Multiwindow Gabor frames on discrete periodic sets

In this paper, we study multiwindow discrete Gabor $(M-D-G)$ systems \(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.