Gabor Orthonormal Bases with Maximal Localization and Gabor Frame Operator on Local Fields

We provide an explicit construction of a Gabor orthonormal bases for a local field K that provides maximal localization in both time and frequency. Such a localization is not true in case of R due to the uncertainty principle. In particular, we construct examples of functions f ∈ L²(K) such that the support of the ambiguity function of f is of minimum measure. Moreover, we establish a quantitative uncertainty principle for local fields, which follows as a consequence of Lieb’s inequalities for general locally compact abelian group. In addition, we develop fundamental operator representations for Gabor systems defined over local fields.