Stable decomposition of homogeneous Mixed-norm Triebel-Lizorkin spaces

TL;DR

This paper develops smooth, localized orthonormal bases for homogeneous mixed-norm Triebel-Lizorkin spaces in anisotropic settings, enabling unconditional decompositions and analyzing nonlinear approximation properties.

Contribution

It introduces tensor product-based brushlet functions as bases for these spaces and studies their nonlinear approximation behavior, highlighting differences from the unmixed case.

Findings

Constructed orthonormal bases compatible with mixed-norm Triebel-Lizorkin spaces.

Proved these bases form unconditional bases in the anisotropic setting.

Derived Jackson and Bernstein inequalities for m-term approximation.

Abstract

We construct smooth localized orthonormal bases compatible with homogeneous mixed-norm Triebel-Lizorkin spaces in an anisotropic setting on $\bR^d$. The construction is based on tensor products of so-called univariate brushlet functions that are constructed using local trigonometric bases in the frequency domain. It is shown that the associated decomposition system form unconditional bases for the homogeneous mixed-norm Triebel-Lizorkin spaces. In the second part of the paper we study nonlinear $m$-term nonlinear approximation with the constructed basis in the mixed-norm setting, where the behaviour, in general, for $d\geq 2$, is shown to be fundamentally different from the unmixed case. However, Jackson and Bernstein inequalities for $m$-term approximation can still be derived.