On Polyharmonic Interpolation

TL;DR

This paper introduces a novel multivariate interpolation method using polyharmonic functions, which are solutions to higher order elliptic equations, for smooth or analytic functions within a ball, and proves key approximation results.

Contribution

It presents a new approach to multivariate interpolation employing polyharmonic functions and establishes theoretical results for approximating smooth or analytic functions.

Findings

Proves approximation theorems for polyharmonic interpolation

Extends interpolation theory to higher order elliptic equations

Provides foundational results for multivariate interpolation methods

Abstract

In the present paper we will introduce a new approach to multivariate interpolation by employing polyharmonic functions as interpolants, i.e. by solutions of higher order elliptic equations. We assume that the data arise from $C^{\infty}$ or analytic functions in the ball $B_{R}.$ We prove two main results on the interpolation of $C^{\infty}$ or analytic functions $f$ in the ball $B_{R}$ by polyharmonic functions $h$ of a given order of polyharmonicity $p.$