Spaces of Besov-Sobolev type and a problem on nonlinear approximation

TL;DR

This paper explores fractional Besov-Sobolev spaces, establishing their properties, embeddings, and applications to harmonic analysis and wavelet approximation, extending classical results in nonlinear approximation theory.

Contribution

It introduces fractional variants of quasi-norms, identifies these as interpolation spaces, and characterizes approximation spaces via wavelet bases, extending classical nonlinear approximation results.

Findings

Spaces are identified as real interpolation spaces.

Equivalence between Fourier and difference operator definitions established.

Characterization of approximation spaces via wavelet bases extended.

Abstract

We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space $\dot W^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodecki\u{\i} spaces. We establish the equivalence between Fourier analytic definitions and definitions via difference operators acting on measurable functions. We prove various new results on embeddings and non-embeddings, and give applications to harmonic and caloric extensions. For suitable wavelet bases we obtain a characterization of the approximation spaces for best $n$-term approximation from a wavelet basis via smoothness conditions on the function; this extends a classical result by DeVore, Jawerth and Popov.