A representation problem for smooth sums of ridge functions

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.