On the Convergence of Kergin and Hakopian Interpolants at Leja Sequences for the Disk

TL;DR

This paper proves that Kergin and Hakopian interpolation polynomials at Leja sequences for the unit disk converge uniformly to smooth functions, including their derivatives, providing theoretical guarantees for these interpolation methods.

Contribution

It establishes the convergence of Kergin and Hakopian interpolants at Leja sequences for the disk, including derivative convergence for smooth functions.

Findings

Kergin and Hakopian interpolants converge uniformly to the target function.

All derivatives of the interpolants converge uniformly when the function is smooth.

The results apply to functions in a neighborhood of the disk.

Abstract

We prove that Kergin interpolation polynomials and Hakopian interpolation polynomials at the points of a Leja sequence for the unit disk $D$ of a sufficiently smooth function $f$ in a neighbourhood of $D$ converge uniformly to $f$ on $D$. Moreover, when $f$ is $C^\infty$ on $D$, all the derivatives of the interpolation polynomials converge uniformly to the corresponding derivatives of $f$.