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 [Proc. Amer. Math. Soc., 121 (1994), pp. 511--517], [SIAM J. Math. Anal., 28 (1997), pp. 213--232] 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 [J. Funct. Anal., 148 (1997), pp. 408--447] 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.