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

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Abstract

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 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.

Keywords

Riemann–Stieltjes integral
Trapezoidal quadrature rule
Selfadjoint operators
Functions of selfadjoint operators
Spectral representation
Inequalities for selfadjoint operators

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