**ABSTRACT:**

Polysplines, christened and promoted by

Ognyan Kounchev [1], are natural multivariate

analogues of classical univariate piecewise polynomial

splines.

In this work we establish properties of

harmonic polysplines that interpolate

arbitrary distributions defined on parallel

hyperplanes.

In particular we show the following:

(a) Such polysplines provide a

solution to the natural minimization problem

involving the class of distributions whose

gradient is square integrable. (b)

In the case when the data are polynomials

such polysplines coincide with harmonic

polynomials on the slabs which are determined

by the parallel hyperplanes. \\

\noindent

[1] O. Kounchev,

{\em Multivariate Polysplines: Applications to Numerical

and Wavelet Analysis}, Academic Press,

San Diego, 2001.

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