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 di lation factor a > 1, that are generated by a family of finitely many functi ons 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 appl ication, we give an example of tight frames generated by one single L-2 fun ction for an arbitrary dilation a > 1 that possess "good" time-frequency lo calization. As another application, we also show that there does not exist an orthonormal wavelet with good time-frequency localization when the dilat ion factor a > 1 is irrational such that a(j) remains irrational for any po sitive integer j. This answers a question in Daubechies' Ten Lectures book for almost all irrational dilation factors. Other applications include a ge neralization of the notion of s-elementary wavelets of Dai and Larson to s- elementary wavelet families with arbitrary dilation factors a > 1. Generali zation to dual frames is also discussed in this paper. (C) 2000 Academic Pr ess.