Orthonormal wavelets and tight frames with arbitrary real dilations

The objective of this paper is to establish a complete characterization of tight frames, and particularly of orthonormal wavelets, for an arbitrary dilation factor a > 1, that are generated by a family of finitely many functions in L-2:= L-2(R). This is a generalization of the fundamental work of G. Weiss and his colleagues who considered only integer dilations. As an application, we give an example of tight frames generated by one single L-2 function for an arbitrary dilation a > 1 that possess "good" time-frequency localization. As another application, we also show that there does not exist an orthonormal wavelet with good time-frequency localization when the dilation factor a > 1 is irrational such that a(j) remains irrational for any positive integer j. This answers a question in Daubechies' Ten Lectures book for almost all irrational dilation factors. Other applications include a generalization of the notion of s-elementary wavelets of Dai and Larson to s-elementary wavelet families with arbitrary dilation factors a > 1. Generalization to dual frames is also discussed in this paper. (C) 2000 Academic Press.