
# 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 sampling smooth functions in the Fourier domain and hence have certain localization properties in time. Especially a generalization of the de la Vallee Poussin mean to the anisotropic multivariate 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. [1]

# References

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.

1. Bergmann, R and Prestin, J (2015). Multivariate periodic wavelets of de la Vallée Poussin type. Journal of Fourier Analysis and Applications. 21 342–69
2. Bergmann, R (2013). The fast Fourier transform and fast wavelet transform for patterns on the torus. Applied and Computational Harmonic Analysis. 35 39–51

[1]

1. Bergmann, R and Prestin, J (2014). Multivariate anisotropic interpolation on the torus. Approximation Theory XIV: San Antonio 2013. 27–44