On Polyharmonic Interpolation
It provides a new theoretical framework for multivariate interpolation using higher-order elliptic equations, offering rigorous convergence guarantees for smooth functions.
This paper introduces polyharmonic interpolation for multivariate functions, proving that any C∞ or analytic function in a ball can be approximated by polyharmonic functions of a given order, with explicit error bounds depending on the order and radius.
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 $p.$