Rationally sampled Gabor frames on the half real line

Abstract

Recently, Gabor analysis on locally compact abelian (LCA) groups has interested some mathematicians. The half real line R+=(0,∞) is an LCA group under multiplication and the usual topology, with the Haar measure dμ=dxx. This paper addresses rationally sampled Gabor frames for L2(R+,dμ). Given a function in L2(R+,dμ), we introduce a new Zak transform matrix associated with it, which is different from the conventional Zibulski-Zeevi matrix. It allows us to define a function by designing its Zak transform matrix. Using our Zak transform matrix method, we characterize and express complete Gabor systems, Bessel sequences, Gabor frames, Riesz bases and Gabor duals of an arbitrarily given Gabor frame for L2(R+,dμ), and prove the minimality of the canonical dual frames in some sense. Some examples are also provided to illustrate the generality of our theory.