On Spectral Accuracy of Quadrature Formulae Based on Piecewise Polynomial Interpolation

Abstract. It is well-known that the trapezoidal rule, while being only second-order accurate in general, improves to spectral accuracy if applied to the integration of a smooth periodic function over an entire period on a uniform grid. More precisely, for the function that has a square integrable derivative of order r the convergence rate is o ( N−(r−1/2) ) , where N is a number of grid nodes. Accordingly, for a C∞-function the trapezoidal quadrature converges with the rate faster than any polynomial. In this paper, we prove that the same property holds for all quadrature formulae obtained by integrating fixed degree piecewise polynomial interpolations of a smooth integrand, such as the midpoint rule, Simpson’s rule, etc.