%0 Journal Article
%T Mean-Value Theorems for Harmonic  Functions on the Cube in  R<sup><i>n</i></sup>
%A Petar Petrov
%J Advances in Pure Mathematics
%P 683-688
%@ 2160-0384
%D 2015
%I Scientific Research Publishing
%R 10.4236/apm.2015.511062
%X Let <img alt=\"\" src=\"Edit_153341d9-1cde-47e7-adc1-623bdecae434.jpg\" />be a hypercube in R<sup>n</sup>. We prove theorems concerning mean-values of harmonic and polyharmonic functions on <em>I</em><sub><em>n</em></sub>(<em>r</em>), 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¡ªthe problem of best canonical one-sided L1-approximation by harmonic functions on <em>I</em><sub><em>n</em></sub>(<em>r</em>).
%K Harmonic Functions
%K Polyharmonic Functions
%K Hypercube
%K Quadrature Domain
%K Best One-Sided Approximation
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=59617