Multivariate Anisotropic Translation Invariant Spaces on the Torus

In my PhD thesis Bergmann-2013-1 we investigated translation invariant spaces on anisotropic lattices, a corresponding fast Fourier transform as well as periodic wavelets and a fast wavelet transform. The corresponding wavelets are anisotropic and they are obtained by sampling smooth functions in the Fourier domain and hence have certain localization properties in time. Especially a generalization of the de la Vallée Poussin mean to the anisotropic multivariate case is derived and their corresponding wavelets are derived.

Furthermore the software package was developed within Mathematica and later transcribed to Matlab in order to be used within the homogenization project.

References

  1. Bergmann-2013-1
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    Bergmann, R.2013Translationsinvariante Räume multivariater anisotroper Funktionen auf dem Torus
    Dissertation, german, Universität zu Lübeck similarily: Shaker Verlag, ISBN 978-3844022667, 2013
  2. Bergmann-2013-2
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    Bergmann, R.2013The fast Fourier transform and fast wavelet transform for patterns on the torus
    Applied and Computational Harmonic Analysis35139–51
  3. BergmannPrestin-2014
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    Bergmann, R., Prestin, J.2014Multivariate anisotropic interpolation on the torus
    in: Fasshauer, G., Schumaker, L.: Approximation Theory XIV: San Antonio 201327–44 arXiv 1309.3432
  4. BergmannPrestin-2015
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    Bergmann, R., Prestin, J.2015Multivariate periodic wavelets of de la Vallée Poussin type
    Journal of Fourier Analysis and Applications212342–369
  5. LangemannPrestin2009 Prestin, J., Langemann, D.2010Multivariate periodic wavelet analysis
    Applied Computational Harmonic Analysis28146–66