Orthonormal Wavelets and Tight Frames with Arbitrary Real Dilations

✍ Scribed by Charles K Chui; Xianliang Shi

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 170 KB
• Volume: 9
• Category: Article
• ISSN: 1063-5203
• DOI: 10.1006/acha.2000.0316

✦ Synopsis

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.