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

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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, and 27 with positive coefficients; it has simple root 1/2 and five double roots in (−1, 1/2). We also consider the dual extremal problem for nonnegative measures on [−1, 1/2] and, in particular, prove that an extremal measure is unique.

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Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia

    N. A. Kuklin

  2. Institute of Mathematics and Computer Science, Ural Federal University, pr. Lenina 51, Yekaterinburg, 620000, Russia

    N. A. Kuklin

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  1. N. A. Kuklin
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Correspondence to N. A. Kuklin.

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Original Russian Text © N.A. Kuklin, 2014, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2014, Vol. 20, No. 1.

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Kuklin, N.A. The extremal function in the Delsarte problem of finding an upper bound for the kissing number in the three-dimensional space. Proc. Steklov Inst. Math. 288 (Suppl 1), 99–111 (2015). https://doi.org/10.1134/S008154381502011X

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