A ridge fuction on \(\mathbb {R}^n\) is of the form \(g(a\cdot x)\), where \(g:\mathbb{R}\to\mathbb{R}\) and \(a\in\mathbb{R}^n\setminus\{ 0\} \). Recently the following problem was extensively studied: Assume \(f\in C^k(\mathbb{R}^n)\) is a finite sum of ridge functions, (1) \(f(x)=\sum_{i=1}^r g_i (a^i\cdot x)\), where \(a^1,\dots,a^r\) are pairwise linearly independent. “What can we say about the smoothness of \(g^i\)?”
It is not difficult to see that if \(a^1,\dots,a^r\) are linearly independent, then all \(g^i\) belong to the class \(C^k (\mathbb{R})\) (see S. V. Konyagin and A. A. Kuleshov [Math. Notes 98, No. 2, 336–338 (2015; Zbl 1329.26027); translation from Mat. Zametki 98, No. 2, 308–309 (2015)]). A. Pinkus [Indag. Math., New Ser. 24, No. 4, 725–738 (2013; Zbl 1305.26036)] proved that if a priori we assume that the functions \(g^i\) satisfy some mild regularity condition (e.g., the Lebesgue measurability), then \(g^i \in C^k(\mathbb{R})\).
In this paper, the authors modify the above question: “Can we write \(f\) as a sum (2) \(f(x)=\sum_{i=1}^r f_i(a^i\cdot x)\) but with the \(f_i \in C^k(\mathbb{R})\), \(i=1,\dots,r\) ?” Let \(p\) be the number of those \(a^i\) which form a maximal linearly independent system. The following two theorems are main results of the paper:
Theorem 2.3. Assume that \(r\geq 3\) and \(p=r-1\). If \(f\geq 2\), then \(f\) has a representation (2), where \(f_i \in C^k(\mathbb{R})\), \(i=1,\dots,r\). It also holds for \(k=1\) if we additionally assume, that the first order partial derivatives of \(f\) are Hölder continuous.
Theorem 3.1. Assume \(k\geq r-p+1\). There exist functions \(f_i \in C^k(\mathbb{R})\) and a polynomial \(P\) of total degree at most \(r-p+1\) such that \(f(x)=\sum_{i=1}^r f_i (a^i\cdot x)+P(x)\).
The authors do not know if the polynomial term is really necessary.