Random Approximants and Neural Networks Original Research Article

LetDbe a set with a probability measureμ,μ(D)=1, and letKbe a compact subset ofLq(D, μ), 1⩽q<∞. Forf∈Lq,n=1, 2, …, letρn(f, K)=inf ‖f−gn‖q, where the infimum is taken over allgnof the formgn=∑ni=1 aiφi, with arbitraryφi∈Kandai∈R. It is shown that for[formula], under some mild restrictions,ρn(f, K)⩽Cqεn(K) n−1/2, whereεn(K)→0 asn→∞. This fact is used to estimate the errors of certain neural net approximations. For the latter, also the lower estimates of errors are given.