Distribution of the points of a design on the sphere. (English. Russian original) Zbl 1106.05020

Yudin, V. A.

Izv. Math. 69, No. 5, 1061-1079 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 5, 205-224 (2005).

Author’s abstract: We determine the size of spherical caps with centres at the points of a design that cover the whole sphere in Euclidean space with a given multiplicity. By projecting \(q\)-designs on one-dimensional subspaces, we obtain the nodes of a Chebyshev-type quadrature formula of the same precision \(q\). For large values of \(q\) we establish that the points of a minimal \(q\)-design are uniformly distributed on the sphere. We construct a weighted cubature formula on the sphere with the minimum number of nodes.

Reviewer: Ioan Tomescu (Bucureşti)

MSC:

05B30	Other designs, configurations
65D32	Numerical quadrature and cubature formulas
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.)

Keywords:

Euclidean space; \(q\)-design; Chebyshev-type quadrature formula; spherical cap; cubature formula; Fourier series; Gegenbauer polynomial