Approximation with best order by Toeplitz-irregular methods of summability of Fourier series
type: Item
instance of: scholarly article
title: Approximation with best order by Toeplitz-irregular methods of summability of Fourier series
zbMATH Open document ID: 0858.42002
DOI: 10.1007/BF02266697
author: V. A. Baskakov
published in: Mathematical Notes
publication date: 1996-01-08T00:00:00Z
review text: Let \(f\) be a continuous \(2\pi\)-periodic function with the Fourier coefficients \(a_k\), \(b_k\) and \(\varepsilon_n=\varepsilon_n(f)\) denote the sequence of best uniform approximations of \(f\) by trigonometric polynomials of order \(n\). It is proved that if there is \(C>0\) independent of \(n\) such that \[ {1\over a_0}+\sum^n_{k=1} \min\Biggl\{{1\over |a_k|},{1\over |b_k|}\Biggr\}\leq {C\over \varepsilon_{n-1}},\quad n=1,2,\dots, \] then \(f\) can be approximated with the best order \(O(\varepsilon_n)\) only by regular summability methods. If this condition fails, then \(f(x)\) can be approximated with the best order \(O(\varepsilon_n)\) both by regular and irregular methods. This result is also valid for approximation in the space \(L^p\), \(1\leq p<\infty\).
Mathematics Subject Classification ID: 42A10
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 830776
zbMATH Keywords: best approximation
zbMATH Keywords: Toeplitz regular and irregular methods
zbMATH Keywords: Fourier series
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: trigonometric polynomials
zbMATH Keywords: summability
reviewed by: D. K. Ugulava
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf02266697
OpenAlex ID: W2014505826
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description: scientific article; zbMATH DE number 830776