Existence of the best possible uniform approximation of a function of several variables by a sum of functions of fewer variables

Let phi(i) be some maps of a set X onto sets X(i), i = 1,..., n, n greater than or equal to 2. Approximations of a real function f on X by sums g(1) circle phi(i) +...+ g(n) circle phi(2) are considered, where the g(i) are real functions on X(i). Under certain constraints on the phi(i) the existence of the best possible approximation is proved in three cases. In the first case the function f and the approximating sums are bounded, but the functions g(i) can be unbounded. In the second case f and the g(i) are bounded. In the third case f and the g(i) are continuous, X and the X(i) are compact sets with metrics, and the maps phi(i) are continuous.