Multiwindow Gabor frames on discrete periodic sets

In this paper, we study multiwindow discrete Gabor (M-D-G) systems \\G (g, L, M, N) on discrete periodic sets S and give some necessary and/or sufficient matrix-conditions for a M-D-G system in ² (S) to be a frame. We characterize, also, which M-D-G frames are Riesz bases by the parameters L, M and N. Matrix-characterizations of Parseval M-D-G frames and M-D-G orthonormal bases are also given. Then, we characterize the existence of M-D-G frames, M-D-G Parseval frames, M-D-G Riesz bases and M-D-G orthonormal bases for ² (S) by the parameters M, N and L. We prseent, also, a matrix-characterization of dual M-D-G frames in ² (S). A perturbation matrix-condition of M-D-G frames is also prsented. We, then, show that a pair of M-D-G Bessel systems can generate pairs of M-D-G dual frames. By the Zak-transform, characterizations of complete M-D-G systems and M-D-G frames in ² (Z) are given in the case of M=N and necssary conditions for a M-D-G system to be a Riesz basis/ orthonormal basis for ² (Z) are also given. We, also, study K-M-D-G frames in ² (S), where K B (² (S) \, ), and presente some sufficient matrix-conditions for a M-D-G system to form a K-frame and give a construction method of K-M-D-G frames which are not M-D-G frames and some examples.