Characterizations of Tight Over-sampled Affine Frame Systems and Over-sampling Rates

Let M be a dilation matrix, Ψ a finite family of L-functions, andP 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. Firstly, for each P in P , we characterize all tight affine frames X(Ψ, M) generated by Ψ such that the over-sampled affine systems X (Ψ, M) relative to the “over-sampling rate” P remain to be tight frames. Secondly, we characterize all over-sampling rates P ∈ P , such that the over-sampled affine systems X (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.