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 \({{f}_{P}}(x)\) 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.
