The Order of Accuracy of Quadrature Formulae for Periodic Functions

The trapezoidal quadrature rule on a uniform grid has spectral accuracy when integrating C ∞ periodic function over a period. The same holds for quadrature formulae based on piecewise polynomial interpolations. In this paper, we prove that these quadratures applied to \({\rm{W}}_{{\rm{per}}}^{{\rm{r,p}}} \) periodic functions with r > 2 and p ≥ 1 have error \({\mathcal O}((\Delta x)^r)\). The order is independent of p, sharp, and for p < ∞ is higher than predicted by best trigonometric approximation. For p=1 it is higher by 1.