The paper is concerned with the approximation of multivariate functions by expressions of the form \((*)\quad \sum^{n}_{i=1}g_ i(a_{i1}x_ 1+...+a_{in}x_ n)\) more precisely, with the exact representations of functions in this form. A typical result is Theorem 3: Let \(f\in C^ n([0,1]^ 2)\). Suppose that the operator \(\sum^{n}_{i=0}c_ i\partial^ n/\partial x^ i\partial y^{n-1}\) annihilates f, where \(c_ 0,...,c_ n\in {\mathbb{R}}\). If the polynomial \(\sum c_ iz^ i\) has distinct real zeros, then f can be represented in the form (*). The viewpoint is not the same as in Kolmogoroff's and Lorentz' investigations for Hilbert's 13th problem.