The characters of multiple dual pseudoframes with multiple refinable functions and a generalized multiresolution structure

The notion of a generalized multiresolution structure (GMS) of L 2 (R) is formulated. The definition of multiple pseudoframes for subspaces of L 2 (R) is presented. The construction of a GMS of Paley-Wiener subspaces of L 2 (R) is investigated. The sufficient condition for the existence of multiple pseudoframes for subspaces of L 2 (R) is obtained based on such a GMS. A necessary and sufficient condition on the existence of biorthogonal multiple vector-valued wavelets is presented by means of paraunitary vector filter bank theory.