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 Trudy Inst. Mat. i Mekh. UrO RAN, 2014, Volume 20, Number 1, Pages 130–141 (Mi timm1036)

The extremal function in the Delsarte problem of finding an upper bound for the kissing number in the three-dimensional space

N. A. Kuklinab

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
b Institute of Mathematics and Computer Science, Ural Federal University

Abstract: We consider an extremal problem for continuous functions that are nonpositive on a closed interval and can be represented by series in Legendre polynomials with nonnegative coefficients. This problem arises from the Delsarte method of finding an upper bound for the kissing number in the three-dimensional Euclidean space. We prove that the problem has a unique solution, which is a polynomial of degree $27$. This polynomial is a linear combination of Legendre polynomials of degrees $0,1,2,3,4,5,8,9,10,20,27$ with positive coefficients; it has simple root $1/2$ and five roots of multiplicity $2$ in $(-1,1/2)$. Also we consider dual problem for nonnegative measures on $[-1,1/2]$. We prove that extremal measure is unique.

Keywords: Delsarte method, infinite-dimensional linear programming, Legendre polynomials, kissing numbers.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2015, 288, suppl. 1, 99–111

Bibliographic databases:

UDC: 517.518.86+519.147

Citation: N. A. Kuklin, “The extremal function in the Delsarte problem of finding an upper bound for the kissing number in the three-dimensional space”, Trudy Inst. Mat. i Mekh. UrO RAN, 20, no. 1, 2014, 130–141; Proc. Steklov Inst. Math. (Suppl.), 288, suppl. 1 (2015), 99–111

Citation in format AMSBIB
\Bibitem{Kuk14} \by N.~A.~Kuklin \paper The extremal function in the Delsarte problem of finding an upper bound for the kissing number in the three-dimensional space \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2014 \vol 20 \issue 1 \pages 130--141 \mathnet{http://mi.mathnet.ru/timm1036} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3364198} \elib{http://elibrary.ru/item.asp?id=21258489} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2015 \vol 288 \issue , suppl. 1 \pages 99--111 \crossref{https://doi.org/10.1134/S008154381502011X} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000352991400010} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84958292235}