Gabor Functional Multiplier in the Higher Dimensions

TL;DR

This paper investigates conditions under which a function acts as a Gabor frame multiplier in higher dimensions, proving a key conjecture in the two-dimensional case that was previously only established in one dimension.

Contribution

It proves that the necessary condition for a function to be a Gabor frame multiplier holds in two dimensions, advancing understanding of Gabor frame multipliers in higher-dimensional spaces.

Findings

The necessary condition for Gabor frame multipliers is valid in 2D.

The conjecture is fully proven in the two-dimensional setting.

This extends previous one-dimensional results to higher dimensions.

Abstract

For two given full-rank lattices $\mathcal{L}=A\mathbb{Z}^d$ and $\mathcal{K}=B\mathbb{Z}^d$ in $\mathbf{R}^d$, where $A$ and $B$ are nonsingular real $d\times d$ matrices, a function $g(\bf{t})\in L^2(\mathbf{R}^d)$ is called a Parseval Gabor frame generator if $\sum_{\bf{l},\bf{k}\in\mathbb{Z}^d}|\langle f, {e^{2\pi i\langle B\bf{k},\bf{t}\rangle}}g(\bf{t}-A\bf{l})\rangle|^2=\|f\|^2$ holds for any $f(\bf{t})\in L^2(\mathbf{R}^d)$. It is known that Parseval Gabor frame generators exist if and only if $|\det(AB)|\le 1$. A function $h\in L^{\infty}(\mathbf{R}^d)$ is called a functional Gabor frame multiplier if it has the property that $hg$ is a Parseval Gabor frame generator for $L^2(\mathbf{R}^d)$ whenever $g$ is. It is conjectured that an if and only if condition for a function $h\in L^{\infty}(\mathbf{R}^d)$ to be a functional Gabor frame multiplier is that $h$ must be unimodular and $h(\bf{x})\overline{h(\bf{x}-(B^T)^{-1}\bf{k})}=h(\bf{x}-A\bf{l})\overline{h(\bf{x}-A\bf{l}-(B^T)^{-1}\bf{k})},\ \forall\ \bf{x}\in \mathbf{R}^d$ {\em a.e.} for any $\bf{l},\bf{k}\in \mathbb{Z}^d$, $\bf{k}\not=\bf{0}$. The if part of this conjecture is true and can be proven easily, however the only if part of the conjecture has only been proven in the one dimensional case to this date. In this paper we prove that the only if part of the conjecture holds in the two dimensional case.