The extremal function in the Delsarte problem of finding an upper bound for the kissing number in the three-dimensional space

We consider an extremal problem for continuous functions that are nonpositive on a closed interval and can be represented by series in Legendre polynomials with nonnegative coefficients. This problem arises from the Delsarte method of finding an upper bound for the kissing number in the three-dimensional Euclidean space. We prove that the problem has a unique solution, which is a polynomial of degree $27$. This polynomial is a linear combination of Legendre polynomials of degrees $0,1,2,3,4,5,8,9,10,20,27$ with positive coefficients; it has simple root $1/2$ and five roots of multiplicity $2$ in $(-1,1/2)$. Also we consider dual problem for nonnegative measures on $[-1,1/2]$. We prove that extremal measure is unique.