Discrete Triebel-Lizorkin spaces and expansive matrices
Jordy Timo van Velthoven, Felix Voigtlaender

TL;DR
This paper characterizes when two expansive matrices define the same discrete anisotropic Triebel-Lizorkin spaces, extending Triebel's results from diagonal to arbitrary matrices and revealing a new classification of dilations.
Contribution
It extends Triebel's classification of dilations for Triebel-Lizorkin spaces from diagonal to general expansive matrices, providing necessary and sufficient conditions.
Findings
Spaces coincide if the set of matrix powers is finite.
Equality holds trivially when p=q and determinants satisfy a specific relation.
The classification differs from previous results for anisotropic Triebel-Lizorkin spaces.
Abstract
We provide a characterization of two expansive dilation matrices yielding equal discrete anisotropic Triebel-Lizorkin spaces. For two such matrices and , it is shown that for all and if and only if the set is finite, or in the trivial case when and . This provides an extension of a result by Triebel for diagonal dilations to arbitrary expansive matrices. The obtained classification of dilations is different from corresponding results for anisotropic Triebel-Lizorkin function spaces.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Advanced Differential Geometry Research · Mathematical Analysis and Transform Methods
