A note on mean-value properties of harmonic functions on the hypercube

TL;DR

This paper explores mean-value properties of harmonic functions on the hypercube, extending classical formulas and contributing to approximation theory within this geometric setting.

Contribution

It establishes analogues of Gauss mean-value formulas for harmonic functions on the hypercube and extends these results to polyharmonic functions.

Findings

Derived mean-value formulas for harmonic functions on the hypercube

Extended results to polyharmonic functions

Improved understanding of harmonic approximation on the hypercube

Abstract

For functions defined on the $n$-dimensional hypercube $I_n (r) = \{{\bm{x}} \in \mathbb{R}^n ~\vert~ \vert x_i \vert \le r,~ i = 1, 2, \ldots , n\}$ and harmonic therein, we establish certain analogues of Gauss surface and volume mean-value formulas for harmonic functions on the ball in $\mathbb{R}^n$ and their extensions for polyharmonic functions. In particular, our results contribute to the best one-sided $L^1$-approximation by harmonic functions on $I_n (r)$.