Speaker
Description
We study the problem of finding lower bounds for the cardinality of weighted designs on compact rank-1 spaces. To solve this problem, P. Delsarte, J. Goethals, and J. Seidel introduced what is known as the linear programming bound, based on a two-point distribution of the design. This bound is based on solving an extremal problem known as the Delsarte problem for Jacobi--Fourier series. Earlier, V.V. Arestov, A.G. Babenko, and their students proposed a solution scheme for a similar problem in the case of spherical codes, based on the primal-dual problem. We adapt this scheme to the case of designs. The scheme is based on convex analysis and consists of several steps, including: formulating the dual problem for the Stieltjes measure, proving the existence of an extremal function and measure, deriving duality relations, characterizing extremal functions and measures based on these relations, reducing the problem to a polynomial system of equations in specific cases, proving the existence of an analytical solution to the system through its certification or by using a special Gröbner basis, and applying the uniform Stieltjes--Bernstein estimate.
The described method has been used to solve several new Delsarte problems. These results are useful in the problem of integral norm discretization when estimating the number of nodes in discrete norms.
