An Interpolation Series for Integral Functions

Abstract

1. The Gontcharoff interpolation serieswherehas been studied in various special cases. For example, if an = a0 (all n), (1.0) reduces to the Taylor expansion of F(z). If an = (−1)n, J. M. Whittaker showed that the series (1.0) converges to F(z) provided F(z) is an integral function whose maximum modulus satisfiesthe constant ¼π being the “best possible”. In the case |an| ≤ 1, I have shown that the series converges to F(z) provided F(z) is an integral function whose maximum modulus satisfiesand that while ·7259 is not the “best possible” constant here, it cannot be replaced by a number as great as ·7378.