Multiple orthogonal polynomials, Jacobi matrices on trees, and generalizations of the classical theorem of Szego.

Given a measure on the real line, one can consider the corresponding orthogonal polynomials that satisfy a three-term recurrence that defines the Jacobi matrix. The classical Szegö theorem gives a condition on the measure for which these polynomials satisfy the strong asymptotics. We will discuss the generalization to the case of multiple orthogonality and show that the corresponding polynomials give rise to a Jacobi matrix on the tree. Finally, we will explain how the classical fixed-point theorems from functional analysis can be used to obtain their strong asymptotics. Based on joint work with A. Aptekarev and M. Yattselev.