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Howard Weiss – One of the best experts on this subject based on the ideXlab platform.

  • How Smooth is Your Wavelet? Wavelet Regularity via Thermodynamic Formalism
    Communications in Mathematical Physics, 2008
    Co-Authors: Mark Pollicott, Howard Weiss
    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.

Mark Pollicott – One of the best experts on this subject based on the ideXlab platform.

  • How Smooth is Your Wavelet? Wavelet Regularity via Thermodynamic Formalism
    Communications in Mathematical Physics, 2008
    Co-Authors: Mark Pollicott, Howard Weiss
    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.

  • How Smooth is Your Wavelet? Wavelet Regularity via Thermodynamic Formalism
    Communications in Mathematical Physics, 2008
    Co-Authors: Mark Pollicott, Howard M. Weiss
    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.

Williams John – One of the best experts on this subject based on the ideXlab platform.

  • Wavelet-Galerkin solution of boundary value problems
    Archives of Computational Methods in Engineering, 2015
    Co-Authors: Amaratunga Kevin, Williams John
    Abstract:

    In this paper we review the application of Wavelets to the solution of partial differential equations. We consider in detail both the single scale and the multiscale Wavelet Galerkin method. The theory of Wavelets is described here using the language and mathematics of signal processing. We show a method of adapting Wavelets to an interval using an extrapolation technique called Wavelet Extrapolation. Wavelets on an interval allow boundary conditions to be enforced in partial differential equations and image boundary problems to be overcome in image processing. Finally, we discuss the fast inversion of matrices arising from differential operators by preconditioning the multiscale Wavelet matrix. Wavelet preconditioning is shown to limit the growth of the matrix condition number, such that Krylov subspace iteration methods can accomplish fast inversion of large matrices.