Lidstone interpolation I. One variable

TL;DR

This paper reviews Lidstone interpolation for a single variable, detailing how entire functions of exponential type less than pi are determined by even derivatives at two points, and sets the stage for multivariate generalizations.

Contribution

It provides a comprehensive survey and complete proofs of univariate Lidstone interpolation theory, forming the foundation for future multivariate extensions.

Findings

Complete proofs of univariate Lidstone interpolation theory

Clarification of how functions are determined by derivatives at specific points

Framework for extending to multivariate Lidstone interpolation

Abstract

According to Lidstone interpolation theory, an entire function of exponential type $<\pi$ is determined by it derivatives of even order at $0$ and $1$. This theory can be generalized to several variables. Here we survey the theory for a single variable. Complete proofs are given. This first paper of a trilogy is devoted to Univariate Lidstone interpolation; Bivariate and Multivariate Lidstone interpolation will be the topic of two forthcoming papers.