On Gabor orthonormal bases over finite prime fields

Abstract

We study Gabor orthonormal windows in \(L^2({\Bbb Z}_p^d)\) for translation and modulation sets \(A\) and \(B\), respectively, where \(p\) is prime and \(d\geq 2\). We prove that for a set \(E\subset \Bbb Z_p^d\), the indicator function \(1_E\) is a Gabor window if and only if \(E\) tiles and is spectral. Moreover, we prove that for any function \(g:\Bbb Z_p^d\to \Bbb C\) with support \(E\), if the size of \(E\) coincides with the size of the modulation set \(B\) or if \(g\) is positive, then \(g\) is a unimodular function, i.e., \(|g|=c1_E\), for some constant \(c>0\), and \(E\) tiles and is spectral. We also prove the existence of a Gabor window \(g\) with full support where neither \(|g|\) nor \(|\hat g|\) is an indicator function and \(|B|<<p^d\). We conclude the paper with an example and open questions.