Interpolation of α-harmonic functions on a circle

Abstract

Similarly to harmonic functions, an $\alpha$-harmonic function $u$ on the unit disc $D$ is uniquely determined by its values on the boundary of the disc $\partial D$. Depending on $\alpha$, we investigate interpolatory formulas for approximating $u\left(\zeta \right)$ for any given $\zeta$, as a weighted sum of values of $u$ at $n$ nodes on $\partial D$. We show how to construct such formulas of highest possible algebraic degree of exactness and discuss their convergence.