Tensor products of Sobolev-Besov spaces and applications to approximation from the hyperbolic cross
type: Item
instance of: scholarly article
title: Tensor products of Sobolev-Besov spaces and applications to approximation from the hyperbolic cross
zbMATH Open document ID: 1194.46056
author: Winfried Sickel
author: Tino Ullrich
published in: Journal of Approximation Theory
publication date: 2010-01-08T00:00:00Z
review text: Let \(S^{r_1, \dots, r_d}_{p,p} B (\mathbb R^d)\), \(0<p<\infty\), \(r_j \in \mathbb R\) and \(S^{r_1, \dots, r_d}_p H(\mathbb R^d)\), \(1<p<\infty\), \(r_j \in \mathbb R\) be the usual Besov-Sobolev spaces with dominating mixed smoothness in \(\mathbb R^d\). It is the main aim of this paper to show that these spaces are tensor products of corresponding spaces in \(\mathbb R\), \[ S^{r_1, \dots, r_d}_{p,p}B (\mathbb R^d) = \bigotimes^d_{j=1} B^{r_j}_{p,p} (\mathbb R), \quad S^{r_1, \dots, r_d}_p H (\mathbb R^d) = \bigotimes^d_{j=1} H^{s_j}_p (\mathbb R). \] It gives the possibility to shift (orthogonal) spline wavelet bases from \(B^s_{p,p} (\mathbb R)\) and \(H^s_p (\mathbb R)\) to these spaces with dominating mixed smoothness. This includes corresponding isomorphisms to adapted sequence spaces. It is used afterwards to study best approximations related to the hyperbolic cross.
reviewed by: Hans Triebel
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 5655046
zbMATH Keywords: Besov-Sobolev spaces with dominating mixed smoothness
zbMATH Keywords: tensor products
zbMATH Keywords: spline expansions
zbMATH Keywords: hyperbolic cross
MaRDI profile type: Publication
full work available at URL: j.jat.2009.01.001
OpenAlex ID: W2135512293
cites work: Q3590614
cites work: Wavelets and Besov spaces
cites work: Q4434436
cites work: Ten Lectures on Wavelets
cites work: Q4023934
cites work: Q4215356
cites work: Hyperbolic wavelet approximation
cites work: Q3772674
cites work: A discrete transform and decompositions of distribution spaces
cites work: Approximation of classes of periodic functions of several variables by nuclear operators
cites work: Q3406671
cites work: Q5557257
cites work: Q5630205
cites work: On hyperbolic summation and hyperbolic moduli of smoothness
cites work: New bases for Triebel-Lizorkin and Besov spaces
cites work: Approximation theory in tensor product spaces
cites work: Q3135829
cites work: Best \(N\) term approximation spaces for tensor product wavelet bases
cites work: Q3945010
cites work: Approximation spaces
cites work: Q4780972
cites work: Q4093889
cites work: Recent developments in the theory of function spaces with dominating mixed smoothness.
cites work: An unconditional basis in periodic spaces with dominating mixed smoothness properties
cites work: Spaces of functions of mixed smoothness and approximation from hyperbolic crosses
cites work: Q3810338
cites work: Q3244672
cites work: Q5864575
cites work: Q4240379
cites work: Q5640160
cites work: Q4348451
cites work: Q3881362
cites work: Spline bases and spline representations in function spaces
cites work: Q5449091
cites work: Q3336200
cites work: Q5673447
cites work: Représentation fonctionnelle des espaces vectoriels toplogiques
cites work: Q3001708
cites work: Q5507931
cites work: Function spaces with dominating mixed smoothness
cites work: A mathematical introduction to wavelets
DOI: 10.1016/J.JAT.2009.01.001
description: scientific article; zbMATH DE number 5655046