On the Establishment of the Order of Smoothness of the Minimal Concave Continuation of a Real Discrete Function Defined on the Vertices of a Parallelepiped

In this paper, we study the order of differentiability of the minimal concave continuation to an n-dimensional coordinate parallelepiped of a real discrete function defined on the vertices of this arbitrary n-dimensional coordinate parallelepiped. As a result of the study, the order of differentiability of the minimal concave continuation to an n-dimensional coordinate parallelepiped of a real discrete function defined on the vertices of this arbitrary n-dimensional coordinate parallelepiped is established, namely, it is proved that if a given real discrete function can be represented as a linear combination of its discrete variables, then its minimal concave continuation is linear and, therefore, infinitely differentiable on an n-dimensional coordinate parallelepiped, and if it cannot be represented as a linear combination of its discrete variables, then its minimal concave continuation on an n-dimensional coordinate parallelepiped is only continuous.