Approximating the Riemann–Stieltjes integral by a trapezoidal quadrature rule with applications

Abstract

In this paper we provide sharp bounds for the error in approximating the Riemann–Stieltjes integral ∫ a b f ( t ) d u ( t ) by the trapezoidal rule f ( a ) + f ( b ) 2 ⋅ [ u ( b ) − u ( a ) ] under various assumptions for the integrand f and the integrator u for which the above integral exists. Applications for continuous functions of selfadjoint operators in Hilbert spaces are provided as well.