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 Zap. Nauchn. Sem. POMI, 2009, Volume 371, Pages 78–108 (Mi znsl3546)

On approximating periodic functions by the Fourier sums

V. V. Zhuk

Saint-Petersburg State University, Saint-Petersburg, Russia

Abstract: Let $L_p$, $1\le p<\infty$, be the space of $2\pi$-periodic functions $f$ with the norm $\|f\|_p=(\int^\pi_{-\pi}|f|^p)^{1/p}$, and let $C=L_\infty$ be the space of continuous $2\pi$-periodic functions with the norm $\|f\|_\infty=\|f\|=\max_{x\in\mathbb R}|f(x)|$. Let $CP$ be the subspace of $C$ with a semi-norm $P$ that is invariant with respect to translation and such that $P(f)\le M\|f\|$ for every $f\in C$. By $\sum^\infty_{k=0}A_k(f)$ we denote the Fourier series of the function $f$, and let $\lambda=\{\lambda_k\}^\infty_{k=0}$ be a sequence of real numbers for which $\sum^\infty_{k=0}\lambda_kA_k(f)$ is the Fourier series of a certain function $f_{\lambda}\in L_p$.
The paper considers questions related to approximating the function $f_\lambda$ by its Fourier sums $S_n(f_\lambda)$ on a point set and on the spaces $L_p$ and $CP$. Estimates of $\|f_\lambda-S_n(f_\lambda)\|_p$ and $P(f_\lambda-S_n(f_\lambda))$ are obtained by using the structural characteristics (the best approximations and the modules of continuity) of the functions $f$ and $f_\lambda$. As a rule, the essential part of deviation is estimated with the use of the structural characteristics of the function $f$. Bibl. – 11 titles.

Key words and phrases: periodic function, Fourier series, Fourier sums, Fejér sums, Vallée-Poussin sums, Riesz sums, best approximation, modulus of continuity.

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English version:
Journal of Mathematical Sciences (New York), 2010, 166:2, 167–185

UDC: 517.5

Citation: V. V. Zhuk, “On approximating periodic functions by the Fourier sums”, Analytical theory of numbers and theory of functions. Part 24, Zap. Nauchn. Sem. POMI, 371, POMI, St. Petersburg, 2009, 78–108; J. Math. Sci. (N. Y.), 166:2 (2010), 167–185

Citation in format AMSBIB
\Bibitem{Zhu09} \by V.~V.~Zhuk \paper On approximating periodic functions by the Fourier sums \inbook Analytical theory of numbers and theory of functions. Part~24 \serial Zap. Nauchn. Sem. POMI \yr 2009 \vol 371 \pages 78--108 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl3546} \transl \jour J. Math. Sci. (N. Y.) \yr 2010 \vol 166 \issue 2 \pages 167--185 \crossref{https://doi.org/10.1007/s10958-010-9857-5} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77952096268}