On the rate of convergence of Berrut's interpolant at equally spaced nodes
Provides a sharp asymptotic error description for a class of practical functions, advancing theoretical understanding of barycentric interpolation.
The authors prove conjectures about the convergence rate of Berrut's barycentric interpolant at equally spaced nodes, showing an error order of 1/n for functions with derivatives of bounded variation.
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.