$\alpha$-Molecules

TL;DR

This paper introduces the concept of $oldsymbol{ extalpha}$-molecules, a unified framework for multiscale systems in harmonic analysis, enabling the transfer of approximation results and deriving optimal sparse approximations for cartoon-like functions.

Contribution

The paper develops a general framework of $oldsymbol{ extalpha}$-molecules that unifies various multiscale systems and provides a methodology for approximation results and sparse approximation of cartoon-like functions.

Findings

Systems of $ extalpha$-molecules are almost orthogonal.

A methodology to transfer approximation results is established.

Optimal sparse approximation results for cartoon-like functions are derived.

Abstract

Within the area of applied harmonic analysis, various multiscale systems such as wavelets, ridgelets, curvelets, and shearlets have been introduced and successfully applied. The key property of each of those systems are their (optimal) approximation properties in terms of the decay of the $L^2$-error of the best $N$-term approximation for a certain class of functions. In this paper, we introduce the general framework of $\alpha$-molecules, which encompasses most multiscale systems from applied harmonic analysis, in particular, wavelets, ridgelets, curvelets, and shearlets as well as extensions of such with $\alpha$ being a parameter measuring the degree of anisotropy, as a means to allow a unified treatment of approximation results within this area. Based on an $\alpha$-scaled index distance, we first prove that two systems of $\alpha$-molecules are almost orthogonal. This leads to a general methodology to transfer approximation results within this framework, provided that certain consistency and time-frequency localization conditions of the involved systems of $\alpha$-molecules are satisfied. We finally utilize these results to enable the derivation of optimal sparse approximation results \msch{for} a specific class of cartoon-like functions by sufficient conditions on the 'control' parameters of a system of $\alpha$-molecules.