Stable decomposition of homogeneous Mixed-norm Triebel-Lizorkin spaces

2022/06/08 by Nielsen, Morten
#41A17 #41A65 #42B35 #42C15 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2206.03724

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≥ 2, is shown to be fundamentally different from the unmixed case. However, Jackson and Bernstein inequalities for m-term approximation can still be derived.

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