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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).

Mean-Value Theorems for Harmonic Functions on the Cube in R&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;

Mean-Value Theorems for Harmonic Functions on the Cube in R<sup><i>n</i></sup>