A representation problem for smooth sums of ridge functions

Abstract

In this paper we prove that if a multivariate function of a certain smoothness class is represented by a sum of k arbitrarily behaved ridge functions, then it can be represented by a sum of k ridge functions of the same smoothness class and a polynomial of degree at most k−1. This solves the problem posed by A. Pinkus in his monograph “Ridge Functions” up to a multivariate polynomial.