p-frames and shift invariant subspaces of L-P

In this article we investigate the frame properties and closedness for the shift invariant space [GRAPHICS] We derive necessary and sufficient conditions for an indexed family {phi (i)( - j) : 1 less than or equal to i less than or equal to r, j is an element of Z(d)} to constitute a p-frame for V-p (Phi), and to generate a closed shift invariant subspace of L-p. A function in the L-p-closure of V-p(Phi) is not necessarily generated by l(p) coefficients. Hence we often hope that V-p(Phi) itself is closed, i.e., a Banach space. For p not equal 2, this issue is complicated, but we show that under the appropriate conditions on the frame vectors, there is an equivalence between the concept of p-frames, Banach frames, and the closedness of the space they generate. The relation between a function f is an element of V-p(Phi) and the coefficients of its representations is neither obvious, nor unique, in general. For the case of p-frames, we are in the context of normed linear spaces. but we are still able to give a characterization of p-frames in terms of the equivalence between the norm of f and an l(p)-norm related to its representations. A Banach frame does not have a dual Banach frame in general, however; for the shift invariant spaces V-p (Phi), dual Banach frames exist and can be constructed.