Theory and use of functionals on the class of nonnegative polynomials and quadrature formulae

Abstract

A functional of special type on classes of nonnegative polynomials is studied, which is helpful for the investigation of packings and cubature formulae. This functional in the classical case plays a very important role in analysis. The corresponding extremal problem is solved with the help of Legendre and Jacobi polynomials. Also, a new set of Gaussian quadrature formulae is obtained and used for two-dimensional integrals with circular symmetry.