Frames and system of translates of Hilbert C⁎-module valued functions

Abstract

In this paper, we investigate three types of problems dealing with frames in L2(Rn,H), where H denotes a Hilbert C⁎-module. First, we obtain a characterization for the system of translates {Tkϕ:k∈Z} to be a frame sequence. Then we investigate the system of generalized translates GJ associated with a non-singular matrix D. Here we obtain a characterization for GJ to be a frame for L2(Rn,H). Given two systems of such translates, we obtain a necessary and sufficient condition for such systems to form a pair of dual frames. Finally, we study semi-discrete frames for L2(Rn,H), where H is a separable Hilbert space.