Gibbs phenomenon on sampling series based on Shannon's and Meyer's wavelet analysis
type: Item
instance of: scholarly article
title: Gibbs phenomenon on sampling series based on Shannon's and Meyer's wavelet analysis
zbMATH Open document ID: 0948.42025
DOI: 10.1007/BF01257192
author: C. Karanikas
author: Nikolaos Atreas
published in: The Journal of Fourier Analysis and Applications
publication date: 2000-11-14T00:00:00Z
full work available at URL: https://eudml.org/doc/59624
review text: Suppose \(f\in L^2(\mathbb R)\) is continuous except in one point \(t= \lambda 2^{- \mu}\), where \(\mu \in {\mathbb N}\), \(\lambda \in {\mathbb Z}\). Assume further that \(f(t^-) \leq f(t) \leq f(t^+)\). Choose \(\alpha \in [0,1]\) such that \(f(t)= \alpha f(t^-)+(1- \alpha)f(t^+)\). It is proved that in this situation, the classical Shannon sampling formula always exhibits Gibbs phenomenon. For Meyer sampling series it is proved that Gibbs phenomenon appears when \(\alpha < 0.12495\) and when \(\alpha > 0.306853\). For some values of \(\alpha\), the overshoot is proved to be shorter than for the Shannon sampling formula.
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 94C10
Mathematics Subject Classification ID: 94A12
zbMATH DE Number: 1441220
zbMATH Keywords: sampling
zbMATH Keywords: multiresolution analysis
zbMATH Keywords: Gibbs phenomenon
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1441220