EXISTENCE OF THE BEST POSSIBLE UNIFORM APPROXIMATION OF A FUNCTION OFSEVERAL 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 gre ater than or equal to 2. Approximations of a real function f on X by s ums 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 t hree cases. In the first case the function f and the approximating sum s 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.