Gabor-type expansions on local fields

Abstract

In this paper, we introduce Gabor-type systems on local fields and study that any square integrable function can be expanded with respect to these systems. We establish necessary and sufficient conditions for two families of Gabor functions $$\Psi =\{\psi _{1},\psi _{2},\dots , \psi _{L}\}$$ and $$\tilde{\Psi }=\{\tilde{\psi }_{1},\tilde{\psi }_{2},\dots , \tilde{\psi }_{L}\}$$ in $$L^2(K)$$ to yield a reproducing identity in $$L^2(K)$$ . Precisely, we give complete characterizations of orthogonal/bi-orthogonal, tight frames and orthonormal bases of Gabor systems on local fields by means of simple equations in frequency domain.