Pointwise variable anisotropic function spaces on \(\mathbb{R}^n\)
type: Item
instance of: scholarly article
title: Pointwise variable anisotropic function spaces on \(\mathbb{R}^n\)
zbMATH Open document ID: 1494.46002
DOI: 10.1515/9783110761795
author: Shai Dekel
published in: De Gruyter Studies in Mathematics
publication date: 2021-12-02T00:00:00Z
review text: This monograph is dedicated to an exhaustive analysis of pointwise variable anisotropic function spaces on \(\mathbb R^n\), including Hardy spaces, Campanato spaces, Besov spacs and Sobolev spaces. In [\textit{M. Bownik}, Anisotropic Hardy spaces and wavelets. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1036.42020)], Bownik first extended the theory of Hardy spaces to the anisotropic setting, where the spaces were defined via the maximal function \(M^0_\varphi f(x)=\sup_{k\in\mathbb Z}|f\ast \varphi_k(x)|\) with \(\varphi_k\) generated by the dilation structure \(\varphi_k(x)=|{\det A}|^{-k}\varphi(A^{-k}x)\) for an appropriate matrix \(A\) and given function \(\varphi\). In this monograph, the author further considers pointwise variable anisotropic function spaces related to affine transformations of the form \(A_{x,t}(y)=M_{x, t}y+x\) for all \(x,y\in \mathbb R^n\) and \(t\in \mathbb R\) such that \(M_{x,t}\) is a positive definite \(n\times n\) matrix (or \(A_{x,m}(y)=M_{x,m}y+x\) for all \(x,y\in\mathbb R^n\) and \(m\in\mathbb Z\)). Here are the topics covered in the book. Chapter 1 contains basic information of local approximation over domains in \(\mathbb R^n\), including \(K\)-functionals, Marchaud inequalities, the Bramble-Hilbert lemma and Whitney theorem for convex domains. Chapter 2 is devoted to the anisotropic multilevel ellipsoid covers of \(\mathbb R^n\), where ellipsoids are defined via affine transformations. In particular, construction and properties of continuous (and discrete) ellipsoid covers are presented. The purpose of Chapter 3 is to present multiresolution analysis constructions that are subordinate to the anisotropic quasi-distance induced by the ellipsoid covers. The multiresolution analysis construction on \(\mathbb R^n\) provides polynomial reproduction of arbitrary order and has arbitrarily high regularization. The author also provides anisotropic locally stable basis consisting of bumps supported over the ellipsoids…
reviewed by: Dongyong Yang
Mathematics Subject Classification ID: 46-01
Mathematics Subject Classification ID: 42-01
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 7438179
zbMATH Keywords: anisotropic
zbMATH Keywords: affine transformation
zbMATH Keywords: ellipsoid
zbMATH Keywords: wavelet
zbMATH Keywords: multiresolution
zbMATH Keywords: Besov space
zbMATH Keywords: Campanato space
zbMATH Keywords: Hardy space
zbMATH Keywords: singular operator
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110761795
OpenAlex ID: W4226044314
description: scientific article; zbMATH DE number 7438179