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₏} (x) at the central point of G, an arbitrary n -dimensional coordinate parallelepiped, is equal to the mean value of the function {f₏} (x) over the set of k -dimensional faces G for any k \ 0, , n\. Based on this, it is justified that just once, by calculating the value of the polylinear continuation {f₏} (x) of an arbitrary Boolean function {f₁} (x) at the central point of an n -dimensional unit cube, one can find the number of Boolean vectors on which the Boolean function {f₁} (x) takes the value 1 and thereby, in particular, determine the satisfiability of the Boolean function {f₁} (x). It is also established that such a property is characteristic only of polylinear functions, that is, it is proved that if for any G, an n -dimensional coordinate parallelepiped and at least for some number k \ 0, , n\, the value of the continuous function f (x) at the central point of 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.