Linear differential operators with real spectrum, and optimal quadrature formulas

M. A. Chahkiev

Izvestiya. Mathematics , Tome 25 (1985) no. 2, pp. 391-417

Résumé

This article deals with an investigation of optimal quadrature formulas on periodic function classes defined by a restriction imposed on the action of a linear differential operator with constant coefficients and real spectrum in the metric of the space $L^p$, $1\leqslant p\leqslant\infty$. It is proved that on each class of this form there is for any $n$ an optimal quadrature formula with $n$ nodes, and the nodes are equally spaced on a period. The uniqueness of an optimal quadrature formula is investigated. Our results, on the one hand, give a direct generalization of previous results obtained by Nikol'skii, Motornyi, Zhensykbaev, Ligun, and Boyanov, and, on the other hand, make it possible to investigate the problem of optimal quadrature formulas and to obtain a result on optimality of equally spaced nodes on certain classes of infinitely differentiable functions that are limits of the aforementioned classes in a definite sense. Bibliography: 22 titles.