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

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