On the rate of convergence of Berrut's interpolant at equally spaced nodes

TL;DR

This paper analyzes the convergence rate of Berrut's barycentric interpolant at equally spaced nodes, proving conjectures and providing sharp asymptotic error descriptions for functions with absolutely continuous derivatives.

Contribution

It proves two conjectures about Berrut's interpolant and characterizes its asymptotic error for a broad class of functions, advancing understanding of its approximation properties.

Findings

Proved the first conjecture of Mastroianni and Szabados.

Provided a sharp asymptotic error description for functions with absolutely continuous derivatives.

Showed the interpolant has order of approximation 1/n for functions with derivatives of bounded variation.

Abstract

We extend the work by Mastroianni and Szabados regarding the barycentric interpolant introduced by J.-P. Berrut in 1988, for equally spaced nodes. We prove fully their first conjecture and present a proof of a weaker version of their second conjecture. More importantly than proving these conjectures, we present a sharp description of the asymptotic error incurred by the interpolants when the derivative of the interpolated function is absolutely continuous, which is a class of functions broad enough to cover most functions usually found in practice. We also contribute to the solution of the broad problem they raised regarding the order of approximation of these interpolants, by showing that they have order of approximation of order 1/n for functions with derivatives of bounded variation.