Non-uniform painless decompositions for anisotropic Besov and Triebel-Lizorkin spaces

TL;DR

This paper develops a flexible atomic decomposition method for anisotropic Besov and Triebel-Lizorkin spaces using non-uniform affine systems with irregular translations, extending previous painless constructions.

Contribution

It introduces a novel non-uniform affine system approach for atomic decompositions in anisotropic function spaces, accommodating irregular translation sets and broadening applicability.

Findings

Constructed affine systems for a wide class of spaces including Lebesgue spaces.

Proved existence of smooth windows for atomic decompositions with irregular translations.

Extended painless construction to Besov and Triebel-Lizorkin spaces with dual systems.

Abstract

In this article we construct affine systems that provide a simultaneous atomic decomposition for a wide class of functional spaces including the Lebesgue spaces $L^p(\Rdst)$, $1<p<+\infty$. The novelty and difficulty of this construction is that we allow for non-lattice translations. We prove that for an arbitrary expansive matrix $A$ and any set $\Lambda$ - satisfying a certain spreadness condition but otherwise irregular- there exists a smooth window whose translations along the elements of $\Lambda$ and dilations by powers of $A$ provide an atomic decomposition for the whole range of the anisotropic Triebel-Lizorkin spaces. The generating window can be either chosen to be bandlimited or to have compact support. To derive these results we start with a known general "painless" construction that has recently appeared in the literature. We show that this construction extends to Besov and Triebel-Lizorkin spaces by providing adequate dual systems.