_{1}

^{*}

Let
be a hypercube in R
^{n}. We prove theorems concerning mean-values of harmonic and polyharmonic functions on
*I*
_{n}(
*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
*I*
_{n}(
*r*).

This note is devoted to formulas for calculation of integrals over the n-dimensional hypercube centered at

and its boundary^{n}. Section 3 deals with the best one-sided L^{1}-approximation by harmonic functions.

Let us remind that a real-valued function f is said to be harmonic ( polyharmonic of degree

For any set

Let

The surface mean-value theorem. If

where

The volume mean-value theorem. If

The balls are known to be the only sets in

such that

open ball centered at

and there is a Borel measure

integrable harmonic function f on^{1}-approximation with respect to

Let us denote by

(see

and

and

The following holds true.

Theorem 1 If

(i) Surface mean-value formula for the hypercube

(ii) Volume mean-value formula for the hypercube

In particular, both surface and volume mean values of h are attained on

Proof. Set

and

Using the harmonicity of h, we get for

Hence, we have

if

if

Clearly, (5) is equivalent to (3) and from (6) it follows

which is equivalent to (4). □

Let

Corollary 1 If

The volume mean-value formula (2) was extended by P. Pizzetti to the following [

The Pizzetti formula. If

Here, we present a similar formula for polyharmonic functions on the hypercube based on integration over the set

Theorem 2 If

where

Proof. Equation (9) is a direct consequence from (8):

Theorem 1 suggests that for a certain positive cone in ^{1}-approximation with respect to

For a given

A harmonic function ^{1}-approximant from below to f with respect to

where

Theorem 1 (ii) readily implies the following ([

Theorem 3 Let ^{1}-approximant from below to f with respect to

Corollary 2 If

is a best one-sided L^{1}-approximant from below to f with respect to

Corollary 3 If

a best one-sided L^{1}-approximant from below to f with respect to

Example 1 Let

^{1}-

appro-ximant from below to f_{1} with respect to

to the positive cone of the partial differential operator

the best harmonic one-sided L^{1}-approximation to f_{1} with the corresponding approximation from the linear sub- space of

The possibility for explicit constructions of best one-sided L^{1}-approximants from^{1}-approximants to f_{1} with respect to ^{1}-approximation by elements of_{1} on the boundary

Example 2 Let

^{1}-approximant from

below to

Remark 1 Let ^{1}-approximation from below with respect to

PetarPetrov, (2015) Mean-Value Theorems for Harmonic Functions on the Cube in R^{n}. Advances in Pure Mathematics,05,683-688. doi: 10.4236/apm.2015.511062