Vol. 294, No. 1, 2018

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ISSN: 0030-8730
Spark deficient Gabor frames

Romanos-Diogenes Malikiosis

Vol. 294 (2018), No. 1, 159–180
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 on the existence of full spark Gabor frames. In a previous paper, we proved the existence of such frames when the underlying group is finite cyclic, and constructed some examples. In this paper, we resolve the noncyclic case; in particular, we show that there can be no full spark Gabor frames of windows defined on finite abelian noncyclic 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.

Keywords
Gabor frames, full spark, finite Weyl–Heisenberg groups, Clifford group, short-time Fourier transform, uncertainty principles
Mathematical Subject Classification 2010
Primary: 15A03, 42C15
Secondary: 11E95
Milestones
Received: 1 March 2016
Revised: 13 September 2017
Accepted: 14 September 2017
Published: 5 January 2018
Authors
Romanos-Diogenes Malikiosis
Institut für Mathematik
Technische Universität Berlin
Berlin
Germany