$\DeclareMathOperator{\arccosh}{arccosh} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator{\Exp}{Exp} \newcommand{\geo}[2]{\gamma_{\overset{\frown}{#1,#2}}} \newcommand{\geoS}{\gamma} \newcommand{\geoD}[2]{\gamma_} \newcommand{\geoL}[2]{\gamma(#2; #1)} \newcommand{\gradM}{\nabla_{\M}} \newcommand{\gradMComp}[1]{\nabla_{\M,#1}} \newcommand{\Grid}{\mathcal G} \DeclareMathOperator{\Log}{Log} \newcommand{\M}{\mathcal M} \newcommand{\N}{\mathcal N} \newcommand{\mat}[1]{\mathbf{#1}} \DeclareMathOperator{\prox}{prox} \newcommand{\PT}[3]{\mathrm{PT}_{#1\to#2}#3} \newcommand{\R}{\mathbb R} \newcommand{\SPD}[1]{\mathcal{P}(#1)} \DeclareMathOperator{\Tr}{Tr} \newcommand{\tT}{\mathrm{T}} \newcommand{\vect}[1]{\mathbf{#1}}$

# 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