On the representability of a continuous multivariate function by sums of ridge functions (Q6632950)

Ridge functions -- much like radial basis functions (both have very similar forms: a set of \(m\), say, univariate [usually at least continuous] functions \(f_k\) applied to [composed with] an inner product plus a shift \(a_k\cdot x + b_k\) in the former case, and a Euclidean norm \(\|x-a_k\|_2\) in the latter case) provide extremely useful methods for the approximation of multivariate functions. This is due to the simplicity of their forms, that is, e.g., \N\[ F(x)=\sum_{k=1}^m f_k(a_k\cdot x),\qquad x\in \mathbb{R}^d,\] and due to the fact that they are readily available in more than one dimension.\N\NIn order to describe the usefulness of ridge function approximations, it is of the essence to know when a multivariate function \(F:\mathbb{R}^d\mapsto \mathbb R\) can be represented in a finite expansion of those ridge functions (for different inner products \(a_k\cdot x\) and shifts; the shifts can be incorporated into the \(f_k\)).\N\NMuch work has been carried out in this direction, notably by Allan Pinkus, and the current authors extend these valuable contributions into a more general setting. In particular they present a new, more concise proof of one of the theorems that answer Pinkus' question about which function classes may be written as a finite sum of ridge functions.