Characterizations of tight over-sampled affine frame systems and over-sampling rates

Let M be a dilation matrix, Ψ a finite family of L2-functions, and P the collection of all nonsingular matrices P such that M, P, and PMP−1 have integer entries. The objective of this paper is two-fold. First, for each P in P, we characterize all tight affine frames X(Ψ,M) generated by Ψ such that the over-sampled affine systems XP(Ψ,M) relative to the “over-sampling rate” P remain to be tight frames. Second, we characterize all over-sampling rates P∈P, such that the over-sampled affine systems XP(Ψ,M) are tight frames whenever the affine system X(Ψ,M) is a tight frame. Our second result therefore provides a general and precise formulation of the second over-sampling theorem for tight affine frames.