Characterization of Schauder basis property of Gabor systems in local fields

Abstract

Let K be a totally disconnected, locally compact and nondiscrete field of positive characteristic and D be its ring of integers. We characterize the Schauder basis property of the Gabor systems in K in terms of A(2) weights on D x D and the Zak transform Zg of the window function g that generates the Gabor system. We show that the Gabor system generated by g is a Schauder basis for L-2(K) if and only if vertical bar Zg vertical bar(2) is an A 2 weight on DxD. Some examples are given to illustrate this result. Moreover, we construct a Gabor system which is complete and minimal, but fails to be a Schauder basis for L-2 (K).