Vector-valued Gabor frames associated with periodic subsets of the real line

The notion of vector-valued frame (also called superframe) was first introduced by Balan in the context of multiplexing. It has significant applications in mobile communication, satellite communication, and computer area network. For vector-valued Gabor analysis, existent literatures mostly focus on L2(R,CL) instead of its subspace. Let a>0, and S be an aZ-periodic measurable set in R (i.e. S+aZ=S). This paper addresses Gabor frames in L2(S,CL) with rational time-frequency product. They can model vector-valued signals to appear periodically but intermittently. And the projections of Gabor frames in L2(R,CL) onto L2(S,CL) cannot cover all Gabor frames in L2(S,CL) if S≠R. By introducing a suitable Zak transform matrix, we characterize completeness and frame condition of Gabor systems, obtain a necessary and sufficient condition on Gabor duals of type I (resp. II) for a general Gabor frame, and establish a parametrization expression of Gabor duals of type I (resp. II). All our conclusions are closely related to corresponding Zak transform matrices. This allows us to easily realize these conclusions by designing the corresponding matrix-valued functions. An example theorem is also presented to illustrate the efficiency of our method.