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AMERICAN MATHEMATICA L SOCIET Y TRANSLATIONS

Ser ies 2

V o l u m e 8 6

FIFTEEN PAPER S O N REA L AN D COMPLE X FUNCTIONS , SERIES ,

DIFFERENTIAL AN D INTEGRA L EQUATION S

N. I. Ahiezer I. S. Arson Dong Guang-chung

(Tung Kuang-ch'ang) V. F. Gaposkin E. K. Godunova

by

L. D. Kudrjavcev V. I. Levin V. I. Mel'nik S. M. Nikol'skii M. K. Potapov

E. M. Saak G. H. Sindalovskif S. A. Teljakovskif R. M. Trigub G. M. Vaihikko A. G. Vituskin

Published by the

AMERICAN MATHEMATICA L SOCIET Y Providence, Rhode Island

1970

http://dx.doi.org/10.1090/trans2/086

Copyright © 197 0 by the American Mathematical Society

Standard Book Number 821-81786-0

Library of Congress Catalog Number A51-5559

Printed in the United States of America All Rights Reserved

No portion of this book may be reproduced without the written permission of the publisher

TABLE OF CONTENTS Page

Ahiezer, N . I . Extrema l propertie s o f entir e function s o f exponentia l type. [Axwe3ep , H . M . 3KCTpeManbHbi e CBoMcTB a i ejibix <|>yHKu;MM SKCnoHeHl^waJIbHOro TMna. ] Teor . Funkci i Funkcional . Anal , i Pri lozen. 1 (1965), 111-13 5 1

Trigub, R . M . Constructiv e characterization s o f som e functio n clas -s e s . [TpwryS , P . M . KoHCTpyKTMBHbi e xapaKTepwcTWK M aeKO -TOpwx KJiaccO B (|)yHKl MM. ] Izv . Akad. Nau k SSS R Ser . Mat. 29 (1965), 615-63 0 3 1

Arson, I . S . O n the decreas e o f harmoni c function s o f thre e variables . [ApmoH, 1/1 . C. 0 6 ySwBaHH H rapMOHtmecKn x <J>yHKijHf t Tpe x nepeMeHHblx.] Izv . Akad. Nau k SSS R Ser . Mat. 29 (1965), 1 2 8 3-1294 5 1

Sindalovskii, G . H. Differentiabilit y relativ e t o congruen t s e t s . [CwHAaJioBCKWM, P. X . /tM^epeHijifpyeMocT b OTHOGWTejibH o KOHrpySHTHblx MHOatecTB. ] Izv . Akad. Nau k SSS R Ser . Mat. 29 (1965), 11-4 0 6 7

Vituskin, A . G. Representabili t y o f function s b y superpositio n o f functions o f a smalle r numbe r o f var iables . [BMTytUKMH , A. F . 0 B03M0>KH0CT M npeACTaBJieHM H (J)yHKLI,wl l Cynepn03MU,M5IM M (J3yHK ~

UMM OT MeHbQier o MMCJi a nepeMeHHblx. ] Proc . Internat . Congr . Math . (Moscow, 1966 ) "Mir", Moscow , 1968 , pp . 322-32 8 10 1

Teljakovskii , S . A . Som e est imate s fo r trigonometri c ser ie s wit| i quasi-convex coefficients . [TeJlflKOBCKWM , C . A. HeKOTOpbi e OIjeHKM AJI H TpwrOHOMeTpM^eCKW X pflAO B C KBa3MBbinyKJIbIM M

K03(|)(|)MimeHTaMM.] Mat . Sb. 63(105) (1964) , 426-44 4 10 9

Levin, V . I. an d Godunova, E . K . Generalizatio n o f th e Carlso n ine -quality. [JleBMH , B . J/1 , H FoflyHOBa, E . K . 06o6meHn e Hepa -BecTBa KapjicoHa. ] Mat . Sb. 67(109) (1965) , 643-64 6 13 3

Mel nik , V. I . A Tauberian theore m o n larg e indice s fo r th e Bore l method. [MejibHHK , B . M. TaySepoB a TeopeM a o ,,6ojibiuw x noKa3aTejiHx" AJI H MeTOA a BopejiH. ] Mat . Sb . 68(110) (1965) , 17-25 13 7

i n

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Potapov, M . K . O n a questio n involvin g equivalenc e o f condition s for convergenc e o f Fourie r se r i e s . [llOTanOB , M . K . K BOnpOC y 0 6 9KBMBaJieHTH0CT M yCJIOBW M CXOAMMOCT M pHAO B $ y p b e . ] Mat .

Sb. 68(110 ) (1965) , 111-12 7 14 7

Gaposkin, V . F . O n serie s relativ e t o th e system \(£>(nx)\.

[ranoojKMH, B . $ . 0 pnaa x n o cwcTeM e \cf>(nx)\.] Mat . Sb . 68(110) (1965) , 328-35 3 16 7

Kudrjavcev, L . D . an d Niko l ski i , S . M . Imbeddin g theorem s an d their applicatio n t o solvin g ellipti c equations . [KyApflBljeB , JI. fl. M HttKOJIbCKHM , C . M . TeOpeMb I BJIO>KeHW H M MX npMJIO -

>KeHwe K peiueHWK ) ajuiwnTuqecKwx ypaBHeHMM. ] Izdat . Sibirsk . Otdel. Akad . Nau k SSSR , Novosibirsk , 1961 , pp. 87-10 9 19 9

Saak, E . M . O n th e regula r approximatio n o f solution s o f comple x elliptic equation s o f arbitrar y order . [CaaK , 3 . M . 0 peryJIHp -HOM annpoKCMMai w w peiueHW M KOMnJieKCHbi x sjuiwnTimecKM X ypaBHeHMM JiioSor o nopHAKa. ] Ukrai n Mat . Z . 1 8 (1966) , no . 6 , 26 -40 22 9

Vaniikko, G . M . O n th e rat e o f convergenc e o f certai n approximatio n methods o f Galerki n typ e i n a n eigenvalu e problem . [BailHMKKO , r . M . 0 6bICTpOT e CXOAMMOCT M HeKOTOpbl X npn6jM)KeHHbI X

MeTOAOB Tun a FajiepKWH a B npoGjieM e co6cTBeHHbi x 3HaqeHMM. ] Izv. Vyss . Ucebn . Zaved . Matematik a 1966 , no . 2 (51) , 37 -4 5 24 9

Dong Guang-chan g (Tun g Kuang-ch'ang) . Uniquenes s theore m fo r th e

Caplygin problem . I [ f * i : 3 E At f t &7? t& ft t — *fe Jg M

(I).] Act a Math . Sinic a 6 (1956) , no . 2 , 242-24 7 26 1

Dong Guang-chan g (Tun g Kuang-c h ang) . Uniquenes s theore m fo r th e

Caplygin problem . I I [ | t % 1 : It * * £ ~fS tlk f# * £ — « • 5 E i §

(II).] Act a Math . Sinic a 6 (1956) , no . 2 , 250-26 1 26 9

IV