On Sobolev and Besov Spaces of Hybrid Regularity

TL;DR

This paper investigates the approximation properties of functions in Sobolev spaces using tensor-product wavelet bases, establishing new inequalities and showing the inclusion of classical Besov spaces within hybrid regularity Besov spaces.

Contribution

It derives Jackson and Bernstein inequalities for tensor-product wavelet approximation and characterizes the embedding of classical Besov spaces into hybrid regularity Besov spaces.

Findings

Functions approximable by classical wavelets are also approximable by tensor-product wavelets at the same rate.

Classical Besov spaces of regularity q+sn are included in hybrid regularity Besov spaces with isotropic regularity q and mixed regularity s.

The paper establishes approximation classes containing Besov spaces of hybrid regularity.

Abstract

The present article is concerned with the nonlinear approximation of functions in the Sobolev space H^q with respect to a tensor-product, or hyperbolic wavelet basis on the unit n-cube. Here, q is a real number, which is not necessarily positive. We derive Jackson and Bernstein inequalities to obtain that the approximation classes contain Besov spaces of hybrid regularity. Especially, we show that all functions that can be approximated by classical wavelets are also approximable by tensor-product wavelets at least at the same rate. In particular, this implies that for nonnegative regularity, the classical Besov spaces of regularity q+sn, integrability and weak index t, with 1/t = s + 1/2, are included in the Besov spaces of hybrid regularity with isotropic regularity q and additional mixed regularity s.