Gibbs phenomenon for Fourier series on finite sets
type: Item
instance of: scholarly article
title: Gibbs phenomenon for Fourier series on finite sets
zbMATH Open document ID: 1206.11144
published in: Functional Analysis and Other Mathematics
publication date: 2010-12-15T00:00:00Z
review text: The classical Gibbs phenomenon describes the behaviour of the partial sums of the Fourier series at the discontinuity points of the function whose Fourier series is studied. The author is interested in the Gibbs phenomenon over the finite circle \(\mathbb Z/p\mathbb Z\), with \(p\) an odd prime. He studies the `jumping function' \(F\) defined by \(F(\varepsilon^m)=m\), at the \(p\)-th roots of unity, where \(-q\leq m\leq q\), \(p=2q+1\) and \(\varepsilon=\exp(2\pi i/p)\). The jump of the function is the large difference \(F(\varepsilon^q)-F(\epsilon^{q+1})=2q\). The Fourier series is in this case, the representation of (any) function \(f:\mathbb Z_p\rightarrow \mathbb C\) in the form of a linear combination of the characters \(f=\sum_{k=0}^{p-1}c_kf_k\), where \(f_k(z)=z^k\). For the function \(F\) the author considers the approximation obtained by removing the ``highest harmonics'' \(f_q\) and \(f_{q+1}\). He then shows that, as in the classical Gibbs phenomenon, the approximation has in some sense a larger jump than the original function (being approximately \(1+1/p\) times greater in this setup).
reviewed by: Pieter Moree
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 43A25
Mathematics Subject Classification ID: 43A50
zbMATH DE Number: 5826925
zbMATH Keywords: jump
zbMATH Keywords: discontinuity
zbMATH Keywords: truncated Fourier series
zbMATH Keywords: characters
zbMATH Keywords: nonuniform convergence
author: Vładimir I. Arnold
MaRDI profile type: Publication
full work available at URL: s11853-010-0043-6
OpenAlex ID: W1963709166
DOI: 10.1007/S11853-010-0043-6
description: scientific article; zbMATH DE number 5826925