Ufimskii Matematicheskii Zhurnal
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Ufimsk. Mat. Zh.: Year: Volume: Issue: Page: Find

 Ufimsk. Mat. Zh., 2018, Volume 10, Issue 1, Pages 96–117 (Mi ufa421)

On two-sided estimate for norm of Fourier operator

I. A. Shakirov

Naberezhnye Chelny State Pedagogical University, Nizametdinova str. 28, 423806, Naberezhnye Chelny, Russia

Abstract: In the work we study the behavior of Lebesgue constant $L_n$ of the Fourier operator defined in the space of continuous $2\pi$-periodic functions. The known integral representations expressed in terms of the improper integrals are too cumbersome. They are complicated both for theoretical and practical purposes.
We obtain a new integral representation for $L_n$ as a sum of Riemann integrals defined on bounded converging domains. We establish equivalent integral representations and provide strict two-sided estimates for their components. Then we provide a two-sided estimate for the Lebesgue constant. We solve completely the problem on the upper bound of the constant $L_n$. We improve its known lower bound.

Keywords: partial sums of Fourier series, norm of Fourier operator, Lebesgue constant, asymptotic formula, estimate for Lebesgue constant, extremal problem.

Full text: PDF file (475 kB)
References: PDF file   HTML file

English version:
Ufa Mathematical Journal, 2018, 10:1, 94–114 (PDF, 408 kB); https://doi.org/10.13108/2018-10-1-94

Bibliographic databases:

UDC: 517.518.83
MSC: 34A25, 22E05

Citation: I. A. Shakirov, “On two-sided estimate for norm of Fourier operator”, Ufimsk. Mat. Zh., 10:1 (2018), 96–117; Ufa Math. J., 10:1 (2018), 94–114

Citation in format AMSBIB
\Bibitem{Sha18} \by I.~A.~Shakirov \paper On two-sided estimate for norm of Fourier operator \jour Ufimsk. Mat. Zh. \yr 2018 \vol 10 \issue 1 \pages 96--117 \mathnet{http://mi.mathnet.ru/ufa421} \elib{https://elibrary.ru/item.asp?id=32705556} \transl \jour Ufa Math. J. \yr 2018 \vol 10 \issue 1 \pages 94--114 \crossref{https://doi.org/10.13108/2018-10-1-94} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000432413800008} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85044285311}