Bounding the errors in constructive function approximation

In this paper we study the theoretical limits of finite constructive convex approximations of a given function in a Hilbert space using elements taken from a reduced subset. We also investigate the trade-off between the global error and the partial error during the iterations of the solution. The results obtained constitute a refinement of well established convergence analysis for constructive iterative sequences in Hilbert spaces with applications in projection pursuit regression and neural network training.