TITLE:
Mean-Value Theorems for Harmonic Functions on the Cube in Rn
AUTHORS:
Petar Petrov
KEYWORDS:
Harmonic Functions, Polyharmonic Functions, Hypercube, Quadrature Domain, Best One-Sided Approximation
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.5 No.11,
September
16,
2015
ABSTRACT: Let be a hypercube in Rn. We prove theorems concerning mean-values of harmonic and polyharmonic functions on In(r), which can be considered as natural analogues of the famous Gauss surface and volume mean-value formulas for harmonic functions on the ball in and their extensions for polyharmonic functions. We also discuss an application of these formulas—the problem of best canonical one-sided L1-approximation by harmonic functions on In(r).