A three point quadrature rule for functions of bounded variation and applications

A three point quadrature rule approximating the Riemann integral for a function of bounded variation $f$ by a linear combination with real coefficients of the values $f\left(a\right),f\left(x\right)$ and $f\left(b\right)$ with $x\in [a,b]$ whose sum is equal to $b‐a$ is given. Applications for special means inequalities and in establishing a priori error bounds for the approximation of selfadjoint operators in Hilbert spaces by spectral families are provided as well.