On the best interval quadrature formulae for classes of differentiable periodic functions

摘要

In this paper we solve the problem about optimal interval quadrature formula for the class W~r F of differentiable periodic functions with rearrangement invariant set F of their derivatives of order r. We prove that the formula with equal coefficients and n node intervals having equidistant midpoints is optimal for considering classes. To this end a sharp inequality for antiderivatives of rearrangements of averaged monosplines is proved.