On approximating periodic functions by the Fourier sums

V. V. Zhuk

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 24, Tome 371 (2009), pp. 78-108

Résumé

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.