Applications of Krein's theory of regular symmetric operators to sampling theory**Research partially supported by CONACYT under project P42553F.

Abstract

The classical Kramer sampling theorem establishes general conditions that allow the reconstruction of functions by mean of orthogonal sampling formulae. One major task in sampling theory is to find concrete, non-trivial realizations of this theorem. In this paper, we provide a new approach to this subject on the basis of Krein's theory of representation of simple regular symmetric operators having deficiency indices (1, 1). We show that the resulting sampling formulae have the form of Lagrange interpolation series. We also characterize the space of function reconstructible by our sampling formulae. Our construction allows a rigorous treatment of certain ideas proposed recently in quantum gravity.