On some properties of smooth sums of ridge functions (Q511441)

Let \(E\) be an open set in \(\mathbb R^n\). Consider a sum of ridge functions \(f\) defined on \(E\) of the form \[ f(\mathbf x) = \sum_{i=1}^m \phi_i(\mathbf a^i \cdot\mathbf x), \] where the \(\phi_i\) are univariate functions. Assume \(f\) belongs to a certain smoothness class. This paper considers the following questions: (a) What can one say about the smoothness of the \(\phi_i\)? (b) Can one always represent the above \(f\) as a sum of ridge functions each of which belongs to a certain smoothness class? The author proves certain generalizations of the results in the references [\textit{R. A. Aliev} and \textit{V. E. Ismailov}, Adv. Appl. Math. 73, 154--169 (2016; Zbl 1342.26037); \textit{S. V. Konyagin} and \textit{A. A. Kuleshov}, Math. Notes 98, No. 2, 336--338 (2015); translation from Mat. Zametki 98, No. 2, 308--309 (2015; Zbl 1329.26027)].