Short time Fourier transform, integral wavelet transform, and wavelet functions associated with splines
type: Item
instance of: scholarly article
title: Short time Fourier transform, integral wavelet transform, and wavelet functions associated with splines
zbMATH Open document ID: 0911.42015
author: K. Appert
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-04-23T00:00:00Z
review text: The author considers a function of the form \(s_m=\sum_j c_j N_m(x-j)\), \(x\in{\mathbb R}\), \(j\in{\mathbb Z}\), where \((c_j)\in\ell_2\) and \(N_m(x)\) is the cardinal B-spline of order \(m\). First \(s_m\) is used as a kernel function in a short time Fourier transform. The time-frequency window can be modulated by the coefficients \((c_j)\). Next, conditions on the \((c_j)\) are derived to make \(s_m\) satisfy the admissibility condition so that it can be used as a wavelet function for an integral (i.e., continuous) wavelet transform. Using the two-scale relation for the \(N_m\), a (bi)orthogonal multiresolution analysis is then derived where the \(s_m\) act as (primal) scaling function. The corresponding wavelet functions are generalizations of B-wavelets. Finally, a recurrence relation is given for the computation of B-wavelets, and hence also for the wavelets considered in this paper.
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A30
zbMATH DE Number: 1218623
zbMATH Keywords: spline wavelets
zbMATH Keywords: B-splines
zbMATH Keywords: multiresolution analysis
zbMATH Keywords: scaling function
reviewed by: Adhemar Bultheel
MaRDI profile type: Publication
full work available at URL: jmaa.1998.5997
OpenAlex ID: W2089092503
cites work: Q3975815
cites work: Q4205895
cites work: Q3198776
cites work: Ten Lectures on Wavelets
cites work: Q4259915
cites work: Q3996824
cites work: Q4888818
DOI: 10.1006/JMAA.1998.5997
description: scientific article; zbMATH DE number 1218623