Interpolation formulas on the unit circle for $1$-harmonic functions

Abstract

A generalization of the deeply investigated harmonic functions, known as $\alpha$-harmonic functions, have recently gained considerable attention. Similarly to the harmonic functions, an $\alpha$-harmonic function $u$ on the unit disc $\mathbb D$ is uniquely determined by its values on the boundary of the disc $\partial\mathbb D$. In fact, for any $z\in\mathbb D$, the value of $u(z)$ can be given as a contour integral over $\partial\mathbb D$ with a modified Poisson kernel. However, this integral can be difficult to evaluate, or the values on the boundary are known only empirically. In such cases, approximating $u(z)$ with an interpolatory formula, as a weighted sum of values of $u$ at $n$ nodes on $\partial\mathbb D$, can be an attractive alternative. The nodes and weights are to be chosen so that the degree $d$ of exactness of the formula is maximized. In other words, the formula should be exact for all basis functions for $\alpha$-harmonic functions of degree up to $d$, with $d$ as large as possible. In the case of harmonic functions, it is known that there is an interpolation formula of degree of exactness as large as $d=n-1$. The objective of this paper are formulas of this type for $\alpha$-harmonic functions. We will prove that, given $n$, in this case the degree of exactness cannot be $n-1$, but there is a unique interpolation formula of degree $n-2$. Finally, we will prove convergence of such formulas to $u(z)$ as $n\to\infty$.