84:
74:
53:
22:
838:
Especially, the contribution of
Bernstein is much more than what you wrote. Bernstein was the first to suggest that approximability is related to properties such as modulus of continuity, and he proved Bernstein's theorem (the reverse of Jackson's thm.), chronologically even before Jackson. Many
834:
I'am certainly not an expert on the subject, almost all I know is from
Akhiezer's book (first edition in 1947, 2-nd - in 1965). If you look through Chapter 5 (on harmonic approximation), you see that already then it was a beautiful and well-developed theory (including L_\infty, L_p and other
831:
regarding the L_2 - L_\infty discussion: it seems I understand what you mean, I agree. Just, please write the precise formulation in the article (if you think it's important enough), with all the definitions.
835:
theorems), to which many people contributed. Therefore I think
Stechkin deserves a part of the name no more than Kornneichuk, or Bernstein, or even de la Vallee Poussin, or Nagy (or btw Akhiezer himself).
703:
140:
583:
839:
other ideas in the "constructive theory of functions" originate in his works. Therefore the collective name "Jackson--Bernstein theorems" that is used sometimes is not so bad, to my taste.
745:
Actually such "mixed norms" Jackson's inequalites usually are not of much interest since functions should be approximated in their spaces where its modulus of continuity are defined
732:
form implies the L_2 form (up to some constant). If you wish to write a more precise L_2 ineq, you should replace the modulus of cont. by the so-called L_2 modulus of continuity.
434:
292:
396:
730:
648:
258:
187:
509:
314:) found by different persons (N. I. Chernych and N. P. Korneychuk respectively). So one cannot say here that one norm is weaker than another. They are simply different.
810:
774:
621:
369:
342:
231:
460:
312:
872:
130:
867:
260:
are extremely different spaces, and they have different
Jackson inequalities proved by absolutely different methods with different sharp constants (
106:
162:
Hi, thanks for the additions to
Jackson's thm. Could you please define E_n(f)? Also, why did you focus on the L_2 norm - it is also true in
97:
58:
653:
33:
522:
21:
705:. Therefore, if on the right-hand side of Jackson's inequality you write the usual modulus of cont., the
39:
83:
401:
263:
374:
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
89:
850:
844:
818:
733:
708:
626:
586:
463:
236:
194:
165:
73:
52:
478:
788:
752:
599:
347:
320:
209:
849:
I have removed the "personal part" about
Stechkin, after deciding that it's irrelevant.
439:
297:
861:
853:
821:
736:
589:
466:
197:
817:
Have you read history which I posted to russian page. Or should I translate it?
102:
749:- for example usual (uniform) modulus of continuity doesn't have much sense in
79:
650:
ineqs. are incomparable), but on a finite interval (e.g. for periodic f-ns),
371:
but AFAIK sharp constants are still not found. Stechkin proved constant
827:
Thanks, your survey is very interesting. Here are a few comments:
511:
usualy means value of best approximation by polynomials of degree
15:
462:. For a brief history see Russian page on the inequality.
698:{\displaystyle \|\cdot \|_{2}\leq C\|\cdot \|_{\infty }}
436:), it is very good estimation but not sharp at least at
791:
785:
And
Stechkin in mentioned 3/2-inequality surely used
755:
711:
656:
629:
602:
525:
481:
442:
404:
377:
350:
323:
300:
266:
239:
212:
189:
norm; for L_2 a weaker condition is sufficient (some
168:
101:, a collaborative effort to improve the coverage of
804:
768:
724:
697:
642:
615:
577:
503:
454:
428:
390:
363:
336:
306:
286:
252:
225:
181:
526:
515:(or, more general, by linear space of dimension
8:
686:
679:
664:
657:
578:{\displaystyle \inf\{\|f-p\|:p\in P_{n-1}\}}
572:
544:
532:
529:
19:
47:
796:
790:
760:
754:
716:
710:
689:
667:
655:
634:
628:
607:
601:
560:
524:
486:
480:
441:
403:
378:
376:
355:
349:
328:
322:
299:
277:
272:
267:
265:
244:
238:
217:
211:
173:
167:
49:
7:
95:This article is within the scope of
38:It is of interest to the following
717:
690:
635:
420:
245:
174:
14:
873:Low-priority mathematics articles
585:. This is a common abbreviation.
115:Knowledge:WikiProject Mathematics
868:Start-Class mathematics articles
429:{\displaystyle p\in [1,\infty )}
118:Template:WikiProject Mathematics
82:
72:
51:
20:
287:{\displaystyle {1}/{\sqrt {2}}}
135:This article has been rated as
596:You are right of course (that
498:
492:
423:
411:
391:{\displaystyle {\frac {3}{2}}}
1:
854:00:58, 10 February 2007 (UTC)
109:and see a list of open tasks.
822:17:14, 8 February 2007 (UTC)
780:is a really wrong word here.
737:15:54, 8 February 2007 (UTC)
590:11:57, 8 February 2007 (UTC)
467:11:57, 8 February 2007 (UTC)
198:12:37, 7 February 2007 (UTC)
725:{\displaystyle L_{\infty }}
643:{\displaystyle L_{\infty }}
253:{\displaystyle L_{\infty }}
182:{\displaystyle L_{\infty }}
889:
193:modulus of cont.) Thanks,
134:
67:
46:
504:{\displaystyle E_{n}(f)}
141:project's priority scale
98:WikiProject Mathematics
806:
770:
726:
699:
644:
617:
579:
505:
456:
430:
392:
365:
338:
308:
288:
254:
227:
183:
28:This article is rated
807:
805:{\displaystyle L_{p}}
771:
769:{\displaystyle L_{2}}
727:
700:
645:
618:
616:{\displaystyle L_{2}}
580:
506:
457:
431:
393:
366:
364:{\displaystyle L_{2}}
344:spaces are closer to
339:
337:{\displaystyle L_{p}}
309:
289:
255:
228:
226:{\displaystyle L_{2}}
184:
789:
753:
709:
654:
627:
600:
523:
479:
440:
402:
375:
348:
321:
298:
264:
237:
210:
166:
121:mathematics articles
455:{\displaystyle p=2}
802:
766:
722:
695:
640:
613:
575:
501:
452:
426:
388:
361:
334:
304:
284:
250:
223:
179:
158:L_2 vers. L_\infty
90:Mathematics portal
34:content assessment
398:for all of them (
386:
307:{\displaystyle 1}
282:
155:
154:
151:
150:
147:
146:
880:
811:
809:
808:
803:
801:
800:
775:
773:
772:
767:
765:
764:
731:
729:
728:
723:
721:
720:
704:
702:
701:
696:
694:
693:
672:
671:
649:
647:
646:
641:
639:
638:
622:
620:
619:
614:
612:
611:
584:
582:
581:
576:
571:
570:
510:
508:
507:
502:
491:
490:
461:
459:
458:
453:
435:
433:
432:
427:
397:
395:
394:
389:
387:
379:
370:
368:
367:
362:
360:
359:
343:
341:
340:
335:
333:
332:
313:
311:
310:
305:
293:
291:
290:
285:
283:
278:
276:
271:
259:
257:
256:
251:
249:
248:
232:
230:
229:
224:
222:
221:
188:
186:
185:
180:
178:
177:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
888:
887:
883:
882:
881:
879:
878:
877:
858:
857:
792:
787:
786:
756:
751:
750:
712:
707:
706:
685:
663:
652:
651:
630:
625:
624:
603:
598:
597:
556:
521:
520:
482:
477:
476:
438:
437:
400:
399:
373:
372:
351:
346:
345:
324:
319:
318:
296:
295:
262:
261:
240:
235:
234:
213:
208:
207:
169:
164:
163:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
886:
884:
876:
875:
870:
860:
859:
841:
840:
836:
832:
825:
824:
814:
813:
799:
795:
782:
781:
763:
759:
741:
719:
715:
692:
688:
684:
681:
678:
675:
670:
666:
662:
659:
637:
633:
610:
606:
595:
593:
592:
574:
569:
566:
563:
559:
555:
552:
549:
546:
543:
540:
537:
534:
531:
528:
500:
497:
494:
489:
485:
471:
451:
448:
445:
425:
422:
419:
416:
413:
410:
407:
385:
382:
358:
354:
331:
327:
316:
315:
303:
281:
275:
270:
247:
243:
220:
216:
206:In this topic
202:
176:
172:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
885:
874:
871:
869:
866:
865:
863:
856:
855:
852:
847:
846:
837:
833:
830:
829:
828:
823:
820:
816:
815:
797:
793:
784:
783:
779:
761:
757:
748:
744:
743:
742:
739:
738:
735:
713:
682:
676:
673:
668:
660:
631:
608:
604:
591:
588:
567:
564:
561:
557:
553:
550:
547:
541:
538:
535:
518:
514:
495:
487:
483:
474:
473:
472:
469:
468:
465:
449:
446:
443:
417:
414:
408:
405:
383:
380:
356:
352:
329:
325:
301:
279:
273:
268:
241:
218:
214:
205:
204:
203:
200:
199:
196:
192:
170:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
848:
842:
826:
777:
746:
740:
594:
516:
512:
470:
317:
201:
190:
161:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
862:Categories
812:-modulus.
747:for sure
519:), i.e.
191:average
139:on the
843:Best,
778:weaker
36:scale.
851:Sasha
845:Sasha
819:Mir76
776:. So
734:Sasha
587:Mir76
475:P.S.
464:Mir76
195:Sasha
623:and
294:and
233:and
527:inf
513:n-1
131:Low
864::
718:∞
691:∞
687:‖
683:⋅
680:‖
674:≤
665:‖
661:⋅
658:‖
636:∞
565:−
554:∈
545:‖
539:−
533:‖
421:∞
409:∈
246:∞
175:∞
798:p
794:L
762:2
758:L
714:L
677:C
669:2
632:L
609:2
605:L
573:}
568:1
562:n
558:P
551:p
548::
542:p
536:f
530:{
517:n
499:)
496:f
493:(
488:n
484:E
450:2
447:=
444:p
424:)
418:,
415:1
412:[
406:p
384:2
381:3
357:2
353:L
330:p
326:L
302:1
280:2
274:/
269:1
242:L
219:2
215:L
171:L
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
