On the Oversampling of Affine Wavelet Frames

Abstract

The properties of oversampled affine frames are considered here with two main goals in mind. The first goal is to generalize the approach of Chui and Shi to the matrix oversampling setting for expanding, lattice-preserving dilations, whereby we obtain a new proof of the Second Oversampling Theorem for affine frames. The Second Oversampling Theorem, proven originally by Ron and Shen via Gramian analysis, states that oversampling an affine frame with dilation M by a matrix P will result in a frame with the same bounds (after renormalization) provided that P and M satisfy a certain relative primality condition. In this case, the matrix P is said to be admissible for M . The second goal of this work is to examine the compatibility of admissible oversampling with the refinable affine frames arising from a certain class of scaling functions. In this setting we show that oversampling dual affine systems by an admissible P preserves the multiresolution structure and, from this fact, that the oversampled systems remain dual. We then show that the admissibility of P is also sufficient to endow the dual oversampled systems with a discrete wavelet transform. The novelty of this work lies both in our approach to the Second Oversampling Theorem as well as our consideration of oversampling in the context of multiresolution analysis.

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

“…This result was later extended and generalized in other works (see, for instance, [4,5,[8][9][10]13,19,[21][22][23], including the second over-sampling theorem in [8]). …”

“…A lot of effort has been devoted to find over-sampling rates P for the purpose of tight-frame preservation over-sampling in the literature (see, for example, [5,[8][9][10]13,19,[21][22][23]25]). …”

“…Hence, the affine system X(Ψ, M, K) is indeed an over-sampled affine system obtained by over-sampling the affine system X(Ψ, M) with "over-sampling rates" governed by the over-sampling set K. We are particularly interested in sets of the form K = [0, 1) d ∩ P −1 Z d , where P , to be called the over-sampling rate of the affine system X(Ψ, M, K), is a nonsingular matrix with integer entries. For this choice of K, X(Ψ, M, K) becomes the familiar over-sampled affine system X P (Ψ, M) := |det M| j/2 ψ M j · − P −1 k : ψ ∈ Ψ, j ∈ Z, k ∈ Z d (2) with over-sampling rate P (see [5,8,19,21,22]). …”

Oversampling generates super-wavelets

“…We have, with But according to the proof of Theorem 3.2 in [9], this quantity is bigger than A T θ r f 2 = A f 2 = A T P θ r S f 2 . This yields the lower bound.…”

“…This type of oversampling was introduced by Chui and Shi in [6] for one dimension and scaling by 2. Since then, the result has been generalized and has become known as "the second oversampling theorem"; see [12,5,11,9]. For details on the history of the second oversampling theorem we refer to [9].…”