Interpolation formulas for 1-harmonic functions on the unit circle

Abstract

Ageneralization of the deeply investigated harmonic functions, known as ?-harmonic functions, have recently gained considerable attention. Similarly to the harmonic functions, an ?-harmonic function u on the unit disc D is uniquely determined by its values on the boundary of the disc ?D. In fact, for any z ? D, the value of u(z) can be given as a contour integral over ?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 ?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 ?-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 ?-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 ? ?.