Fundamentality of Ridge Functions

Abstract

For a given integer d, 1 ≤ d ≤ n − 1, let Ω be a subset of the set of all d × n real matrices. Define the subspace M(Ω) = span { g(Ax): A ∈ Ω, g ∈ Ω, g ∈ C(Rd, R)}. We give necessary and sufficient conditions on Ω so that M(Ω) is dense in C(Rn, R) in the topology of uniform convergence on compact subsets. This generalizes work of Vostrecov and Kreines. We also consider some related problems.