  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
$p.$
