Criteria for hierachical bases in Sobolev spaces
type: Item
instance of: scholarly article
title: Criteria for hierachical bases in Sobolev spaces
zbMATH Open document ID: 0957.46026
published in: Applied and Computational Harmonic Analysis
publication date: 2001-03-25T00:00:00Z
review text: The object of this paper is to determine, both theoretically and numerically, the range of \(s\) for which a hierarchical system \[ \Psi:= \{\Phi(x- \alpha),\;2^{jd/2} \Psi^\lambda(2^jx- \alpha): \alpha\in \mathbb{Z}^d,\;j= 0,1,2,\dots,\;\lambda= 1,\dots, 2^d- 1\} \] forms a Riesz basis for the Sobolev space \(H^s(\mathbb{R}^d)\). This system is derived by a dyadic multiresolution analysis \[ V_0\subset V_1\subset\cdots\subset V_j\subset\cdots \] of closed subspaces of \(L_2(\mathbb{R}^d)\) with scaling function \(\phi\in V_0\) of compact support by specifying \(2^d- 1\) functions \(\Psi^\lambda\in V_1\), \(\lambda= 1,\dots, 2^d\), of compact support. This involves determining the smoothness of the dual system. The elements of the dual system consist of noncompactly supported functions. The authors show how to determine the exact range of Sobolev exponents \(s\) in the multivariant case using spectral properties of transfer operators. For Stevenson's hierarchical basis they find the estimation \(-0,990236\dots< s< 3/2\).
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46B15
zbMATH DE Number: 1415638
zbMATH Keywords: dual symbol
zbMATH Keywords: hierarchical system
zbMATH Keywords: Riesz basis
zbMATH Keywords: Sobolev space
zbMATH Keywords: dyadic multiresolution analysis
zbMATH Keywords: smoothness of the dual system
zbMATH Keywords: noncompactly supported functions
zbMATH Keywords: exact range of Sobolev exponents
zbMATH Keywords: Stevenson's hierarchical basis
author: Rudolph Lorentz
author: Peter Oswald
reviewed by: Vladimír Ďurikovič
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1415638