Decomposition of Triebel-Lizorkin and Besov spaces in the context of Laguerre expansions

TL;DR

This paper constructs dual frames with localized elements on + using Laguerre functions and characterizes Triebel-Lizorkin and Besov spaces via needlet coefficients, advancing harmonic analysis tools.

Contribution

It introduces a new dual frame construction with almost exponential localization based on Laguerre functions and characterizes related function spaces through needlet coefficients.

Findings

Constructed dual frames with localized needlets on +

Characterized Triebel-Lizorkin and Besov spaces via needlet coefficients

Established equivalence between function spaces and sequence spaces

Abstract

A pair of dual frames with almost exponentially localized elements (needlets) are constructed on $\RR_+^d$ based on Laguerre functions. It is shown that the Triebel-Lizorkin and Besov spaces induced by Laguerre expansions can be characterized in terms of respective sequence spaces that involve the needlet coefficients.