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Covering a sphere and extremal properties of orthogonal polynomials. (English. Russian original) Zbl 0842.05021

Discrete Math. Appl. 5, No. 4, 371-379 (1995); translation from Diskretn. Mat. 7, No. 3, 81-88 (1995).
The author considers coverings of a multidimensional sphere by spherical caps centered around the points of a spherical \(s\)-design. He analyzes the quality of these coverings by spherical harmonics and Gegenbauer polynomials, obtaining a covering version of the ‘linear programming bound’ of Delsarte et al. For this he determines for each \(s\) the greatest \(\eta\) for which there is a polynomial of degree \(s\) which is non-negative on \([- 1, \eta]\) and has the zero coefficient in the Gegenbauer system equal to 0; this \(\eta\) gives an upper bound for the angular covering radius of a spherical \(s\)-design.

MSC:

05B40 Combinatorial aspects of packing and covering
05E35 Orthogonal polynomials (combinatorics) (MSC2000)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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