Approximating the Stieltjes integral via a weighted trapezoidal rule with applications

In this paper we provide sharp error bounds in approximating the weighted Riemann–Stieltjes integral ∫abf(t)g(t)dα(t) by the weighted trapezoidal rule f(a)+f(b)2∫abg(t)dα(t). Applications for continuous functions of selfadjoint operators in complex Hilbert spaces are given as well.