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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 ∞ or analytic functions in the ball B R . We prove two main results on the interpolation of C ∞ or analytic functions f in the ball B R by polyharmonic functions h of a given order of polyharmonicity p.

Let m,n(N,l

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