A Short Note on the Frame Set of Odd Functions

TL;DR

This paper proves that Gabor systems with odd functions and symplectic lattices of density 2^d cannot form a Gabor frame, extending previous results with a new algebraic approach that removes the separability assumption.

Contribution

It introduces a novel algebraic method using the ambiguity function and Wigner distribution to analyze Gabor frames with odd functions, removing the separability constraint.

Findings

Gabor systems with odd functions and certain lattice densities cannot be frames.

The approach exploits the relation between ambiguity function and Wigner distribution.

New restrictions are established on the frame set of odd functions.

Abstract

In this work we derive a simple argument which shows that Gabor systems consisting of odd functions of $d$ variables and symplectic lattices of density $2^d$ cannot constitute a Gabor frame. In the 1--dimensional, separable case, this is a special case of a result proved by Lyubarskii and Nes, however, we use a different approach in this work exploiting the algebraic relation between the ambiguity function and the Wigner distribution as well as their relation given by the (symplectic) Fourier transform. Also, we do not need the assumption that the lattice is separable and, hence, new restrictions are added to the full frame set of odd functions.