Linear independence of compactly supported separable shearlet systems

TL;DR

This paper proves that compactly supported separable shearlet systems are linearly independent, using a sampling strategy that leverages the structure of an associated oversampled wavelet system and the support shapes of shearlet elements.

Contribution

It provides the first proof of linear independence for this class of shearlet systems, advancing understanding of their mathematical properties.

Findings

Linear independence of compactly supported separable shearlet systems established.

Sampling strategy effectively utilized the structure of oversampled wavelet systems.

Supports of shearlet elements play a key role in the proof.

Abstract

This paper examines linear independence of shearlet systems. This property has already been studied for wavelets and other systems such as, for instance, for Gabor systems. In fact, for Gabor systems this problem is commonly known as the HRT conjecture. In this paper we present a proof of linear independence of compactly supported separable shearlet systems. For this, we employ a sampling strategy to utilize the structure of an implicitly given underlying oversampled wavelet system as well as the shape of the supports of the shearlet elements.