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Solving the Delsarte problem for 4-designs on the sphere S^2

https://doi.org/10.22405/2226-8383-2021-22-3-154-165

Abstract

An important problem in discrete geometry and computational mathematics is the estimation of the minimum number of nodes 𝑁(𝑠) of a quadrature formula (weighted 𝑠- design) of the form 1
|S2| ∫︀ S2 𝑓(𝑥) 𝑑𝑥 = Σ︀𝑁 𝜈=1 𝜆𝜈𝑓(𝑥𝜈) with positive weights, exact for all spherical polynomials of degree at most 𝑠. P. Delsarte, J.M Goethals, and J.J. Seidel (1977) to estimate 𝑁(𝑠) from below formulated an extremal problem 𝐴𝑠 for expansions in terms
of orthogonal Gegenbauer (Legendre for S2) polynomials with restrictions on the sign of the Fourier–Gegenbauer coefficients. Using a version of this problem 𝐴𝑠,𝑛 on polynomials of degree
𝑛 = 𝑠, they proved the classical estimate for tight designs. This estimate is sharp and gives a solution to 𝐴𝑠 only in exceptional cases (𝑠 = 0, 1, 2, 3, 5 for S2). For general dimensions, there
are cases when 𝐴𝑠,𝑛 > 𝐴𝑠,𝑠 for 𝑛 > 𝑠, which leads to better estimates for 𝑁(𝑠). In particular, N.N. Andreev (2000) proved in this way the minimality of an 11-design on the sphere S3. Related
Delsarte problems are also formulated for estimating the cardinality of spherical codes. In this direction, V.V. Arestov and A.G. Babenko (1997), based on the methods of infinite-dimensional linear programming, solved an analog of the 𝐴𝑠 problem for the case of spherical 0.5-codes on the sphere S3 (the kissing number problem). Then this method was developed in the works of D.V. Shtrom, N.A. Kuklin. A.V. Bondarenko and D.V. Gorbachev (2012) showed that 𝑁(4) = 10. This fact follows from the estimate 𝐴4,7 > 9, previously obtained by P. Boyvalenkov and S. Nikova (1998), and the existence of weighted 4-designs of 10 nodes. Nevertheless, it is of interest to solve the problem 𝐴4 exactly, aiming to transfer the method of calculating 𝐴𝑠 to the general dimensions and orders of designs. In this paper, it is proved that 𝐴4 = 𝐴4,22 = 9.31033 . . .
For this, the Arestov–Babenko–Kuklin method is adapted and the problem is reduced to the construction of a special quadrature formula for [−1, 1], consistent with the form of the assumed extremal function (polynomial). The proposed method is based on the use of nonlinear programming, in particular, semidefinite programming, and the solution of a polynomial system of equations arising from a quadrature formula. To prove the existence of an analytical solution of such a system in the neighborhood of the numerical solution, interval Krawczyk’s method from HomotopyContinuation.jl is used.

About the Author

Ivan Anatol’evich Martyanov
Tula State University
Russian Federation

graduate student



References

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Martyanov I.A. Solving the Delsarte problem for 4-designs on the sphere S^2. Chebyshevskii Sbornik. 2021;22(3):154-165. https://doi.org/10.22405/2226-8383-2021-22-3-154-165

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