Interpolation by division

Abstract

Employing a generalization of the remainder theorem for polynomials P nx , we obtain by division both polynomial and rational interpolants for P nx , including all osculatory forms. Ordinary and non-linear divided differences of P nx are found as by-products. Applied to P n ( O) when O is an operator, we obtain “operational polynomial interpolants”, which for O ≡ Δ (difference operator) or O ≡ D (derivative operator) reproduce the operation of P n ( O) upon polynomial-exponential functions.