Approximate complexity and functional representation

Abstract : Results are obtained dealing with the exact and the approximate representation of a function F as a superposition, in designated formats, of functions of fewer variables. Two main cases are considered. In the classical nomographic case one seeks criteria for deciding if a function can be expressed in the form f(phi(x) + psi(y)), or as a uniform limit of such functions. The second case is also related to the solution of Hilbert's 13th problem, and deals with the format F(x) = f(phi(x)) where x lies in an n-cell I and phi is a real valued continuous function on I, and f is a function on R taking values in a chosen normed space epsilon. The use of these criteria is illustrated with several specific functions.