Some problems in the theory of ridge functions

Let $d\ge2$ and $E\subset\mathbb R^d$ be a set. A ridge function on $E$ is a function of the form $\varphi(\mathbf a\cdot\mathbf x)$, where $\mathbf x=(x_1,\dots,x_d)\in E$, $\mathbf a=(a_1,\dots,a_d)\in\mathbb R^d\setminus\{\mathbf0\}$, $\mathbf a\cdot\mathbf x=\sum_{j=1}^da_jx_j$, and $\varphi$ is a real-valued function. Ridge functions play an important role both in approximation theory and mathematical physics and in the solution of applied problems. The present paper is of survey character. It addresses the problems of representation and approximation of multidimensional functions by finite sums of ridge functions. Analogs and generalizations of ridge functions are also considered.