Information transmission with automorphic functions

Departing from the classical example of the Shannon sampling theorem, which compresses the
information contained on a real function (with compactly supported Fourier transform-a so-called band-limited signal), in the set of integer numbers (a discrete subgroup of the real numbers), we
will see how discrete subgroups and their associated automorphic functions can be used to compress
the information of a function defined on a continuous space. This will lead us to Weierstrass
functions and lattices in the Euclidean case of the complex plane and to modular functions and Fuchsian groups in the richer case of the hyperbolic half-plane. This last correspondence can be lifted to the
eigenspaces of the Maass diferential equation, revealing a connection between wavelets, electron
distributions in higher Landau levels and Maass forms that remains to be fully understood.