:separable
12950:γ
12923:Γ
12777:⟩
12748:⟨
12693:γ
12685:→
12680:Γ
12598:γ
12558:Γ
12525:Λ
12522:×
12516:∈
12423:γ
12384:γ
12360:Λ
12357:∈
12345:∈
12318:γ
12270:→
12267:Λ
12264:×
12258:×
12255:Λ
12252:×
12243:γ
12218:Λ
12215:×
12183:…
12168:Λ
12106:∈
12094:∈
12076:Γ
12040:⊺
12014:⟩
12001:Γ
11991:⟨
11962:∈
11948:∈
11927:Γ
11903:Γ
11877:∈
11842:∈
11807:→
11728:Γ
11680:−
11671:π
11660:−
11648:
11516:¯
11505:−
11488:π
11339:unit disc
11226:≠
11022:σ
10995:σ
10972:∫
10954:‖
10947:‖
10875:σ
10864:σ
10860:‖
10854:−
10848:‖
10842:−
10777:σ
10759:σ
10744:‖
10737:−
10731:‖
10725:−
10661:‖
10655:−
10649:‖
10587:∈
10573:∈
10570:α
10547:⟩
10535:⟨
10532:α
10474:‖
10470:β
10467:‖
10450:‖
10443:‖
10423:⟩
10420:β
10411:⟨
10368:The RKHS
10353:⟩
10341:⟨
10260:∈
10254:∣
10176:∈
10109:∈
10046:∈
10034:∀
9988:≤
9927:∈
9921:∀
9896:−
9870:‖
9856:−
9843:‖
9788:∈
9691:×
9635:⊂
9588:×
9585:⋯
9582:×
9503:⋯
9439:…
9401:…
9177:⟩
9164:φ
9149:φ
9146:⟨
9103:φ
9066:∈
9055:∀
9043:⟩
9030:φ
9021:⟨
8997:∈
8982:‖
8975:‖
8961:φ
8957:‖
8950:‖
8925:φ
8888:∈
8877:∀
8865:⟩
8852:φ
8843:⟨
8819:∈
8813:∃
8810:∣
8802:→
8782:φ
8703:φ
8691:σ
8670:φ
8644:ℓ
8623:. Then (
8608:∈
8563:φ
8526:) holds.
8443:⟩
8430:φ
8415:φ
8412:⟨
8339:→
8333::
8330:φ
8322:is a map
8276:σ
8247:φ
8207:φ
8186:∞
8171:∑
8130:given by
8090:∞
8076:σ
8043:φ
8023:∞
8008:∑
7976:∈
7847:φ
7827:φ
7816:σ
7795:∑
7791:−
7747:∞
7744:→
7710:∈
7663:φ
7643:φ
7632:σ
7626:∞
7611:∑
7511:∈
7502:φ
7390:σ
7311:φ
7272:φ
7262:σ
7240:φ
7197:σ
7191:∞
7188:→
7157:≥
7138:σ
6963:μ
6895:μ
6867:⋅
6852:∫
6842:⋅
6778:→
6721:→
6715:×
6686:μ
6566:⊥
6471:⟩
6444:⟨
6432:⟩
6405:⟨
6393:⟩
6385:⊥
6359:⟨
6347:⟩
6320:⟨
6308:⟩
6288:⟨
6235:∈
6224:. Now if
6210:⊥
6202:∈
6195:⊥
6162:∈
6128:⊥
6022:⊂
5993:∈
5948:⟩
5944:⋅
5938:⋅
5935:⟨
5923:⟩
5919:⋅
5913:⋅
5910:⟨
5878:⟩
5851:⟨
5818:⟩
5791:⟨
5684:∞
5669:∑
5596:∞
5581:∑
5568:⟩
5548:⟨
5442:∑
5427:≥
5414:∞
5411:→
5355:∞
5340:∑
5109:∑
5088:∑
5019:∑
4968:∑
4906:, define
4879:on a set
4806:−
4797:δ
4687:ω
4675:ω
4664:⋅
4658:ω
4639:−
4635:∫
4628:π
4604:¯
4578:⋅
4561:∞
4556:∞
4553:−
4549:∫
4529:⟩
4509:⟨
4463:otherwise
4434:−
4428:∈
4425:ω
4410:ω
4404:−
4373:ω
4367:−
4338:∞
4333:∞
4330:−
4326:∫
4251:−
4242:π
4228:−
4213:
4190:−
4179:π
4166:
4158:π
4039:‖
4032:‖
4026:π
4012:ω
3990:ω
3974:∞
3969:∞
3966:−
3962:∫
3953:π
3931:ω
3909:ω
3885:−
3881:∫
3866:π
3854:≤
3775:ω
3766:ω
3749:ω
3730:−
3726:∫
3719:π
3680:From the
3673:, by the
3516:¯
3497:⋅
3480:∞
3475:∞
3472:−
3468:∫
3448:⟩
3435:⟨
3376:ω
3370:−
3348:∞
3343:∞
3340:−
3336:∫
3326:ω
3244:−
3238:⊂
3226:
3220:∣
3196:∈
3161:∞
3090:⊂
3027:∈
2980:μ
2937:μ
2842:∈
2826:…
2799:∈
2783:…
2759:∈
2730:≥
2667:∑
2599:∑
2548:∑
2480:∑
2423:∑
2341:∑
2286:→
2280:×
2239:⟩
2209:⟨
2138:→
2132:×
2030:∈
1988:⟩
1958:⟨
1724:∈
1718:∀
1708:⟩
1685:⟨
1411:∞
1365:∈
1342:∞
1321:Although
1286:∈
1280:∀
1270:‖
1263:‖
1249:≤
1058:at every
958:∈
952:∀
932:↦
720:functions
587:∞
584:→
552:
498:−
418:‖
412:−
406:‖
256:⟩
237:⟨
212:∈
186:∈
153:∈
14803:Category
14673:(1971).
14562:(1950).
14363:See also
13859:′
13692:′
13632:′
13469:′
13452:′
13106:for all
11029:′
10681:Gaussian
9777:for all
8597:for all
8480:Clearly
8354:, where
8257:⟩
8236:⟨
8217:⟩
8196:⟨
8157:⟩
8143:⟨
8053:⟩
8032:⟨
7693:for all
7535:for all
7414:for all
6083:. Since
6037:because
5648:⟩
5613:⟨
5496:‖
5437:‖
5257:⟩
5229:⟨
5067:⟩
4963:⟨
4935: :
4421:if
2906:Examples
2715:‖
2662:‖
2647:⟩
2543:⟨
2528:⟩
2455:⟨
708:harmonic
14775:, 1990.
14747:2914365
14704:0290013
14644:1864085
14600:0051437
14592:1990404
14538:Rosasco
14529:Alvarez
14511:De Vito
14478:Rosasco
14469:Rosasco
14460:Durrett
14451:Paulson
12834:is the
12375:where
11315:(where
9949:By the
6011:. Then
5298:be the
4869:Theorem
3400:is the
3069:is the
3016:. For
2992:is the
2329:, i.e.
2012:where
50:) is a
14753:
14745:
14702:
14642:
14598:
14590:
13896:
13749:where
12807:where
12547:) via
11742:matrix
11094:. Let
9059:
8881:
6919:where
6144:where
6059:. Let
6055:is in
3274:where
2952:where
2225:
1974:
1876:is in
1758:Since
1694:
948:
753:, and
734:, and
614:spaces
609:While
14784:(PDF)
14751:S2CID
14678:(PDF)
14652:arXiv
14588:JSTOR
14520:Zhang
14395:Notes
13933:i.e.
10088:as a
5398:where
4896:Proof
1125:is a
983:is a
450:and
378:and
98:(to
14773:SIAM
14319:ReLU
14289:ReLU
14055:<
13842:and
13800:<
13552:and
13209:The
13152:and
12937:and
11980:and
11866:and
11716:and
10878:>
10780:>
9617:Let
9211:Let
8662:and
8555:and
8087:<
7399:>
7221:and
6698:and
6258:and
6177:and
5766:and
5294:Let
4163:sinc
3797:and
3223:supp
3158:<
3152:<
2303:(or
2155:(or
1827:(or
1339:<
1167:>
813:and
789:Let
710:and
706:for
335:at
120:or
48:RKHS
42:, a
14767:,
14735:doi
14690:doi
14630:doi
14578:doi
14262:min
14088:min
13236:max
13178:in
13132:in
11770:in
11645:sin
11451:is
11391:on
10683:or
9673:to
8969:inf
8939:by
7931:of
7753:sup
7737:lim
7181:lim
7005:of
6809:as
6619:in
6582:in
5774:, (
5770:in
5537:):
5420:sup
5404:lim
5302:of
4950:by
4902:in
4210:sin
3404:of
2996:on
1615:of
1568:in
1500:in
1389:sup
1129:on
1078:in
1054:is
1007:in
841:on
837:of
811:set
714:.
540:sin
478:sup
38:In
14805::
14792:13
14790:.
14786:.
14771:,
14749:.
14743:MR
14741:.
14731:35
14729:.
14700:MR
14698:.
14686:33
14684:.
14680:.
14640:MR
14638:.
14626:39
14624:.
14618:.
14596:MR
14594:.
14586:.
14574:68
14572:.
14566:.
14186:.
12964:.
12847:th
11284:.
10804::
10687::
10492:.
9953:,
8740:.
8318:A
8310:.
7908:.
7873:0.
6975:.
6262::
5516:0.
4939:∈
4913:=
4864:.
3821:,
3677:.
3105:.
2092:.
1423:.
1224::=
1027:,
905:,
833:a
781:.
749:,
730:,
603:.
227:,
14757:.
14737::
14721:.
14706:.
14692::
14654::
14646:.
14632::
14602:.
14580::
14340:.
14337:)
14334:x
14328:y
14325:(
14313:y
14310:=
14307:)
14304:y
14298:x
14295:(
14283:x
14280:=
14277:)
14274:y
14271:,
14268:x
14265:(
14239:)
14233:,
14230:0
14227:[
14221:)
14215:,
14212:0
14209:[
14206:=
14203:X
14197:X
14174:f
14154:)
14151:y
14148:,
14142:(
14139:K
14136:=
14131:y
14127:K
14103:)
14100:y
14097:,
14094:x
14091:(
14085:=
14068:,
14065:y
14058:y
14052:x
14046:0
14041:,
14038:x
14032:{
14027:=
14024:z
14021:d
14017:)
14014:y
14011:,
14008:z
14005:(
14002:G
13997:x
13992:0
13984:=
13981:)
13978:x
13975:(
13970:y
13966:K
13962:=
13959:)
13956:y
13953:,
13950:x
13947:(
13944:K
13921:0
13918:=
13915:)
13912:0
13909:(
13904:y
13900:K
13893:,
13890:)
13887:y
13884:,
13881:x
13878:(
13875:G
13872:=
13869:)
13866:x
13863:(
13855:y
13851:K
13813:,
13810:0
13803:y
13797:x
13792:,
13789:1
13783:{
13778:=
13775:)
13772:y
13769:,
13766:x
13763:(
13760:G
13731:f
13728:,
13723:y
13719:K
13712:=
13709:x
13706:d
13702:)
13699:x
13696:(
13689:f
13685:)
13682:y
13679:,
13676:x
13673:(
13670:G
13660:0
13652:=
13649:x
13646:d
13642:)
13639:x
13636:(
13629:f
13623:y
13618:0
13610:=
13607:)
13604:y
13601:(
13598:f
13575:0
13572:=
13569:)
13566:0
13563:(
13560:f
13540:)
13534:,
13531:0
13528:[
13523:1
13519:C
13512:f
13489:.
13486:x
13483:d
13479:)
13476:x
13473:(
13466:g
13462:)
13459:x
13456:(
13449:f
13438:0
13430:=
13424:H
13415:g
13412:,
13409:f
13381:2
13377:L
13356:0
13353:=
13350:)
13347:0
13344:(
13341:f
13321:)
13315:,
13312:0
13309:[
13306:)
13303:0
13300:(
13295:1
13290:2
13286:L
13282:=
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13248:x
13245:,
13242:0
13239:{
13233:=
13230:)
13227:x
13224:(
13221:f
13186:T
13166:s
13163:,
13160:t
13140:X
13120:y
13117:,
13114:x
13091:)
13088:s
13085:,
13082:t
13079:(
13074:T
13070:K
13066:)
13063:y
13060:,
13057:x
13054:(
13051:K
13048:=
13045:)
13042:)
13039:s
13036:,
13033:y
13030:(
13027:,
13024:)
13021:t
13018:,
13015:x
13012:(
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12983:T
12946:H
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12760:)
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12745:=
12742:)
12739:t
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12733:x
12730:(
12727:)
12724:f
12721:D
12718:(
12689:H
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12672::
12669:D
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12640:.
12637:)
12634:)
12631:s
12628:,
12625:y
12622:(
12619:,
12616:)
12613:t
12610:,
12607:x
12604:(
12601:(
12595:=
12590:)
12587:s
12584:,
12581:t
12578:(
12574:)
12570:y
12567:,
12564:x
12561:(
12544:4
12537:.
12519:X
12513:)
12510:s
12507:,
12504:y
12501:(
12498:,
12495:)
12492:t
12489:,
12486:x
12483:(
12462:)
12459:)
12456:s
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12450:y
12447:(
12444:,
12441:)
12438:t
12435:,
12432:x
12429:(
12426:(
12420:=
12417:)
12414:s
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12408:y
12405:(
12400:)
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12391:x
12388:(
12363:}
12354:t
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12348:X
12342:x
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12331:t
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12325:x
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12314:{
12299:)
12297:4
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12274:R
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12249:X
12246::
12212:X
12192:}
12189:T
12186:,
12180:,
12177:1
12174:{
12171:=
12141:H
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12103:c
12100:,
12097:X
12091:x
12088::
12085:c
12080:x
12072:{
12048:.
12045:c
12036:)
12032:x
12029:(
12026:f
12023:=
12018:H
12010:c
12005:x
11997:,
11994:f
11965:X
11959:y
11951:H
11945:c
11942:)
11939:y
11936:,
11933:x
11930:(
11924:=
11921:)
11918:y
11915:(
11912:c
11907:x
11880:X
11874:x
11852:T
11847:R
11839:c
11817:T
11812:R
11804:X
11801::
11798:f
11778:X
11758:y
11755:,
11752:x
11692:.
11686:)
11683:y
11677:x
11674:(
11666:)
11663:y
11657:x
11654:(
11651:a
11639:=
11636:)
11633:y
11630:,
11627:x
11624:(
11621:K
11598:a
11595:2
11575:)
11571:R
11567:(
11562:2
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11534:.
11526:2
11522:)
11513:y
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11502:1
11499:(
11495:1
11485:1
11480:=
11477:)
11474:y
11471:,
11468:x
11465:(
11462:K
11439:)
11435:D
11431:(
11426:2
11422:H
11400:D
11371:)
11367:D
11363:(
11358:2
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11324:D
11302:D
11298:=
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11223:x
11218:0
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11208:=
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11189:=
11186:)
11183:y
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11177:x
11174:(
11171:K
11148:)
11145:y
11142:,
11139:x
11136:(
11133:K
11123:x
11118:x
11116:K
11108:H
11104:X
11100:H
11096:X
11064:.
11061:x
11057:d
11051:)
11044:2
11040:)
11036:x
11033:(
11026:f
11019:+
11014:2
11010:)
11006:x
11003:(
11000:f
10992:1
10985:(
10977:R
10968:=
10963:2
10958:H
10950:f
10925:H
10905:f
10881:0
10871:,
10857:y
10851:x
10838:e
10834:=
10831:)
10828:y
10825:,
10822:x
10819:(
10816:K
10783:0
10773:,
10763:2
10755:2
10748:2
10740:y
10734:x
10721:e
10717:=
10714:)
10711:y
10708:,
10705:x
10702:(
10699:K
10664:)
10658:y
10652:x
10646:(
10643:K
10640:=
10637:)
10634:y
10631:,
10628:x
10625:(
10622:K
10591:N
10584:d
10581:,
10577:R
10566:,
10561:d
10557:)
10553:1
10550:+
10544:y
10541:,
10538:x
10529:(
10526:=
10523:)
10520:y
10517:,
10514:x
10511:(
10508:K
10478:2
10464:=
10459:2
10454:H
10446:f
10417:,
10414:x
10408:=
10405:)
10402:x
10399:(
10396:f
10376:H
10350:y
10347:,
10344:x
10338:=
10335:)
10332:y
10329:,
10326:x
10323:(
10320:K
10298:.
10286:H
10266:}
10263:X
10257:x
10249:x
10245:K
10241:{
10214:)
10211:y
10208:,
10205:x
10202:(
10199:K
10179:X
10173:y
10170:,
10167:x
10147:)
10144:y
10141:,
10138:x
10135:(
10132:K
10112:X
10106:y
10103:,
10100:x
10076:K
10052:.
10049:X
10043:y
10040:,
10037:x
10030:1
10027:=
10024:)
10021:y
10018:,
10015:y
10012:(
10009:K
10006:)
10003:x
10000:,
9997:x
9994:(
9991:K
9983:2
9979:)
9975:y
9972:,
9969:x
9966:(
9963:K
9933:.
9930:X
9924:x
9917:)
9914:)
9911:y
9908:,
9905:x
9902:(
9899:K
9893:1
9890:(
9887:2
9884:=
9879:2
9874:H
9864:y
9860:K
9851:x
9847:K
9840:=
9837:)
9834:y
9831:,
9828:x
9825:(
9820:K
9816:d
9791:X
9785:x
9765:1
9762:=
9759:)
9756:x
9753:,
9750:x
9747:(
9744:K
9724:K
9699:0
9695:X
9686:0
9682:X
9661:K
9641:,
9638:X
9630:0
9626:X
9601:.
9596:p
9592:X
9577:1
9573:X
9569:=
9566:X
9542:)
9537:p
9533:y
9529:,
9524:p
9520:x
9516:(
9511:p
9507:K
9500:)
9495:1
9491:y
9487:,
9482:1
9478:x
9474:(
9469:1
9465:K
9461:=
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9455:)
9450:p
9446:y
9442:,
9436:,
9431:1
9427:y
9423:(
9420:,
9417:)
9412:p
9408:x
9404:,
9398:,
9393:1
9389:x
9385:(
9382:(
9379:K
9355:.
9350:p
9345:1
9342:=
9339:i
9335:)
9329:i
9325:X
9321:(
9299:p
9294:1
9291:=
9288:i
9284:)
9278:i
9274:K
9270:(
9248:p
9243:1
9240:=
9237:i
9233:)
9227:i
9223:X
9219:(
9181:F
9173:)
9170:y
9167:(
9161:,
9158:)
9155:x
9152:(
9143:=
9140:)
9137:y
9134:,
9131:x
9128:(
9125:K
9099:H
9075:.
9072:}
9069:X
9063:x
9052:,
9047:F
9039:)
9036:x
9033:(
9027:,
9024:w
9018:=
9015:)
9012:x
9009:(
9006:f
9003:,
9000:F
8994:w
8991::
8986:F
8978:w
8972:{
8966:=
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8921:H
8897:.
8894:}
8891:X
8885:x
8874:,
8869:F
8861:)
8858:x
8855:(
8849:,
8846:w
8840:=
8837:)
8834:x
8831:(
8828:f
8825:,
8822:F
8816:w
8806:R
8799:X
8796::
8793:f
8790:{
8787:=
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8751:H
8726:i
8722:)
8718:)
8715:x
8712:(
8707:i
8695:i
8685:(
8682:=
8679:)
8676:x
8673:(
8648:2
8640:=
8637:F
8626:3
8611:X
8605:x
8583:x
8579:K
8575:=
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8569:x
8566:(
8543:H
8540:=
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8523:3
8508:F
8488:K
8473:)
8471:3
8469:(
8452:.
8447:F
8439:)
8436:y
8433:(
8427:,
8424:)
8421:x
8418:(
8409:=
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8403:y
8400:,
8397:x
8394:(
8391:K
8362:F
8342:F
8336:X
8287:.
8280:i
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8243:,
8240:g
8227:2
8223:L
8211:i
8203:,
8200:f
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8147:f
8118:H
8094:}
8080:i
8070:2
8063:2
8059:L
8047:i
8039:,
8036:f
8018:1
8015:=
8012:i
8001:|
7995:)
7992:X
7989:(
7984:2
7980:L
7973:f
7969:{
7965:=
7962:H
7939:K
7919:H
7896:K
7870:=
7866:|
7862:)
7859:v
7856:(
7851:j
7842:)
7839:u
7836:(
7831:j
7820:j
7810:n
7805:1
7802:=
7799:j
7788:)
7785:v
7782:,
7779:u
7776:(
7773:K
7769:|
7763:v
7760:,
7757:u
7741:n
7713:X
7707:y
7704:,
7701:x
7678:)
7675:y
7672:(
7667:j
7658:)
7655:x
7652:(
7647:j
7636:j
7621:1
7618:=
7615:j
7607:=
7604:)
7601:y
7598:,
7595:x
7592:(
7589:K
7566:K
7546:.
7543:i
7523:)
7520:X
7517:(
7514:C
7506:i
7481:)
7478:X
7475:(
7472:C
7450:K
7446:T
7425:.
7422:i
7402:0
7394:i
7386:,
7381:K
7377:T
7356:)
7353:X
7350:(
7345:2
7341:L
7320:}
7315:i
7307:{
7287:)
7284:x
7281:(
7276:i
7266:i
7258:=
7255:)
7252:x
7249:(
7244:i
7234:K
7230:T
7209:0
7206:=
7201:i
7185:i
7160:0
7152:i
7148:)
7142:i
7134:(
7108:K
7104:T
7080:K
7058:K
7054:T
7033:K
7013:K
6991:K
6987:T
6943:)
6940:X
6937:(
6932:2
6928:L
6904:)
6901:t
6898:(
6892:d
6888:)
6885:t
6882:(
6879:f
6876:)
6873:t
6870:,
6864:(
6861:K
6856:X
6848:=
6845:)
6839:(
6836:]
6833:f
6828:K
6824:T
6820:[
6797:)
6794:X
6791:(
6786:2
6782:L
6775:)
6772:X
6769:(
6764:2
6760:L
6756::
6751:K
6747:T
6725:R
6718:X
6712:X
6709::
6706:K
6663:X
6639:K
6621:G
6605:H
6601:f
6597:=
6594:f
6584:G
6562:H
6557:f
6546:H
6530:x
6526:K
6502:,
6499:)
6496:x
6493:(
6488:H
6484:f
6480:=
6475:H
6465:H
6461:f
6457:,
6452:x
6448:K
6441:=
6436:G
6426:H
6422:f
6418:,
6413:x
6409:K
6402:=
6397:G
6381:H
6376:f
6372:,
6367:x
6363:K
6356:+
6351:G
6341:H
6337:f
6333:,
6328:x
6324:K
6317:=
6312:G
6304:f
6301:,
6296:x
6292:K
6285:=
6282:)
6279:x
6276:(
6273:f
6260:H
6256:G
6252:K
6238:X
6232:x
6206:H
6191:H
6186:f
6165:H
6157:H
6153:f
6124:H
6119:f
6115:+
6110:H
6106:f
6102:=
6099:f
6089:G
6085:H
6081:G
6067:f
6057:H
6053:G
6046:0
6043:H
6039:G
6025:G
6019:H
5999:}
5996:X
5990:x
5987::
5982:x
5978:K
5974:{
5952:G
5941:,
5932:=
5927:H
5916:,
5887:.
5882:G
5872:y
5868:K
5864:,
5859:x
5855:K
5848:=
5845:)
5842:y
5839:,
5836:x
5833:(
5830:K
5827:=
5822:H
5812:y
5808:K
5804:,
5799:x
5795:K
5777:2
5772:X
5768:y
5764:x
5760:K
5756:G
5739:.
5736:)
5733:x
5730:(
5727:f
5724:=
5721:)
5718:x
5715:,
5710:i
5706:x
5702:(
5699:K
5694:i
5690:a
5679:1
5676:=
5673:i
5665:=
5658:0
5654:H
5642:x
5638:K
5634:,
5627:i
5623:x
5618:K
5606:i
5602:a
5591:1
5588:=
5585:i
5577:=
5572:H
5562:x
5558:K
5554:,
5551:f
5534:2
5513:=
5506:0
5502:H
5488:i
5484:x
5479:K
5473:i
5469:a
5463:p
5460:+
5457:n
5452:n
5449:=
5446:i
5430:0
5424:p
5408:n
5393:)
5390:x
5387:(
5380:i
5376:x
5371:K
5365:i
5361:a
5350:1
5347:=
5344:i
5336:=
5333:)
5330:x
5327:(
5324:f
5311:H
5307:0
5304:H
5296:H
5289:K
5285:K
5267:0
5263:H
5251:y
5247:K
5243:,
5238:x
5234:K
5224:=
5221:)
5218:y
5215:,
5212:x
5209:(
5206:K
5183:,
5180:)
5175:i
5171:x
5167:,
5162:j
5158:y
5154:(
5151:K
5146:j
5142:b
5135:i
5131:a
5124:n
5119:1
5116:=
5113:j
5103:m
5098:1
5095:=
5092:i
5084:=
5077:0
5073:H
5059:i
5055:x
5050:K
5044:i
5040:a
5034:m
5029:1
5026:=
5023:i
5015:,
5008:j
5004:y
4999:K
4993:j
4989:b
4983:n
4978:1
4975:=
4972:j
4948:0
4945:H
4941:X
4937:x
4932:x
4930:K
4926:0
4923:H
4919:x
4917:(
4915:K
4910:x
4908:K
4904:X
4900:x
4889:K
4885:X
4881:X
4873:K
4832:a
4812:)
4809:x
4803:y
4800:(
4777:)
4774:y
4771:(
4766:x
4762:K
4735:x
4731:K
4705:.
4702:)
4699:x
4696:(
4693:f
4690:=
4684:d
4678:x
4672:i
4668:e
4661:)
4655:(
4652:F
4647:a
4642:a
4625:2
4621:1
4616:=
4613:y
4610:d
4600:)
4597:y
4594:(
4589:x
4585:K
4575:)
4572:y
4569:(
4566:f
4545:=
4538:2
4534:L
4523:x
4519:K
4515:,
4512:f
4468:,
4456:0
4449:,
4446:]
4443:a
4440:,
4437:a
4431:[
4413:x
4407:i
4400:e
4393:{
4388:=
4385:y
4382:d
4376:y
4370:i
4363:e
4359:)
4356:y
4353:(
4348:x
4344:K
4302:)
4299:y
4296:(
4291:x
4287:K
4263:.
4257:)
4254:x
4248:y
4245:(
4237:)
4234:)
4231:x
4225:y
4222:(
4219:a
4216:(
4204:=
4200:)
4196:)
4193:x
4187:y
4184:(
4176:a
4170:(
4155:a
4150:=
4147:)
4144:y
4141:(
4136:x
4132:K
4106:x
4102:K
4078:H
4055:.
4048:2
4044:L
4035:f
4023:a
4017:=
4009:d
4003:2
3998:|
3993:)
3987:(
3984:F
3980:|
3950:2
3945:a
3942:2
3936:=
3928:d
3922:2
3917:|
3912:)
3906:(
3903:F
3899:|
3893:a
3888:a
3877:a
3874:2
3863:2
3859:1
3850:|
3846:)
3843:x
3840:(
3837:f
3833:|
3809:x
3778:.
3772:d
3763:x
3760:i
3756:e
3752:)
3746:(
3743:F
3738:a
3733:a
3716:2
3712:1
3707:=
3704:)
3701:x
3698:(
3695:f
3656:R
3635:H
3610:R
3589:H
3569:)
3565:R
3561:(
3556:2
3552:L
3528:.
3525:x
3522:d
3512:)
3509:x
3506:(
3503:g
3494:)
3491:x
3488:(
3485:f
3464:=
3457:2
3453:L
3444:g
3441:,
3438:f
3412:f
3388:t
3385:d
3379:t
3373:i
3366:e
3362:)
3359:t
3356:(
3353:f
3332:=
3329:)
3323:(
3320:F
3300:)
3296:R
3292:(
3287:2
3283:L
3259:}
3256:]
3253:a
3250:,
3247:a
3241:[
3235:)
3232:F
3229:(
3217:)
3213:R
3209:(
3204:2
3200:L
3193:f
3190:{
3187:=
3184:H
3155:a
3149:0
3126:H
3093:X
3087:}
3084:x
3081:{
3055:x
3051:K
3030:X
3024:x
3004:X
2960:X
2940:)
2934:,
2931:X
2928:(
2923:2
2919:L
2890:X
2870:K
2850:.
2846:R
2837:n
2833:c
2829:,
2823:,
2818:1
2814:c
2805:,
2802:X
2794:n
2790:x
2786:,
2780:,
2775:1
2771:x
2767:,
2763:N
2756:n
2733:0
2725:2
2720:H
2707:i
2703:x
2698:K
2692:i
2688:c
2682:n
2677:1
2674:=
2671:i
2657:=
2652:H
2639:j
2635:x
2630:K
2624:j
2620:c
2614:n
2609:1
2606:=
2603:j
2595:,
2588:i
2584:x
2579:K
2573:i
2569:c
2563:n
2558:1
2555:=
2552:i
2538:=
2533:H
2520:j
2516:x
2511:K
2505:j
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2495:n
2490:1
2487:=
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2476:,
2469:i
2465:x
2460:K
2448:i
2444:c
2438:n
2433:1
2430:=
2427:i
2419:=
2416:)
2411:j
2407:x
2403:,
2398:i
2394:x
2390:(
2387:K
2382:j
2378:c
2372:i
2368:c
2362:n
2357:1
2354:=
2351:j
2348:,
2345:i
2312:C
2290:R
2283:X
2277:X
2274::
2271:K
2248:.
2243:H
2233:y
2229:K
2222:,
2217:x
2213:K
2206:=
2203:)
2200:y
2197:,
2194:x
2191:(
2188:K
2164:C
2142:R
2135:X
2129:X
2126::
2123:K
2103:H
2078:y
2074:L
2053:H
2033:H
2025:y
2021:K
1997:,
1992:H
1982:y
1978:K
1971:,
1966:x
1962:K
1955:=
1952:)
1947:x
1943:K
1939:(
1934:y
1930:L
1926:=
1923:)
1920:y
1917:(
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1908:K
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1836:C
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1751:)
1749:2
1747:(
1730:.
1727:H
1721:f
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1702:x
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1691:,
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1664:L
1660:=
1657:)
1654:x
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1648:f
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1461:f
1440:H
1429:1
1408:=
1403:x
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1393:x
1368:X
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1334:x
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1312:1
1310:(
1292:.
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1170:0
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612:L
581:n
561:)
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555:(
547:n
544:2
514:|
510:)
507:x
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501:g
495:)
492:x
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262:f
259:=
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243:,
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215:H
209:f
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181:x
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156:X
150:x
129:C
107:R
86:X
66:H
46:(
20:)
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