Sampling theory on Bernstein spaces

Abstract

Sampling theory is studied by many people because of its mathematical interest and because of its importance for applications in the engineering. The main concerns of sampling theory are the followings: the space of functions which can be expressed as a series, sampling points, convergence, error estimation. Sampling theory is based on Shannon-Whittaker-Kotel`nikov sampling theorem. In this thesis, the main topic is to survey the important results of the sampling theory in Bernstein space. Bernstein space is closely related with Paley-Wiener space and a function of Paley-Wiener space is represented by some Fourier transform of functions with compact support. Kramer`s lemma gives some generalization for sampling formula for functions represented by some integral transform. And we give a convergence principle by the properties of Paley-Wiener space. And we survey the important results for irregular sampling with nonuniform sample. These results are developed by the theory of Riesz basis and we get a sampling formula that is similar to Lagrange interpolation formula. Finally, we will give a brief introduction for 1-channel and 2-channel sampling with some transformed data.