On Homogeneous Decomposition Spaces and Associated Decompositions of Distribution Spaces

TL;DR

This paper introduces a new framework for homogeneous decomposition spaces using structured frequency decompositions, constructs adaptive tight frames for $L_2$, and applies these to define homogeneous $ extalpha$-modulation spaces.

Contribution

It develops a novel construction of decomposition smoothness spaces on homogeneous type and creates universal frames for tempered distributions with convergence in distribution space.

Findings

Constructed simple tight frames for $L_2(R^d)$.

Characterized smoothness norms via frame coefficient sparseness.

Introduced homogeneous $ extalpha$-modulation spaces.

Abstract

A new construction of decomposition smoothness spaces of homogeneous type is considered. The smoothness spaces are based on structured and flexible decompositions of the frequency space $\mathbb{R}^d\backslash\{0\}$. We construct simple adapted tight frames for $L_2(\mathbb{R}^d)$ that can be used to fully characterise the smoothness norm in terms of a sparseness condition imposed on the frame coefficients. Moreover, it is proved that the frames provide a universal decomposition of tempered distributions with convergence in the tempered distributions modulo polynomials. As an application of the general theory, the notion of homogeneous $\alpha$-modulation spaces is introduced.