Interpolation of infinite sequences by entire functions (Q952801)

For an arbitrary infinite sequence of real numbers \(\{u_n\} (n \in \mathbb{Z}; u_n \in \mathbb{R})\), and a real strictly positive number \(R\), the authors constructively prove that there are uncountable numbers of interpolants \(u(x) (x \in \mathbb{R})\) satisfying: \(u(n) = u_n, \forall n \in \mathbb{Z}\) analytic over the real line and the radius of convergence of the series locally expressing \(u(x)\) is uniformly greater than \(R\). Further, if the sequence \(\{u_n\}\) is bounded, then there are entire interpolants \(u(x)\) uniformly satisfying the same bound, and such that all their derivatives are uniformly bounded over the real line. The application of the results in the stability and accuracy analysis of finite-difference methods is discussed.