Best quadrature formulas on classes of differentiable periodic functions

Abstract

A solution is given to the problem of finding the best quadrature formula among formulas of the form

$$\int_0^{2\pi } {f(x)dx \approx \sum\nolimits_{k = 0}^{m - 1} {\sum\nolimits_{l = 0}^\rho {pk,l} f^{(l)} (x_k ),} } $$

which are exact in the case of a constant, for p = r − 1, r = 1, 2, 3... and p = r − 2, r even, for the classes W(r) LqM of 2π-periodic functions.