Optimal rules with polynomial precision for Hilbert spaces possessing reproducing kernel functions

Abstract

Optimal linear rules with polynomial precision, based on preassigned abscissas, are considered over a Hilbert space possessing a reproducing kernel function; the weights are determined by minimizing the norm of the error of the rule. The resulting optimal rules are characterized by the property that they are interpolatory for a certain linear manifold of functions of the Hilbert space. For illustration and numerical examples we consider a Hilbert space of functions analytic in a circle. We also obtain explicitly optimal rules for the numerical integration of periodic analytic functions.