Wavelets of Wilson type with arbitrary shapes
type: Item
instance of: scholarly article
title: Wavelets of Wilson type with arbitrary shapes
zbMATH Open document ID: 0947.42022
published in: Applied and Computational Harmonic Analysis
publication date: 2000-05-01T00:00:00Z
review text: Let \(\{ \omega_j \}_{j\in \mathbb Z}\) be a family of complex-valued functions on \(\mathbb R\) and consider a sinusoidal family of functions given by \(T_{j,k}(x)= \sqrt{2} \cos(k+ 1/2)\pi (x-j)\), \(j \in {\mathbb Z}\), \(k=0,1,\dots\) (resp. \(T_{j,k}(x)=\sqrt{2} \sin(k+ 1/2)\pi (x-j) \), or an appropriate combination of sine and cosine). Define a two-parameter family of functions by \(\psi_{j,k}(x)= \omega_j(x) T_{j,k}(x)\). Equivalent conditions for \(\{ \psi_{j,k}\}\) to be a Schauder basis for \(L^2(\mathbb R)\) are given. Under extra conditions on \(\omega_j\), a condition for the existence of a dual Schauder basis of the same type is given. Other assumptions on \(\omega_j\) imply that \(\{ \psi_{j,k}\}\) is a frame if and only if \(\{ \psi_{j,k}\}\) is a Riesz basis.
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1415636
zbMATH Keywords: wavelets of Wilson type
zbMATH Keywords: Schauder basis
zbMATH Keywords: Riesz basis
zbMATH Keywords: frames
author: Charles K. Chui
author: Xianliang Shi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1415636