Unconditional bases for homogeneous $\alpha$-modulation type spaces

TL;DR

This paper constructs orthonormal bases for homogeneous alpha-modulation and Triebel-Lizorkin spaces using tensor products of brushlet functions, providing unconditional bases for these complex function spaces.

Contribution

It introduces a novel construction of unconditional bases for homogeneous alpha-modulation and Triebel-Lizorkin spaces using alpha-coverings and tensor products of brushlet functions.

Findings

Established orthonormal bases for homogeneous alpha-modulation spaces.

Proved these bases are unconditional for the associated Triebel-Lizorkin spaces.

Provided a method for constructing bases compatible with bi-variate alpha-modulation spaces.

Abstract

In this article we construct orthonormal bases compatible with bi-variate homogeneous $\alpha$-modulation spaces and the associated spaces of Triebel-Lizorkin type. The construction is based on generating a separable $\alpha$-covering and using carefully selected tensor products of univariate brushlet functions with regards to this covering. We show that the associated systems form an unconditional bases for the homogeneous $\alpha$-spaces of Triebel-Lizorkin type.