Direct and Inverse Mean Value Properties for Polylinear Functions and Their Applications

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 $${{f}_{P}}(x)$$ at the central point of $$\mathbb{G}$$ , an arbitrary $$n$$ -dimensional coordinate parallelepiped, is equal to the mean value of the function $${{f}_{P}}(x)$$ over the set of $$k$$ -dimensional faces $$\mathbb{G}$$ for any $$k \in \{ 0, \ldots ,n\} $$ . Based on this, it is justified that just once, by calculating the value of the polylinear continuation $${{f}_{P}}(x)$$ of an arbitrary Boolean function $${{f}_{B}}(x)$$ at the central point of an $$n$$ -dimensional unit cube, one can find the number of Boolean vectors on which the Boolean function $${{f}_{B}}(x)$$ takes the value 1 and thereby, in particular, determine the satisfiability of the Boolean function $${{f}_{B}}(x)$$ . It is also established that such a property is characteristic only of polylinear functions, that is, it is proved that if for any $$\mathbb{G}$$ , an $$n$$ -dimensional coordinate parallelepiped and at least for some number $$k \in \{ 0, \ldots ,n\} $$ , the value of the continuous function $$f(x)$$ at the central point of $$\mathbb{G}$$ is equal to the mean value of the function $$f(x)$$ over the set of $$k$$ -dimensional faces of $$\mathbb{G}$$ , then the function $$f(x)$$ is polylinear.