Pollicott, Mark and Weiss, Howard. (2008) How smooth is your wavelet? Wavelet regularity via thermodynamic formalism. Communications in Mathematical Physics, Vol.281 (No.1). pp. 1-21. ISSN 0010-3616

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Abstract

A popular wavelet reference [W] states that "in theoretical and practical studies, the notion of (wavelet) regularity has been increasing in importance." Not surprisingly, the study of wavelet regularity is currently a major topic of investigation. Smoother wavelets provide sharper frequency resolution of functions. Also, the iterative algorithms to construct wavelets converge faster for smoother wavelets. The main goals of this paper are to extend, refine, and unify the thermodynamic approach to the regularity of wavelets and to devise a faster algorithm for estimating regularity. The thermodynamic approach works equally well for compactly supported and non-compactly supported wavelets, and also applies to non-analytic wavelet filters. We present an algorithm for computing the Sobolev regularity of wavelets and prove that it converges with super-exponential speed. As an application we construct new examples of wavelets that are smoother than the Daubechies wavelets and have the same support. We establish smooth dependence of the regularity for wavelet families, and we derive a variational formula for the regularity. We also show a general relation between the asymptotic regularity of wavelet families and maximal measures for the doubling map. Finally, we describe how these results generalize to higher dimensional wavelets.

Item Type:	Journal Article
Subjects:	Q Science > QC Physics
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	Communications in Mathematical Physics
Publisher:	Springer
ISSN:	0010-3616
Date:	July 2008
Volume:	Vol.281
Number:	No.1
Number of Pages:	21
Page Range:	pp. 1-21
Identification Number:	10.1007/s00220-008-0457-x
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
URI:	http://wrap.warwick.ac.uk/id/eprint/30035

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