A new technique to estimate the regularity of refinable functions
type: Item
instance of: scholarly article
title: A new technique to estimate the regularity of refinable functions
zbMATH Open document ID: 0879.65102
DOI: 10.4171/RMI/207
author: Ingrid Daubechies
author: Albert Cohen
published in: Revista Matemática Iberoamericana
publication date: 1997-12-11T00:00:00Z
full work available at URL: https://eudml.org/doc/39499
review text: The regularity of refinable functions satisfying \[ \varphi(x)= 2\sum_n c_n \varphi(2x-n) \] is studied. The regularity of \(\varphi\) is measured by estimating the \(L^p\)-Sobolev exponent \[ s_p= \sup\Biggl\{\gamma: \int|\widehat{\varphi}(\omega)|^p (1+ |\omega|^p)^{\gamma} d\omega<\infty \Biggr\}. \] A new technique for estimating \(s_p\) is proposed which is independent of whether the \(h_n\) are finite in number or not and it generalizes easily to the multidimensional case. The paper proposes three different methods to compute the spectral radius of the transfer operator which is known to be closely related to the Sobolev regularity index \(s_p\). In the third procedure, \(s_p\) is found by determining the zero with the smallest absolute value of the corresponding Fredholm determinant. The proposed technique can be extended to a direct computation of the Hölder exponent \(\mu\) of \(\varphi\).
Mathematics Subject Classification ID: 65T40
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 933294
zbMATH Keywords: refinable functions
zbMATH Keywords: spectral properties of transfer operator
zbMATH Keywords: Fredholm determinant theory
zbMATH Keywords: Sobolev exponent
zbMATH Keywords: Hölder exponent
reviewed by: Gerlind Plonka
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 933294