558:
971:
746:
317:
637:
423:
836:
166:
458:
882:
863:
784:
660:
222:
573:
1020:
380:
1058:
789:
865:
inner product on this space is manifestly invariant under biholomorphisms of D, the
Bergman kernel and the associated
41:
92:
207:
in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function
1041:
1031:
1012:
441:
429:
553:{\displaystyle \operatorname {ev} _{z}f=\int _{D}f(\zeta ){\overline {\eta _{z}(\zeta )}}\,d\mu (\zeta ).}
1036:
21:
204:
841:
762:
48:
8:
356:
33:
1016:
45:
966:{\displaystyle K(z,\zeta )={\frac {1}{\pi }}{\frac {1}{(1-z{\bar {\zeta }})^{2}}}.}
869:
are therefore automatically invariant under the automorphism group of the domain.
991:
981:
866:
29:
444:, this functional can be represented as the inner product with an element of
1052:
1004:
986:
37:
336:
17:
192:
873:
63:
741:{\displaystyle f(z)=\int _{D}K(z,\zeta )f(\zeta )\,d\mu (\zeta ).}
82:) denote the subspace consisting of holomorphic functions in L(
347:. Thus convergence of a sequence of holomorphic functions in
312:{\displaystyle \sup _{z\in K}|f(z)|\leq C_{K}\|f\|_{L^{2}(D)}}
632:{\displaystyle K(z,\zeta )={\overline {\eta _{z}(\zeta )}}.}
70:
be the
Hilbert space of square integrable functions on
786:
holomorphic (n,0)-forms on D, via multiplication by
885:
844:
792:
765:
663:
576:
461:
418:{\displaystyle \operatorname {ev} _{z}:f\mapsto f(z)}
383:
225:
95:
965:
857:
830:
778:
740:
631:
552:
417:
311:
160:
831:{\displaystyle dz^{1}\wedge \cdots \wedge dz^{n}}
359:, and so the limit function is also holomorphic.
1050:
227:
751:One key observation about this picture is that
654:and antiholomorphic in ζ, and satisfies
1009:Function Theory of Several Complex Variables
284:
277:
161:{\displaystyle L^{2,h}(D)=L^{2}(D)\cap H(D)}
179:) is the space of holomorphic functions in
719:
531:
759:) may be identified with the space of
1051:
1029:
1003:
216:
13:
14:
1070:
948:
941:
923:
901:
889:
732:
726:
716:
710:
704:
692:
673:
667:
617:
611:
592:
580:
544:
538:
522:
516:
500:
494:
412:
406:
400:
304:
298:
260:
256:
250:
243:
191:) is a Hilbert space: it is a
155:
149:
140:
134:
118:
112:
1:
1013:American Mathematical Society
997:
621:
526:
442:Riesz representation theorem
430:continuous linear functional
7:
1037:Encyclopedia of Mathematics
975:
872:The Bergman kernel for the
650:,ζ) is holomorphic in
363:
325:
10:
1075:
1059:Several complex variables
1032:"Bergman kernel function"
22:several complex variables
452:), which is to say that
362:Another consequence of (
1030:Chirka, E.M. (2001) ,
967:
859:
832:
780:
742:
633:
554:
419:
313:
162:
968:
860:
858:{\displaystyle L^{2}}
833:
781:
779:{\displaystyle L^{2}}
743:
634:
555:
420:
314:
163:
49:holomorphic functions
1011:, Providence, R.I.:
883:
842:
790:
763:
661:
574:
459:
381:
366:) is that, for each
223:
93:
563:The Bergman kernel
370: ∈
357:compact convergence
195:linear subspace of
963:
855:
828:
776:
738:
629:
550:
415:
309:
241:
158:
34:reproducing kernel
1022:978-0-8218-2724-6
1005:Krantz, Steven G.
958:
944:
915:
624:
529:
374:, the evaluation
333:
332:
226:
203:), and therefore
46:square integrable
1066:
1044:
1025:
972:
970:
969:
964:
959:
957:
956:
955:
946:
945:
937:
918:
916:
908:
879:is the function
864:
862:
861:
856:
854:
853:
837:
835:
834:
829:
827:
826:
805:
804:
785:
783:
782:
777:
775:
774:
747:
745:
744:
739:
688:
687:
638:
636:
635:
630:
625:
620:
610:
609:
599:
559:
557:
556:
551:
530:
525:
515:
514:
504:
490:
489:
471:
470:
424:
422:
421:
416:
393:
392:
365:
327:
318:
316:
315:
310:
308:
307:
297:
296:
276:
275:
263:
246:
240:
217:
167:
165:
164:
159:
133:
132:
111:
110:
1074:
1073:
1069:
1068:
1067:
1065:
1064:
1063:
1049:
1048:
1023:
1000:
978:
951:
947:
936:
935:
922:
917:
907:
884:
881:
880:
849:
845:
843:
840:
839:
822:
818:
800:
796:
791:
788:
787:
770:
766:
764:
761:
760:
683:
679:
662:
659:
658:
605:
601:
600:
598:
575:
572:
571:
510:
506:
505:
503:
485:
481:
466:
462:
460:
457:
456:
388:
384:
382:
379:
378:
355:) implies also
292:
288:
287:
283:
271:
267:
259:
242:
230:
224:
221:
220:
128:
124:
100:
96:
94:
91:
90:
62:In detail, let
12:
11:
5:
1072:
1062:
1061:
1047:
1046:
1027:
1021:
999:
996:
995:
994:
989:
984:
982:Bergman metric
977:
974:
962:
954:
950:
943:
940:
934:
931:
928:
925:
921:
914:
911:
906:
903:
900:
897:
894:
891:
888:
867:Bergman metric
852:
848:
825:
821:
817:
814:
811:
808:
803:
799:
795:
773:
769:
749:
748:
737:
734:
731:
728:
725:
722:
718:
715:
712:
709:
706:
703:
700:
697:
694:
691:
686:
682:
678:
675:
672:
669:
666:
640:
639:
628:
623:
619:
616:
613:
608:
604:
597:
594:
591:
588:
585:
582:
579:
567:is defined by
561:
560:
549:
546:
543:
540:
537:
534:
528:
524:
521:
518:
513:
509:
502:
499:
496:
493:
488:
484:
480:
477:
474:
469:
465:
426:
425:
414:
411:
408:
405:
402:
399:
396:
391:
387:
331:
330:
321:
319:
306:
303:
300:
295:
291:
286:
282:
279:
274:
270:
266:
262:
258:
255:
252:
249:
245:
239:
236:
233:
229:
169:
168:
157:
154:
151:
148:
145:
142:
139:
136:
131:
127:
123:
120:
117:
114:
109:
106:
103:
99:
30:Stefan Bergman
28:, named after
26:Bergman kernel
9:
6:
4:
3:
2:
1071:
1060:
1057:
1056:
1054:
1043:
1039:
1038:
1033:
1028:
1024:
1018:
1014:
1010:
1006:
1002:
1001:
993:
990:
988:
987:Bergman space
985:
983:
980:
979:
973:
960:
952:
938:
932:
929:
926:
919:
912:
909:
904:
898:
895:
892:
886:
878:
875:
870:
868:
850:
846:
823:
819:
815:
812:
809:
806:
801:
797:
793:
771:
767:
758:
754:
735:
729:
723:
720:
713:
707:
701:
698:
695:
689:
684:
680:
676:
670:
664:
657:
656:
655:
653:
649:
645:
626:
614:
606:
602:
595:
589:
586:
583:
577:
570:
569:
568:
566:
547:
541:
535:
532:
519:
511:
507:
497:
491:
486:
482:
478:
475:
472:
467:
463:
455:
454:
453:
451:
447:
443:
439:
435:
431:
409:
403:
397:
394:
389:
385:
377:
376:
375:
373:
369:
360:
358:
354:
350:
346:
342:
338:
329:
322:
320:
301:
293:
289:
280:
272:
268:
264:
253:
247:
237:
234:
231:
219:
218:
215:
214:
210:
206:
202:
198:
194:
190:
186:
182:
178:
174:
152:
146:
143:
137:
129:
125:
121:
115:
107:
104:
101:
97:
89:
88:
87:
85:
81:
77:
73:
69:
67:
60:
58:
54:
50:
47:
43:
39:
38:Hilbert space
35:
31:
27:
23:
19:
1035:
1008:
992:Szegő kernel
876:
871:
838:. Since the
756:
752:
750:
651:
647:
643:
641:
564:
562:
449:
445:
437:
433:
427:
371:
367:
361:
352:
348:
344:
340:
334:
323:
212:
208:
200:
196:
188:
184:
180:
176:
172:
170:
86:): that is,
83:
79:
75:
71:
65:
61:
56:
52:
51:on a domain
25:
18:mathematical
15:
642:The kernel
440:). By the
998:References
335:for every
74:, and let
1042:EMS Press
942:¯
939:ζ
930:−
913:π
899:ζ
874:unit disc
813:∧
810:⋯
807:∧
730:ζ
724:μ
714:ζ
702:ζ
681:∫
622:¯
615:ζ
603:η
590:ζ
542:ζ
536:μ
527:¯
520:ζ
508:η
498:ζ
483:∫
473:
401:↦
285:‖
278:‖
265:≤
235:∈
144:∩
44:) of all
32:, is the
20:study of
1053:Category
1007:(2002),
976:See also
205:complete
183:. Then
55:in
36:for the
339:subset
337:compact
16:In the
1019:
209:ƒ
193:closed
171:where
24:, the
428:is a
1017:ISBN
42:RKHS
432:on
343:of
228:sup
211:in
1055::
1040:,
1034:,
1015:,
464:ev
386:ev
64:L(
59:.
1045:.
1026:.
961:.
953:2
949:)
933:z
927:1
924:(
920:1
910:1
905:=
902:)
896:,
893:z
890:(
887:K
877:D
851:2
847:L
824:n
820:z
816:d
802:1
798:z
794:d
772:2
768:L
757:D
755:(
753:L
736:.
733:)
727:(
721:d
717:)
711:(
708:f
705:)
699:,
696:z
693:(
690:K
685:D
677:=
674:)
671:z
668:(
665:f
652:z
648:z
646:(
644:K
627:.
618:)
612:(
607:z
596:=
593:)
587:,
584:z
581:(
578:K
565:K
548:.
545:)
539:(
533:d
523:)
517:(
512:z
501:)
495:(
492:f
487:D
479:=
476:f
468:z
450:D
448:(
446:L
438:D
436:(
434:L
413:)
410:z
407:(
404:f
398:f
395::
390:z
372:D
368:z
364:1
353:D
351:(
349:L
345:D
341:K
328:)
326:1
324:(
305:)
302:D
299:(
294:2
290:L
281:f
273:K
269:C
261:|
257:)
254:z
251:(
248:f
244:|
238:K
232:z
213:D
201:D
199:(
197:L
189:D
187:(
185:L
181:D
177:D
175:(
173:H
156:)
153:D
150:(
147:H
141:)
138:D
135:(
130:2
126:L
122:=
119:)
116:D
113:(
108:h
105:,
102:2
98:L
84:D
80:D
78:(
76:L
72:D
68:)
66:D
57:C
53:D
40:(
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