2017/12/25 by Iosevich, A., Kolountzakis, M., Lyubarskii, Yu. +2 · 1 citation
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1712.09120
We study Gabor orthonormal windows in L2(\Bbb Zpd) for translation and modulation sets A and B, respectively, where p is prime and d≥ 2. We prove that for a set E⊂ \Bbb Zpd, the indicator function 1E is a Gabor window if and only if E tiles and is spectral. Moreover, we prove that for any function g:\Bbb Zpd→ \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|=c1E, 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 | g| is an indicator function and |B|<