On progressive functions
type: Item
instance of: scholarly article
title: On progressive functions
zbMATH Open document ID: 1304.42086
author: Anca Deliu
author: Baiqiao Deng
published in: Journal of Mathematical Analysis and Applications
publication date: 2014-10-31T00:00:00Z
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 39B22
zbMATH DE Number: 6363002
zbMATH Keywords: progressive functions
zbMATH Keywords: shift-invariant spaces
zbMATH Keywords: orthonormal sequences
zbMATH Keywords: Riesz sequence
zbMATH Keywords: dual functions
reviewed by: Peter Massopust
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.10.001
OpenAlex ID: W2002139859
cites work: The structure of shift-invariant subspaces of \(L^2(\mathbb{R}^n)\)
cites work: Q4003151
cites work: Q4027556
cites work: Biorthogonal bases of compactly supported wavelets
cites work: Orthonormal bases of compactly supported wavelets
cites work: Ten Lectures on Wavelets
cites work: A practical guide to splines
cites work: The structure of finitely generated shift-invariant spaces in \(L_ 2(\mathbb{R}^ d)\)
cites work: Approximation from Shift-Invariant Subspaces of L 2 (ℝ d )
cites work: Q5488964
cites work: Four short courses on harmonic analysis. Wavelets, frames, time-frequency methods, and applications to signal and image analysis
cites work: Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape
cites work: Q3645522
cites work: Q4490829
cites work: Shift-invariant spaces on the real line
cites work: Shift-invariant spaces and linear operator equations
cites work: Shorter Notes: A Simple Proof of Wiener's 1/f Theorem
cites work: Q3159056
cites work: Q3508686
cites work: Q5667036
cites work: Q4945748
DOI: 10.1016/J.JMAA.2014.10.001
review text: The authors investigate functions \(\phi:\mathbb R\to\mathbb R\) that are solutions of functional equations of the form NEWLINE\[NEWLINE \phi = C s + \sum_{k=1}^{N-1} \alpha_k \phi (\cdot - k), NEWLINE\]NEWLINE where \(s\in L^2(\mathbb R)\) with \(\mathrm{supp} s \subset [0, \infty)\), \(1 < N\in \mathbb N\), \(C\in \mathbb R\setminus\{0\}\), and \(\alpha_k\in \mathbb R\), \(k = 1, \dots, N-1\). A function \(g\in L^2(\mathbb R)\) is called \textit{progressive} iff the support of its Fourier-Plancherel transform \(\widehat{g}\) is \([0, \infty)\). The space of progressive functions is identical to the Hardy space \(H^2 (\mathbb R)\).NEWLINENEWLINEThe authors derive sufficient conditions and provide explicit formulas for the construction of progressive functions with prescribed exponential decay. Moreover, conditions for the positive integer translates of a progressive function to form an orthonormal or Riesz basis are derived. It is further shown that progressive functions may be employed to construct wavelets of arbitrary smoothness on \([0,\infty)\) provided that they are generated by a smooth compactly supported refinable function.
description: scientific article; zbMATH DE number 6363002