Mean-Value Theorems for Harmonic Functions on the Cube in n

Let ( ) { } n n i I r x x r i n  , 1, 2, , = ∈ ≤ =  be a hypercube in n  . We prove theorems concerning mean-values of harmonic and polyharmonic functions on ( ) n I 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 n  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 ( ) n I r .

, based on integration over hyperplanar subsets of n I and exact for harmonic or polyharmonic functions.They are presented in Section 2 and can be considered as natural analogues on n I of Gauss surface and volume mean-value formulas for harmonic functions ( [1]) and Pizzetti formula [2], ( [3], Part IV, Ch. 3, pp.287-288) for polyharmonic functions on the ball in  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 2 m ≥ ) in a given domain the linear space of all functions that are harmonic (polyharmonic of degree m) in a domain containing D. The notation d n λ will stand for the Lebesgue measure in n  .

Mean-Value Theorems
Let ( ) : be the ball and the hypersphere in n  with center 0 and radius r.The following famous formulas are basic tools in harmonic function theory and state that for any function h which is harmonic on  ( ) where  ( ) .
The balls are known to be the only sets in n  satisfying the surface or the volume mean-value theorem.This means that if n Ω ⊂  is a nonvoid domain with a finite Lebesgue measure and if there exists a point 0 ∈ Ω x such that ( ) ( ) x for every function h which is harmonic and integrable on Ω , then Ω is an open ball centered at 0 x (see [4]).The mean-value properties can also be reformulated in terms of quadrature domains [5].Recall that n Ω ⊂  is said to be a quadrature domain for , if Ω is a connected open set and there is a Borel measure dµ with a compact support K µ ⊂ Ω such that d d for every n λ - integrable harmonic function f on Ω .Using the concept of quadrature domains, the volume mean-value property is equivalent to the statement that any open ball in n  is a quadrature domain and the measure dµ is the Dirac measure supported at its center.On the other hand, no domains having "corners" are quadrature domains [6].From this point of view, the open hypercube n I  is not a quadrature domain.Nevertheless, it is proved in Theorem 1 below that the closed hypercube n I is a quadrature set in an extended sense, that is, we find explicitly a measure dµ with a compact support K µ having the above property with Ω replaced by n I but the condition n K I µ ⊂  is violated exactly at the "corners" (for the existence of quadrature sets see [7]).This property of n I is of crucial importance for the best one-sided L 1 -approximation with respect to ( ) Let us denote by ij n D the ( ) It can be calculated that ( )


The following holds true.
, then h satisfies: (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 n D .Proof.Set ( ) : , , , , , , .
Hence, we have 5) is equivalent to (3) and from (6) it follows which is equivalent to (4).□ Let ( ) : . Analogously to the proof of Theorem 1 (ii), Equation ( 7) is generalized to: is such that ( ) The volume mean-value formula (2) was extended by P. Pizzetti to the following Here, we present a similar formula for polyharmonic functions on the hypercube based on integration over the set n D .Theorem 2 If ( ) , then the following identity holds true for any 0 k ≥ : where ( ) ( ) ( ) Proof.Equation ( 9) is a direct consequence from ( 8):

A Relation to Best One-Sided L 1 -Approximation by Harmonic Functions
Theorem 1 suggests that for a certain positive cone in ( ) n C I the set n D is a characteristic set for the best one-sided L 1 -approximation with respect to
for every , , h is a best one-sided L 1 -approximant from below to f with respect to ( ) is a best one-sided L 1 -approximant from below to f with respect to ( ) : ), one can compare the best harmonic one-sided L 1 -approximation to f 1 with the corresponding approximation from the linear subspace of ( ) The possibility for explicit constructions of best one-sided L 1 -approximants from ( ) , is studied in [10].The functions   .The smoothness requirements were used for brevity and wherever possible they can be weakened in a natural way.
This note is devoted to formulas for calculation of integrals over the n-dimensional hypercube centered at 0

Figure 1
Figure 1.The sets

.
Assume further that the set n D belongs to the zero set of the function

b
are the unique best one-sided L 1 -approximants to f 1 above, respectively, play the role of basic error functions of the canonical one-sided L 1 -approximation by elements of

Figure 3 .
Figure 3.The graphs of the function