The problem of representation as a sum of smooth ridge functions

Recent years the theory of approximation of functions of several variables by ridge functions has been undergoing rapid development. This is due to the great importance of these functions in computed tomography, in statistics, and in the theory of neural networks. The report is devoted to the problem of representing functions of several variables as a sum of smooth ridge functions. It is shown that if a function of many variables of a certain smoothness class is represented as a sum of ridge functions of arbitrary behavior, then it can be represented by a sum of ridge functions of the same smoothness class and a polynomial. This solves the problem posed by A. Pinkus in his monograph "Ridge function", up to a polynomial of many variables.