Polylinear continuations of some discrete functions and an algorithm for finding them

In this paper, we study the existence and uniqueness of polylinear continuations of some discrete functions. It is proved that for any Boolean function, there exists a corresponding polylinear continuation and it is unique. An algorithm for finding a polylinear continuation of a Boolean function is proposed and its correctness is proved. Based on the result of the proposed algorithm, explicit forms of polylinear continuations are found first for a Boolean function and then for an arbitrary function defined only at the vertices of an $n$-dimensional unit cube, an arbitrary cube, and a parallelepiped, and in each particular case the uniqueness of the corresponding polylinear continuations is proved.