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 Izv. Akad. Nauk SSSR Ser. Mat., 1984, Volume 48, Issue 5, Pages 1078–1108 (Mi izv1508)

Linear differential operators with real spectrum, and optimal quadrature formulas

M. A. Chahkiev

Abstract: 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.

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English version:
Mathematics of the USSR-Izvestiya, 1985, 25:2, 391–417

Bibliographic databases:

UDC: 517.98
MSC: Primary 41A55, 47E05, 65D32; Secondary 26C10, 41A25

Citation: M. A. Chahkiev, “Linear differential operators with real spectrum, and optimal quadrature formulas”, Izv. Akad. Nauk SSSR Ser. Mat., 48:5 (1984), 1078–1108; Math. USSR-Izv., 25:2 (1985), 391–417

Citation in format AMSBIB
\Bibitem{Cha84} \by M.~A.~Chahkiev \paper Linear differential operators with real spectrum, and optimal quadrature formulas \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1984 \vol 48 \issue 5 \pages 1078--1108 \mathnet{http://mi.mathnet.ru/izv1508} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=764310} \zmath{https://zbmath.org/?q=an:0594.41017} \transl \jour Math. USSR-Izv. \yr 1985 \vol 25 \issue 2 \pages 391--417 \crossref{https://doi.org/10.1070/IM1985v025n02ABEH001289} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Nguyên Th{\d i} Thiêu Hoa, “Some extremal problems on classes of functions determined by linear differential operators”, Math. USSR-Sb., 68:1 (1991), 213–255
2. Kholmat M. Shadimetov, Farhod A. Nuraliev, “Optimal formulas of numerical integration with derivatives in Sobolev space”, Zhurn. SFU. Ser. Matem. i fiz., 11:6 (2018), 764–775
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