ACTA issues

Characterization of Schauder basis property of Gabor systems in local fields

Biswaranjan Behera, Md. Nurul Molla

Acta Sci. Math. (Szeged) 87:3-4(2021), 517-539

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 \times \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 $|Zg|^2$ is an $A_2$ weight on $\D \times \D $. 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 $L^2(K)$.

DOI: 10.14232/actasm-021-120-8

AMS Subject Classification (1991): 43A70; 42B25, 43A25

Keyword(s): local field, Gabor system, Zak transform, $A_p$-weight, Schauder basis

received 20.1.2021, accepted 20.3.2021. (Registered under 120/2021.)