Strong approximation by Fourier transforms and Fourier series in \(L^ \infty\)-norm
type: Item
instance of: scholarly article
title: Strong approximation by Fourier transforms and Fourier series in \(L^ \infty\)-norm
zbMATH Open document ID: 0851.42005
author: Dang Vu Giang
author: Ferenc Móricz
published in: Journal of Approximation Theory
publication date: 1996-11-21T00:00:00Z
review text: The question of the strong approximation of functions \(f\) belonging to the space \(L^p(R)\), \(1 < p < \infty\), by its Dirichlet integral \(s_\nu (f,x) = {1\over \pi} \int_R f(x - t)\;{\sin \nu t\over t} dt \) is studied. With the help of quotient \[ d_\nu (f,p):= \Biggl|\biggl\{ {1\over \nu} \int^\nu_0 |s_\mu(f,\cdot) - f(\cdot)|^p d\mu \biggr\}^{1/p} \Biggr|_\infty, \quad \nu \in R_+, \] the class \({\mathcal L}_p(R) := \{f \in L^p(R): d_\nu(f,p) = O(\nu^{-1/p})\) as \(\nu \to \infty\}\) is introduced. Sufficient conditions for \(f\) to belong to the class \({\mathcal L}_p(R)\) in the case \(2 \leq p <\infty\), and necessary conditions for \(f\) to belong to \({\mathcal L}_p(R)\) in the case \(1 < p \leq 2\) are established. In the case, when the Dirichlet integral is replaced by the Riesz means similar results are obtained. Besides the problem of strong approximation of a periodic function by the partial sum or Fejér mean of its Fourier series is studied.
Mathematics Subject Classification ID: 42A10
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 41A35
zbMATH DE Number: 827930
zbMATH Keywords: Fourier transform
zbMATH Keywords: \(L^ p\) spaces
zbMATH Keywords: strong approximation
zbMATH Keywords: Dirichlet integral
zbMATH Keywords: Riesz means
zbMATH Keywords: Fejér mean
zbMATH Keywords: Fourier series
reviewed by: D. K. Ugulava
MaRDI profile type: Publication
full work available at URL: jath.1995.1114
OpenAlex ID: W2051798071
DOI: 10.1006/JATH.1995.1114
description: scientific article; zbMATH DE number 827930