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

Abstract

For functions defined on the $n$-dimensional hypercube $I_n(r)=\{\mathbf{x}\in {\mathbb{R}}^n||x_i|\le r,i=1,2,\dots ,n\}$ 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 $I_n(r)$.