Uniform Approximation by Neural Networks Original Research Article

LetD⊂Rdbe a compact set and letΦbe a uniformly bounded set ofD→Rfunctions. For a given real-valued functionfdefined onDand a given natural numbern, we are looking for a good uniform approximation tofof the form ∑ni=1 a1φi, withφi∈Φ,ai∈R. Two main cases are considered: (1) whenDis a finite set and (2) when the setΦis formed by the functionsφv, b(x)≔s(v·x+b), wherev∈Rd,b∈R, andsis a fixedR→Rfunction.