To compute the optimal interpolation formula

Abstract

The purpose of this paper is to explain how to compute the function Ω \Omega which interpolates values of a function of one variable f ( x ) f(x) at n distinct points x 1 > x 2 > ⋯ > x n − 1 > x n {x_1} > {x_2} > \cdots > {x_{n - 1}} > {x_n} and which, whenever ‖ f ( k ) ‖ ∞ \left \|{f^{(k)}}\right \|_\infty is bounded and the value of the bound is unknown, provides the smallest possible value of B ( x ) B(x) in the error bound \[ | f ( x ) − Ω ( x ) | ⩽ B ( x ) ‖ f ( k ) ‖ ∞ . |f(x) - \Omega (x)| \leqslant B(x)\left \|{f^{(k)}}\right \|_{\infty .} \]