On Approximation by Ridge Functions

Abstract.

Ridge functions are defined as functions of the form $f(\mbox{\footnotesize\bf a}\cdot \mbox{\footnotesize\bf x})$ , where $f\colon\ {\Bbb R}\rightarrow {\Bbb R}$ , $\mbox{\footnotesize\bf x}\in {\Bbb R}^k$, and $\mbox{\footnotesize\bf a}$ belongs to the given ``direction'' set $A\subset {\Bbb R}^k$ . In this paper we study the fundamentality of ridge functions for variable directions sets A and discuss the rate of approximation by ridge functions.