On Polyharmonic Interpolation

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 $p.$