The convergence of Fourier series and a K-functional
type: Item
instance of: scholarly article
title: The convergence of Fourier series and a K-functional
zbMATH Open document ID: 0731.42007
DOI: 10.1016/0022-247X(91)90075-B
published in: Journal of Mathematical Analysis and Applications
publication date: 1991-01-01T00:00:00Z
review text: For \(f\in B[-\pi,\pi]\) and I a subinterval of [-\(\pi\),\(\pi\) ] let \(\omega (f,I)=\sup_{t,u\in I}| f(t)-f(u)|.\) If [a,b]\(\subset [- \pi,\pi]\) and \(\delta >0\) let \(\Omega (f,\delta;[a,b])=\sup \delta \sum_{I\in {\mathcal I}}\omega (f,I),\) where the sup is taken over all families \({\mathcal I}\) of subintervals of [a,b] such that card(\({\mathcal I})\) \(=\) number of intervals in \({\mathcal I}\leq \delta^{-1}\), \(\cup_{I\in {\mathcal I}}I=[a,b]\) and the members of \({\mathcal I}\) are pairwise disjoint. The main result of the author is the following interesting theorem. Theorem. Let f be a bounded, measurable, \(2\pi\)-periodic function, \(x\in [-\pi,\pi]\) and \(f(x+)\), f(x-) exist. Then, for \(n\geq 1\), \[ | S_ n(f,x)-(1/2)(f(x+)+f(x-))| \leq C\sum^{n}_{k=1}(1/k)\Omega (g,k/n;[0,\pi /k]), \] where \(g(t)=f(x+t)+f(x-t)-f(x+)-f(x-)\) and \(S_ n(f)\) denotes the nth partial sum of the Fourier series of f(x).
Mathematics Subject Classification ID: 42A20
Mathematics Subject Classification ID: 42A10
zbMATH DE Number: 4208764
zbMATH Keywords: degree of approximation
zbMATH Keywords: convergence of Fourier series
zbMATH Keywords: K-functional
author: H. N. Mhaskar
reviewed by: S. M. Mazhar
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4208764