Publication:
ON BARY-STECHKIN THEOREM

Дата
2019
Авторы
Rubinshtein, A. I.
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Научные группы
Организационные подразделения
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Аннотация
In the beginning of the past century, N.N. Luzin proved almost everywhere convergence of an improper integral representing the function (f) over bar conjugated to a 2 pi-periodic summable with a square function f (x). A few years later I.I. Privalov proved a similar fact for a summable function. V.I. Smirnov showed that if (f) over bar is summable, then its Fourier series is conjugate to the Fourier series for f (x). It is easy to see that if f (x) is an element of Lip alpha, 0 < alpha < 1, then (f) over bar (x)is an element of Lip alpha. The Hilbert transformation for f (x) differs from (f ) over bar (x) by a bounded function and has a simpler kernel. It is easy to show that the Hilbert transformation of f (x)is an element of Lip alpha, 0 < alpha < 1, also belongs to Lip alpha. In 1956 N.K. Bari and S.B. Stechkin found the necessary and sufficient condition on the modulus of continuity f (x) for the function (f) over bar (x) to have the same modulus of continuity. In 2016, the author introduced the concept of conjugate function as Hilbert transformation for functions defined on a dyadic group. In the present paper we show an analogue of the Bari-Stechkin (and Privalov) theorem fails that for a conjugated in this sense function.
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Цитирование
Rubinshtein, A. I. ON BARY-STECHKIN THEOREM / Rubinshtein, A, I // Ufa Mathematical Journal. - 2019. - 11. - № 1. - P. 70-74;. - 10.13108/2019-11-1-70
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https://www.doi.org/10.13108/2019-11-1-70
 https://www.scopus.com/record/display.uri?eid=2-s2.0-85066054017&origin=resultslist
 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcAuth=Alerting&SrcApp=Alerting&DestApp=WOS_CPL&DestLinkType=FullRecord&UT=WOS:000466964100006
 https://openrepository.mephi.ru/handle/123456789/17921
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