Generalized Vrctor-valued Multi-window Gabor Frames Associated With Periodic Subsets Of The Real Line

Finite-length vector-valued frames(also called super frames)have significant application-s in multiplexing.And this notion was first introduced by Balan.Since then,vector-valued wavelet and Gabor frames in L2(R,CL)have interested many mathematicians.In recent years,some progress on vector-valued wavelet and Gabor frames in subspaces of L2(R,CL)has been achieved.But we have seen no progress on vector-valued multi-window Gabor frames with arbitrary length,even if restricted on the whole space L2(R,l2(L)).This paper focuses on arbitrary-length vector-valued multi-window Gabor frames under the setting of L2(S,l2(L))with S being a shift-periodic subset of R.L2(S,l2(L))models the periodic and intermittent signal space.Under the setting of L2(S,l2(1L)),we characterize complete multi-window Gabor systems,multi-window Gabor frames,Riesz bases,and type I and II Gabor duals of multi-window Gabor frames.And we also obtain an explicit expression of the cannonical Gabor dual.