205:
435:
811:
732:
606:
330:
878:
89:
355:
747:
656:
530:
940:
261:
826:
896:
741:
diverges. For the Dini test, the statement of precision is slightly longer: it says that for any function Ω such that
930:
966:
200:{\displaystyle \left.\right.\omega _{f}(\delta ;t)=\max _{|\varepsilon |\leq \delta }|f(t)-f(t+\varepsilon )|}
961:
636:
Both tests are the best of their kind. For the Dini-Lipschitz test, it is possible to construct a function
430:{\displaystyle \int _{0}^{\pi }{\frac {1}{\delta }}\omega _{f}(\delta ;t)\,\mathrm {d} \delta <\infty .}
44:
252:
926:
625:
806:{\displaystyle \int _{0}^{\pi }{\frac {1}{\delta }}\Omega (\delta )\,\mathrm {d} \delta =\infty }
936:
52:
901:
906:
641:
40:
955:
727:{\displaystyle \omega _{f}(\delta )=O\left(\log {\frac {1}{\delta }}\right)^{-1}.}
601:{\displaystyle \omega _{f}(\delta )=o\left(\log {\frac {1}{\delta }}\right)^{-1}.}
48:
28:
17:
932:
Introduction to
Partial Differential Equations and Hilbert Space Methods
325:{\displaystyle \omega _{f}(\delta )=\max _{t}\omega _{f}(\delta ;t)}
39:
are highly precise tests that can be used to prove that the
97:
94:
873:{\displaystyle \omega _{f}(\delta ;0)<\Omega (\delta )}
640:
with its modulus of continuity satisfying the test with
47:
converges at a given point. These tests are named after
335:
With these definitions we may state the main results:
829:
750:
659:
533:
358:
264:
92:
872:
805:
726:
600:
429:
324:
199:
288:
129:
8:
935:, Courier Dover Publications, p. 121,
834:
828:
789:
788:
766:
760:
755:
749:
712:
697:
664:
658:
624:In particular, any function that obeys a
586:
571:
538:
532:
410:
409:
388:
374:
368:
363:
357:
301:
291:
269:
263:
192:
154:
141:
133:
132:
104:
91:
918:
7:
464:For example, the theorem holds with
75:be a positive number. We define the
628:satisfies the Dini–Lipschitz test.
214:to be a periodic function, e.g. if
858:
800:
790:
776:
519:Theorem (the Dini–Lipschitz test):
421:
411:
25:
867:
861:
852:
840:
785:
779:
676:
670:
550:
544:
406:
394:
319:
307:
281:
275:
193:
189:
177:
168:
162:
155:
142:
134:
122:
110:
1:
897:Convergence of Fourier series
210:Notice that we consider here
249:global modulus of continuity
612:Then the Fourier series of
441:Then the Fourier series of
225:is negative then we define
77:local modulus of continuity
983:
883:and the Fourier series of
737:and the Fourier series of
816:there exists a function
616:converges uniformly to
492:but does not hold with
67:be a function on , let
874:
807:
728:
602:
431:
340:Theorem (Dini's test):
326:
201:
71:be some point and let
875:
808:
729:
603:
432:
346:satisfies at a point
327:
253:modulus of continuity
202:
827:
748:
657:
531:
356:
262:
90:
37:Dini–Lipschitz tests
765:
373:
927:Gustafson, Karl E.
870:
803:
751:
724:
598:
521:Assume a function
427:
359:
342:Assume a function
322:
296:
197:
153:
967:Convergence tests
942:978-0-486-61271-3
774:
705:
579:
382:
287:
128:
96:
16:(Redirected from
974:
946:
945:
923:
886:
879:
877:
876:
871:
839:
838:
819:
812:
810:
809:
804:
793:
775:
767:
764:
759:
740:
733:
731:
730:
725:
720:
719:
711:
707:
706:
698:
669:
668:
648:
644:
639:
626:Hölder condition
619:
615:
607:
605:
604:
599:
594:
593:
585:
581:
580:
572:
543:
542:
524:
513:
511:
509:
508:
503:
500:
491:
489:
487:
486:
481:
478:
459:
448:
444:
436:
434:
433:
428:
414:
393:
392:
383:
375:
372:
367:
349:
345:
331:
329:
328:
323:
306:
305:
295:
274:
273:
255:) is defined by
243:
224:
220:
213:
206:
204:
203:
198:
196:
158:
152:
145:
137:
109:
108:
99:
82:
74:
70:
66:
53:Rudolf Lipschitz
21:
18:Dini's test
982:
981:
977:
976:
975:
973:
972:
971:
952:
951:
950:
949:
943:
925:
924:
920:
915:
902:Dini continuity
893:
887:diverges at 0.
884:
830:
825:
824:
817:
746:
745:
738:
690:
686:
685:
660:
655:
654:
646:
642:
637:
634:
617:
613:
564:
560:
559:
534:
529:
528:
522:
504:
501:
498:
497:
495:
493:
482:
479:
476:
475:
473:
470:
465:
450:
446:
442:
384:
354:
353:
347:
343:
297:
265:
260:
259:
251:(or simply the
226:
222:
215:
211:
100:
93:
88:
87:
80:
72:
68:
64:
61:
23:
22:
15:
12:
11:
5:
980:
978:
970:
969:
964:
962:Fourier series
954:
953:
948:
947:
941:
917:
916:
914:
911:
910:
909:
907:Dini criterion
904:
899:
892:
889:
881:
880:
869:
866:
863:
860:
857:
854:
851:
848:
845:
842:
837:
833:
814:
813:
802:
799:
796:
792:
787:
784:
781:
778:
773:
770:
763:
758:
754:
735:
734:
723:
718:
715:
710:
704:
701:
696:
693:
689:
684:
681:
678:
675:
672:
667:
663:
633:
630:
622:
621:
610:
609:
608:
597:
592:
589:
584:
578:
575:
570:
567:
563:
558:
555:
552:
549:
546:
541:
537:
468:
462:
461:
439:
438:
437:
426:
423:
420:
417:
413:
408:
405:
402:
399:
396:
391:
387:
381:
378:
371:
366:
362:
333:
332:
321:
318:
315:
312:
309:
304:
300:
294:
290:
286:
283:
280:
277:
272:
268:
208:
207:
195:
191:
188:
185:
182:
179:
176:
173:
170:
167:
164:
161:
157:
151:
148:
144:
140:
136:
131:
127:
124:
121:
118:
115:
112:
107:
103:
98:
95:
60:
57:
41:Fourier series
24:
14:
13:
10:
9:
6:
4:
3:
2:
979:
968:
965:
963:
960:
959:
957:
944:
938:
934:
933:
928:
922:
919:
912:
908:
905:
903:
900:
898:
895:
894:
890:
888:
864:
855:
849:
846:
843:
835:
831:
823:
822:
821:
797:
794:
782:
771:
768:
761:
756:
752:
744:
743:
742:
721:
716:
713:
708:
702:
699:
694:
691:
687:
682:
679:
673:
665:
661:
653:
652:
651:
649:
631:
629:
627:
611:
595:
590:
587:
582:
576:
573:
568:
565:
561:
556:
553:
547:
539:
535:
527:
526:
520:
517:
516:
515:
507:
485:
471:
457:
453:
445:converges at
440:
424:
418:
415:
403:
400:
397:
389:
385:
379:
376:
369:
364:
360:
352:
351:
341:
338:
337:
336:
316:
313:
310:
302:
298:
292:
284:
278:
270:
266:
258:
257:
256:
254:
250:
245:
241:
237:
233:
229:
218:
186:
183:
180:
174:
171:
165:
159:
149:
146:
138:
125:
119:
116:
113:
105:
101:
86:
85:
84:
79:at the point
78:
58:
56:
54:
50:
46:
42:
38:
34:
30:
19:
931:
921:
882:
815:
736:
635:
623:
518:
505:
483:
466:
463:
455:
451:
339:
334:
248:
246:
239:
235:
231:
227:
216:
209:
76:
62:
36:
32:
26:
820:such that
645:instead of
49:Ulisse Dini
29:mathematics
956:Categories
913:References
525:satisfies
59:Definition
865:δ
859:Ω
844:δ
832:ω
801:∞
795:δ
783:δ
777:Ω
772:δ
762:π
753:∫
714:−
703:δ
695:
674:δ
662:ω
632:Precision
588:−
577:δ
569:
548:δ
536:ω
422:∞
416:δ
398:δ
386:ω
380:δ
370:π
361:∫
311:δ
299:ω
279:δ
267:ω
187:ε
172:−
150:δ
147:≤
139:ε
114:δ
102:ω
929:(1999),
891:See also
45:function
650:, i.e.
510:
496:
488:
474:
939:
472:= log(
238:(2Ď€ +
31:, the
350:that
43:of a
937:ISBN
856:<
494:log(
419:<
247:The
234:) =
221:and
63:Let
51:and
35:and
33:Dini
692:log
566:log
449:to
289:max
244:.
219:= 0
130:max
83:by
27:In
958::
514:.
55:.
885:f
868:)
862:(
853:)
850:0
847:;
841:(
836:f
818:f
798:=
791:d
786:)
780:(
769:1
757:0
739:f
722:.
717:1
709:)
700:1
688:(
683:O
680:=
677:)
671:(
666:f
647:o
643:O
638:f
620:.
618:f
614:f
596:.
591:1
583:)
574:1
562:(
557:o
554:=
551:)
545:(
540:f
523:f
512:)
506:δ
502:/
499:1
490:)
484:δ
480:/
477:1
469:f
467:ω
460:.
458:)
456:t
454:(
452:f
447:t
443:f
425:.
412:d
407:)
404:t
401:;
395:(
390:f
377:1
365:0
348:t
344:f
320:)
317:t
314:;
308:(
303:f
293:t
285:=
282:)
276:(
271:f
242:)
240:ε
236:f
232:ε
230:(
228:f
223:ε
217:t
212:f
194:|
190:)
184:+
181:t
178:(
175:f
169:)
166:t
163:(
160:f
156:|
143:|
135:|
126:=
123:)
120:t
117:;
111:(
106:f
81:t
73:δ
69:t
65:f
20:)
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