The interpolation formula for a class of meromorphic functions

In this paper we consider a class of functions f (z) (z is an element of C) meromorphic in the half-plane Re z >= 1/2, holomorphic in 0 < Re z < 1/2, continuous on Re z = 0, and satisfying a suitable Carlson-type asymptotic growth condition. First we prove that the position and the residue of the poles of f (z) can be obtained from the samples of f (z) taken at the positive half-integers. In particular, the positions of the poles are shown to be the roots of an algebraic equation. Then we give an interpolation formula for f (x + 1/2) (x = Re z) that incorporates the information on the poles (i.e., position and residue) and which is proved to converge to the true function uniformly on x >= x(0) > -1/2 as the number of samples tends to infinity and the error on the samples goes to zero. An illustrative numerical example of interpolation of a Runge-type function is also given. (c) 2013 Elsevier Inc. All rights reserved.