Uniform approximation by neural networks

Let D subset of R-d be a compact set and let Phi be a uniformly bounded set of D --> R functions. For a given real-valued function f defined on D and a given natural number n, we are looking for a good uniform approximation to f of the form Sigma(i=1)(n) a(i)phi(i), with phi(i) is an element of Phi, a(i) is an element of R. Two main cases are considered: (1) when D is a finite set and (2) when the set Phi is formed by the functions phi(upsilon, b)(x) := s(upsilon . x + b), where upsilon is an element of R-d, b is an element of R, and s is a fixed R --> R function. (C) 1998 Academic Press.