610:
423:
454:
155:
310:
605:{\displaystyle \mu (-\infty ,\xi _{j}]\leq \rho _{m-1}(\xi _{1})+\cdots +\rho _{m-1}(\xi _{j})\leq \mu (-\infty ,\xi _{j+1}).}
102:
653:
27:
347:
199:
658:
47:
20:
51:
43:
8:
35:
31:
647:
629:
168: − 1 (and in particular the integral is defined and finite).
39:
418:{\displaystyle \rho _{m-1}(z)=1{\Big /}\sum _{k=0}^{m-1}|P_{k}(z)|^{2}}
634:
The
Classical Moment Problem and Some Related Questions in Analysis
457:
313:
105:
604:
417:
149:
645:
50:from above and from below in terms of its first
235:. It is not hard to see that the polynomials
46:. Informally, they provide sharp bounds on a
283:, and therefore are determined uniquely by
628:
150:{\displaystyle \int x^{k}d\mu (x)=m_{k}}
646:
25:Chebyshev–Markov–Stieltjes
34:that were formulated in the 1880s by
13:
574:
467:
14:
670:
30:are inequalities related to the
622:
596:
568:
559:
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518:
505:
483:
461:
405:
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380:
336:
330:
131:
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57:
1:
615:
38:and proved independently by
7:
10:
675:
84:, consider the collection
42:and (somewhat later) by
275:are the same for every
606:
419:
378:
200:orthogonal polynomials
151:
607:
420:
352:
152:
21:mathematical analysis
654:Theorems in analysis
636:. Oliver & Boyd.
455:
311:
103:
44:Thomas Jan Stieltjes
16:Mathematical theorem
602:
415:
147:
32:problem of moments
36:Pafnuty Chebyshev
666:
638:
637:
626:
611:
609:
608:
603:
595:
594:
558:
557:
545:
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517:
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482:
481:
424:
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421:
416:
414:
413:
408:
393:
392:
383:
377:
366:
351:
350:
329:
328:
259:and the numbers
226:be the zeros of
202:with respect to
156:
154:
153:
148:
146:
145:
118:
117:
674:
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669:
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627:
623:
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584:
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477:
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409:
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388:
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367:
356:
346:
345:
318:
314:
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289:
274:
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258:
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193:
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141:
137:
113:
109:
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100:
79:
68:
60:
17:
12:
11:
5:
672:
662:
661:
656:
640:
639:
630:Akhiezer, N.I.
620:
619:
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614:
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601:
598:
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590:
587:
583:
579:
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376:
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365:
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355:
349:
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327:
324:
321:
317:
294:
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270:
263:
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189:
182:
175:
158:
157:
144:
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136:
133:
130:
127:
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121:
116:
112:
108:
73:
66:
59:
56:
15:
9:
6:
4:
3:
2:
671:
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635:
631:
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621:
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581:
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562:
554:
550:
541:
538:
535:
531:
527:
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513:
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497:
494:
490:
486:
478:
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464:
458:
451:
450:
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447:
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439:
435:
431:
410:
397:
389:
385:
374:
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368:
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353:
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333:
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293:
286:
282:
278:
273:
269:
262:
256:
252:
245:
238:
233:
229:
224:
220:
213:
209:
205:
201:
197:
194:be the first
192:
188:
181:
174:
169:
167:
163:
142:
138:
134:
128:
122:
119:
114:
110:
106:
99:
98:
97:
95:
91:
87:
83:
77:
72:
65:
55:
53:
49:
45:
41:
40:Andrey Markov
37:
33:
29:
26:
22:
659:Inequalities
633:
624:
445:
441:
437:
433:
429:
428:
303:
296:
291:
284:
280:
276:
271:
267:
260:
254:
250:
243:
236:
231:
227:
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218:
211:
207:
203:
195:
190:
186:
179:
172:
170:
165:
161:
159:
93:
89:
88:of measures
85:
81:
75:
70:
63:
61:
28:inequalities
24:
18:
164:= 0,1,...,2
58:Formulation
648:Categories
616:References
440:, and any
436:= 1,2,...,
210:, and let
96:such that
582:ξ
575:∞
572:−
566:μ
563:≤
551:ξ
539:−
532:ρ
525:⋯
510:ξ
498:−
491:ρ
487:≤
475:ξ
468:∞
465:−
459:μ
372:−
354:∑
323:−
316:ρ
123:μ
107:∫
632:(1965).
430:Theorem
304:Denote
52:moments
48:measure
442:μ
277:μ
268:ξ
261:ξ
249:, ...,
219:ξ
212:ξ
204:μ
185:, ...,
90:μ
62:Given
23:, the
290:,...,
69:,...,
432:For
266:,...
217:,...
198:+ 1
171:Let
160:for
92:on
19:In
650::
448:,
444:∈
301:.
299:-1
279:∈
257:-1
206:∈
80:∈
78:-1
54:.
600:.
597:)
592:1
589:+
586:j
578:,
569:(
560:)
555:j
547:(
542:1
536:m
528:+
522:+
519:)
514:1
506:(
501:1
495:m
484:]
479:j
471:,
462:(
446:C
438:m
434:j
425:.
411:2
406:|
401:)
398:z
395:(
390:k
386:P
381:|
375:1
369:m
364:0
361:=
358:k
348:/
343:1
340:=
337:)
334:z
331:(
326:1
320:m
297:m
295:2
292:m
288:0
285:m
281:C
272:m
264:1
255:m
251:P
247:1
244:P
242:,
240:0
237:P
232:m
228:P
223:m
215:1
208:C
196:m
191:m
187:P
183:1
180:P
178:,
176:0
173:P
166:m
162:k
143:k
139:m
135:=
132:)
129:x
126:(
120:d
115:k
111:x
94:R
86:C
82:R
76:m
74:2
71:m
67:0
64:m
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