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Analytic wavelets generated by radial functions

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Abstract

In this paper, we deal with a class of non-stationary multiresolution analysis and wavelets generated by certain radial basis functions. These radial basis functions are noted for their effectiveness in terms of “projection”, such as interpolation and least-squares approximation, particularly when the data structure is scattered or the dimension of ℝs is large. Thus projecting a functionf onto a suitable multiresolution space is relatively easy here. The associated multiresolution spaces approximate sufficiently smooth functions exponentially fast. The non-stationary wavelets satisfy the Littlewood-Paley identity so that perfect reconstruction of wavelet decompositions is achieved. For the univariate case, we give a detailed analysis of the time-frequency localization of these wavelets. Two numerical examples for the detection of singularities with analytic wavelets are provided.

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Authors and Affiliations

  1. Center for Approximation Theory, Department of Mathematics, Texas A&M University, 77843, College Station, TX, USA

    C. K. Chui & J. D. Ward

  2. Fachbereich Mathematik, Universität Duisburg, D-47048, Duisburg, Germany

    J. Stöckler

Authors
  1. C. K. Chui
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  2. J. Stöckler
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  3. J. D. Ward
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Additional information

Communicated by R.A. DeVore

Supported by NSF Grant #DMS-92-06928, Air Force Grant #F49620-92-J-0403DEF and NATO Grant #CR-900158.

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Chui, C.K., Stöckler, J. & Ward, J.D. Analytic wavelets generated by radial functions. Adv Comput Math 5, 95–123 (1996). https://doi.org/10.1007/BF02124736

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  • DOI: https://doi.org/10.1007/BF02124736

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