In this paper we consider a class of functions $f(z)$ ($z\in\C$) meromorphic in the half-plane $\Real z >= 1/2$, holomorphic in $0 < \Real z < 1/2$, continuous on $\Real z = 0$, and satisfying a suitable Carlson-type asymptotic growth condition. First we prove that position and 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 $f(x+1/2)$ ($x=\Real z$) 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.
The interpolation formula for a class of meromorphic functions
De Micheli Enrico;
2013
Abstract
In this paper we consider a class of functions $f(z)$ ($z\in\C$) meromorphic in the half-plane $\Real z >= 1/2$, holomorphic in $0 < \Real z < 1/2$, continuous on $\Real z = 0$, and satisfying a suitable Carlson-type asymptotic growth condition. First we prove that position and 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 $f(x+1/2)$ ($x=\Real z$) 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
