SHANNON SAMPLING ON MANIFOLDS AND GRAPHS

One of the most interesting properties of the so called bandlimited functions
(=Palley-Wiener functions), i. e. functions whose Fourier transform has compact
support, is that they are uniquely determined by their values on some countable sets
of points and can be reconstructed from such values in a stable way. The sampling
problem for band limited functions had attracted attention of many mathematicians.
The mathematical theory of reconstruction of band limited functions from discrete
sets of samples was introduced to the world of signal analysis and information
theory by Shannon. Later the concept of bandlimitedness and the Sampling Theorem
became the theoretical foundation of many branches of the information theory.
In the talk I will show how these ideas can be extended to the setting of Riemannian
manifolds and combinatorial graphs. It is an active field of research which
found numerous applications in machine learning, astrophysics, and statistics.