Constructive description of function classes on surfaces in $\mathbb{R}^3$ and $\mathbb{R}^4$

Functional classes on a curve in a plane (a partial case of a spatial curve) can be described by the approximation speed by functions that are harmonic in three-dimensional neighbourhoods of the curve. No constructive description of functional classes on rather general surfaces in $\mathbb{R}^3$ and $\mathbb{R}^4$ has been presented in literature so far. The main result of the paper is Theorem 1.