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.
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References
Cohen A., Conze J.-P.: Régularité des bases d’ondelettes et mesures ergodiques. Rev. Mat. Iberoamericana 8, 351–365 (1992)
Cohen A., Daubechies I.: A new technique to estimate the regularity of refinable functions. Rev. Mat. Iberoamericana 12, 527–591 (1996)
Daubechies, I.: Ten lectures on wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1992
Daubechies, I.: Using Fredholm determinants to estimate the smoothness of refinable functions. Approximation theory VIII, Vol. 2 (College Station, TX, 1995), Ser. Approx. Decompos. 6 River Edge, NJ: World Sci. Publishing, 1995, pp. 89–112
Daubechies I.: Orthonormal Bases of Compactly Supported Wavelets II. SIAM J. Math. Anal. 24, 499–519 (1993)
Eirola T.: Sobolev characterization of solutions of dilation equations. SIAM J. Math. Anal. 23, 1015–103 (1992)
Fan A., Sun Q.: Regularity of Butterworth Refinable Functions. Asian J. Math. 5(3), 433–440 (2001)
Gohberg I., Goldberg S., Kaashoek M.A.: Classes of linear operators. Vol 1. Birthauser Verlag, Basel (1990)
Halmos, P.: Introduction to Hilbert space and the theory of spectral multiplicity. Reprint of the Second (1957) edition, Providence, RI: AMS/Chelsea Publishing, 1998
Hervé L.: Comportement asymptotique dans l’algorithme de transformée en ondelettes. Rev. Mat. Iberoamericana 11, 431–451 (1985)
Hervé L.: Construction et régularité des fonctions dćhelle. SIAM J. Math. Anal. 26, 1361–1385 (1995)
Jenkinson O., Pollicott M.: Computing invariant densities and metric entropy. Commun. Math. Phys. 211, 687–703 (2000)
Jenkinson O., Pollicott M.: Orthonormal expansions of invariant densities for expanding maps. Adv. in Math. 192, 1–34 (2005)
Ojanen H.: Orthonormal compactly supported wavelets with optimal sobolev regularity. Applied and Computational Harmonic Analysis 10, 93–98 (2001)
Oppenheim, A., Shaffer, R.: Digital Signal Processing. Upper Saddie River, NJ: Pearson Higher Education, 1986
Parry, W., Pollicott, M.: Zeta Functions and the Periodic Orbit Structures of Hyperbolic Dynamics. Astérisque 187–188 (1990)
Pollicott M.: Meromorphic extensions for generalized zeta functions. Invent. Math 85, 147–164 (1986)
Ruelle, D.: Thermodynamic Formalism. Reading, MA: Addison-Wesley, 1978
Ruelle D.: An extension of the theory of Fredholm determinants. Inst. Hautes Etudes Sci. Publ. Math. 72, 175–193 (1990)
Simon, B.: Trace ideals and their applications. LMS Lecture Note Series, Vol 35, Cambridge: Cambridge University Press, 1979
Sun Q.: Sobolev exponent estimate and asymptotic regularity of M band Daubechies scaling function. Constructive Approximation 15, 441–465 (1999)
Tangerman, F.: Meromorphic continuation of Ruelle zeta functions. Ph.D. Thesis, Boston University, 1988
Villemoes L.: Energy moments in time and frequency for two-scale difference equation solutions and wavelets. SIAM J. Math. Anal. 23, 1519–1543 (1992)
Matlab, Wavelet Toolbox Documentation. http://www.mathworks.com/access/helpdesk/help/tool-box/wavelet/
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Communicated by J.L. Lebowitz
The work of the second author was partially supported by a National Science Foundation grant DMS-0355180.
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Pollicott, M., Weiss, H. How Smooth is Your Wavelet? Wavelet Regularity via Thermodynamic Formalism. Commun. Math. Phys. 281, 1–21 (2008). https://doi.org/10.1007/s00220-008-0457-x
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DOI: https://doi.org/10.1007/s00220-008-0457-x