Method for solving the Delsarte problem for weighted designs on compact homogeneous spaces

D. V. Gorbachev

Čebyševskij sbornik, Tome 25 (2024) no. 4, pp. 53-73

Résumé

A method for solving the Delsarte extremal problem $A_{s}$ for weighted designs is proposed. The value of $A_{s}$ is defined as the supremum of $f(1)$ over the class of continuous nonnegative functions $f$ on $[-1,1]$ that can be represented by a Fourier–Jacobi expansion with a unit zero coefficient and nonpositive coefficients starting from the $(s+1)$th term. The main application of the problem $A_{s}$ lies in providing a lower bound for the number of nodes of weighted $s$-designs (or quadrature formulas, exact on a subspace of polynomials of degree at most $s$) in compact homogeneous Riemannian spaces of rank 1, where the zonal functions are Jacobi polynomials. The method for solving $A_{s}$ is based on convex analysis and builds on the results of V.V. Arestov and A.G. Babenko for the case of spherical codes, as well as on specific cases developed by I.A. Martyanov and the author. The method involves several steps, including the formulation of a dual problem for the Stieltjes measure, proving the existence of the extremal function and measure, deriving the duality relations, characterizing the extremal function and measure based on these relations, reducing the problem to a polynomial system of equations, and demonstrating the existence of a unique real analytic solution of the system in the vicinity of a numerical solution in the studied cases. This step is carried out by certifying the solution using the HomotopyContinuation.jl package, which implements the Krawczyk interval method. A uniform estimate for Jacobi polynomials of the Stieltjes–Bernstein type is also applied. Using this approach, two new Delsarte problems were solved as examples. Additionally, for the case corresponding to projective spaces, it is proven that the extremal function is a polynomial. For the case corresponding to the sphere, this remains an open problem. These results are useful for the problem of discretizing the integral norm when estimating the number of nodes in discrete norms.

Détail

Keywords: Delsarte problem, homogeneous space, weighted design, polynomial system, certification of the solution.
Mots-clés : Jacobi polynomial