Weak Convergence of Cardaliaguet-Euvrard Neural Network Operators Studied Asymptotically

An Nth order asymptotic expansion is established for the error of weak approximation of a special class of functions by the well-known Cardaliaguet-Euvrard neural network operators. This class is made out of functions f that are N times continuously differentiable over R, so that all f,f′,…, f ^(N) have the same compact support and f ^(N) is of bounded variation. This asymptotic expansion involves products of integrals of the network activation bell-shaped function b and f. The rate of the above convergence depends only on the first derivative of involved functions.