Multivariate Anisotropic Translation Invariant Spaces on the Torus

In my PhD thesis (PDF, 4.5 MB, in german) I 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 smapling smooth functions in the Fourier domain and hence have certain lkocalization properties in time. Especially a generalization of the de la Vallee Poussin mean to the anisotropic mutlivariate case is derived and their corresponding wavelets are derived.

Furthermore a software package was developed within Mathematica and later transcribed to Matlab in order to be used within the homogenization project. (Bergmann, 2013)


  1. Bergmann, R. (2013). Translationsinvariante Räume multivariater anisotroper Funktionen auf dem Torus (Dissertation). Universität zu Lübeck. in german. Similarily: Shaker Verlag, ISBN 978-3844022667, 2013.

(Bergmann, 2013)(Bergmann & Prestin, 2015)

  1. Bergmann, R., & Prestin, J. (2015). Multivariate periodic wavelets of de la Vallée Poussin type. Journal of Fourier Analysis and Applications, 21(2), 342–369.
  2. Bergmann, R. (2013). The fast Fourier transform and fast wavelet transform for patterns on the torus. Applied and Computational Harmonic Analysis, 35(1), 39–51.

(Bergmann & Prestin, 2014)

  1. Bergmann, R., & Prestin, J. (2014). Multivariate anisotropic interpolation on the torus. In G. Fasshauer & L. Schumaker (Eds.), Approximation Theory XIV: San Antonio 2013 (pp. 27–44).

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