Approximation by Superpositions of a Sigmoidal Function

Abstract

We generalize a result of B. Gao and Y. Xu [J. Math. Anal. Appl. 178 (1993) 221–226] concerning the approximation of functions of bounded variation by linear combinations of a fixed sigmoidal function to the class of functions of bounded f-variation. Also, in the case of one variable, a proposition of A. R. Barron [IEEE Trans. Inf. Theory 36 (1993) 930–945] is improved. Our proofs are similar to that of Gao and Xu [loc. cit.].