Restricted non-linear approximation in sequence spaces and applications to wavelet bases and interpolation

TL;DR

This paper explores restricted non-linear approximation in sequence spaces, establishing key inequalities and properties, and applies these findings to wavelet bases and interpolation in Triebel-Lizorkin and Besov spaces.

Contribution

It introduces new equivalences for approximation spaces and extends the Temlyakov property and interpolation results to wavelet bases in advanced function spaces.

Findings

Equivalence of approximation space embeddings with Jackson and Bernstein inequalities

Verification of the Temlyakov property for wavelet bases in Triebel-Lizorkin spaces

New interpolation results for Triebel-Lizorkin and Besov spaces

Abstract

Restricted non-linear approximation is a type of N-term approximation where a measure $\nu$ on the index set (rather than the counting measure) is used to control the number of terms in the approximation. We show that embeddings for restricted non-linear approximation spaces in terms of weighted Lorentz sequence spaces are equivalent to Jackson and Bernstein type inequalities, and also to the upper and lower Temlyakov property. As applications we obtain results for wavelet bases in Triebel-Lizorkin spaces by showing the Temlyakow property in this setting. Moreover, new interpolation results for Triebel-Lizorkin and Besov spaces are obtained.