Spark deficient Gabor frames

TL;DR

This paper investigates the existence of full spark Gabor frames over finite abelian groups, proving their non-existence in non-cyclic groups and analyzing eigenvector properties related to spark deficiency.

Contribution

It extends the theory by resolving the non-cyclic case and shows that full spark Gabor frames cannot exist on finite abelian non-cyclic groups.

Findings

Full spark Gabor frames exist only on cyclic groups.

No full spark Gabor frames on finite abelian non-cyclic groups.

Eigenvectors of certain unitary matrices generate spark deficient frames.

Abstract

The theory of Gabor frames of functions defined on finite abelian groups was initially developed in order to better understand the properties of Gabor frames of functions defined over the reals. However, during the last twenty years the topic has acquired an interest of its own. One of the fundamental questions asked in this finite setting is the existence of full spark Gabor frames. The author proved the existence, as well as constructed such frames, when the underlying group is finite cyclic. In this paper, we resolve the non-cyclic case; in particular, we show that there can be no full spark Gabor frames of windows defined on finite abelian non-cyclic groups. We also prove that all eigenvectors of certain unitary matrices in the Clifford group in odd dimensions generate spark deficient Gabor frames. Finally, similarities between the uncertainty principles concerning the finite dimensional Fourier transform and the short-time Fourier transform are discussed.