Direct and inverse mean value properties for polylinear functions and their application

Abstract

This paper is devoted to the formulation and proof of the theorems on the mean value of a polylinear function, similar to the direct and inverse theorems on the mean value of harmonic functions. It is proved that the value of an arbitrary polylinear function fP (x) at the central point of G—an arbitrary n-dimensional coordinate parallelepiped—is equal to the mean value of the function fP (x) over the set of k-dimensional faces G for any k \in \{ 0, . . . , n\} . Based on this, it is justified that just once, by calculating the value of the polylinear continuation fP (x) of an arbitrary Boolean function fB(x) at the central point of an n-dimensional unit cube, one can find the number of Boolean vectors on which the Boolean function fB(x) takes the value 1 and thereby, in particular, determine the satisfiability of the Boolean function fB(x). It has also been established that such a property is characteristic only of polylinear functions, i.e., it has been proven that if for any G — n-dimensional coordinate parallelepiped and at least for some number k \in \{ 0, . . . , n\} , the value of the continuous function f(x) at the central point G is equal to the mean value of the function f(x) over the set of k-dimensional faces of G, then the function f(x) is polylinear.