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

Abstract

Let M be a dilation matrix, Ψ a finite family of L 2 -functions, and P the collection of all nonsingular matrices P such that M, P, and P M P −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 X P ( 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 X P ( 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.