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New upper bounds on sphere packings. II. (English) Zbl 1028.52011

H. Cohn and N. Elkies proved in part I of this paper [Ann. Math. (2), 157, 689-714 (2003)] an upper bound of Delsarte type for density of sphere packings in \(n\) dimensions. Using this bound, they obtained numerically new upper bounds for density of sphere packings which are best known in dimensions 4 through 36.
In the present paper H. Cohn gives another proof of this upper bound via theta series of lattices, using properties of Bessel functions and Laguerre polynomials. He rediscovers the function which is optimal in the bound among radial functions of certain exponential type, which was originally proved by D. V. Gorbachev [Extremal problem for entire functions of exponential spherical type, connected with the Levenshtein bound on the sphere packing density in \(R^n\) (Russian), Izvestiya of the Tula State University, Ser. Mathematics. Mechanics. Informatics. 6, 71-78 (2002)]. H. Cohn uses a quadrature formula of R. B. Ghanem [J. Approximation Theory 92, No. 2, 267-279 (1998; Zbl 0949.41018)] for which he gives a new proof.

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
90C05 Linear programming
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Citations:

Zbl 0949.41018
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References:

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