A corner point Gibbs phenomenon for Fourier series in two dimensions
type: Item
instance of: scholarly article
title: A corner point Gibbs phenomenon for Fourier series in two dimensions
zbMATH Open document ID: 0949.42010
author: Gilbert Helmberg
published in: Journal of Approximation Theory
publication date: 1999-12-20T00:00:00Z
review text: Let \[ A=\biggl\{(x,y): 0\leq y\leq \pi,\;{y\over c}\leq x\leq {y\over c}+\pi \biggr\}\quad (0\neq c\in {\mathbb R}) \] and \(\chi_{_A}\) be the indicator function of \(A\). Denote by \(f\) the function with period \(2\pi\) in each variable which extends \(\chi_{_{A}}\). Denote by \(S_m(f)(x,y)\) the square partial sum of the Fourier series of \(f\), i.e., \[ S_m(f)(x,y)=\sum_{|\mu|\leq m, |\nu|\leq m} c_{\mu,\nu}(f)e^{i(\mu x+\nu y)}\quad (m\in {\mathbb Z}_+) \] where \(c_{\mu,\nu}(f)\) denote the Fourier coefficients of \(f\). The author shows the Gibbs phenomenon of the Fourier series of \(f\) at the corner point \(O=(0,0)\) of \(A\) by finding the limit of \( S_{2m+1}(f)(P_m)\) as \(m\to \infty\) where \(P_m={\pi\over 2m+1}(x+{y\over c}, y)\). The calculation is very careful and rather complicated.
Mathematics Subject Classification ID: 42B05
zbMATH DE Number: 1356329
zbMATH Keywords: double Fourier series
zbMATH Keywords: Gibbs phenomenon
zbMATH Keywords: corner point
reviewed by: Wang Kunyang
MaRDI profile type: Publication
full work available at URL: jath.1998.3320
OpenAlex ID: W1979836930
DOI: 10.1006/JATH.1998.3320
description: scientific article; zbMATH DE number 1356329