Exponentially convergent trapezoidal rules to approximate fractional powers of operators

In this paper we are interested in the approximation of fractional powers of self-adjoint positive operators. Starting from the integral representation of the operators, we apply the trapezoidal rule combined with a single-exponential and a double-exponential transform of the integrand function. For the first approach our aim is only to review some theoretical aspects in order to refine the choice of the parameters that allow a faster convergence. As for the double exponential transform, in this work we show how to improve the existing error estimates for the scalar case and also extend the analysis to operators. We report some numerical experiments to show the reliability of the estimates obtained.