Approximation to continuous functions by Cesàro means of double Fourier series and conjugate series
type: Item
instance of: scholarly article
title: Approximation to continuous functions by Cesàro means of double Fourier series and conjugate series
zbMATH Open document ID: 0617.42013
DOI: 10.1016/0021-9045(87)90074-8
published in: Journal of Approximation Theory
publication date: 1987-01-01T00:00:00Z
review text: Let f(x,y) be of period \(2\pi\) in each of x,y. given \(\alpha\),\(\beta\) with \(0<\alpha,\beta \leq 2\), we say that \(f\in Z(\alpha,\beta)\) if \[ f(x+u,y)+f(x-u,y)-2f(x,y)=O(u^{\alpha}); \] \[ f(x,y+v)+f(x,y-v)- 2f(x,y)=O(v^{\beta}); \] uniformly in x,y. The class Z(\(\alpha\),\(\beta)\) is the same as Lip(\(\alpha\),\(\beta)\) when \(0<\alpha,\beta <1\); but Z(\(\alpha\),\(\beta)\) is larger than Lip(\(\alpha\),\(\beta)\) when either \(\alpha\) or \(\beta\) (or both) is equal to 1. Let \(S_{mn}(f,x,y)\) denote the partial sum of the Fourier series of f, and let \(\sigma_{nm}^{\lambda \delta}\) denote its (C,\(\lambda\),\(\delta)\) mean; that is to say \[ \sigma_{mn}^{\lambda \delta}(f,x,y)=\frac{1}{A_ m^{\lambda}A_ n^{\delta}}\sum^{m}_{j=0}\sum^{n}_{k=0}A_{m-j}^{\lambda - 1}A_{n-k}^{\delta -1}S_{jk}(f,x,y). \] Let \(f\in Z(\alpha,\beta)\), where \(0<\alpha,\beta \leq 1\). It is proved that if \(0<\lambda,\delta\), then (uniformly) in (x,y) \[ \sigma_{mn}^{\lambda \delta}(f,x,y)- f(x,y)=O((m+1)^{-\alpha})+O((n+1)^{-\beta}), \] except that when \(\alpha =1\) the first term on the right has an extra factor \(\log (m+2)\), and when \(\beta =1\) the second term has an extra factor \(\log (n+2)\). It is to be noted that the order terms are independent of \(\lambda\),\(\delta\). Similar results are obtained for the various conjugate series and functions, but here the order terms sometimes have extra logarithmic factors. In the case of most of the results, examples are given to show that they are best possible.
Mathematics Subject Classification ID: 42B99
Mathematics Subject Classification ID: 42B05
Mathematics Subject Classification ID: 42A10
zbMATH DE Number: 4000673
zbMATH Keywords: Fourier series
zbMATH Keywords: conjugate series
zbMATH Keywords: logarithmic factors
zbMATH Keywords: examples
author: Ferenc Móricz
author: Xianliang Shi
MaRDI profile type: Publication
full work available at URL: 0021-9045(87)90074-8
OpenAlex ID: W2026084941
cites work: Q3249881
cites work: Q5804330
cites work: Q5727762
cites work: Q5815726
cites work: Smooth functions
cites work: On the boundary values of functions of several complex variables, I
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description: scientific article; zbMATH DE number 4000673