The Markov sequence problem for the ultra-spherical polynomials was solved in the mid-fifties by Bochner. The solution amounts to the determination and analysis of the extreme points in the space of selfadjoint Markov operators that have the ultra-spherical polynomials as eigenfunctions.
A simple and entirely elementary
proof of this and the corresponding theorem of Gasper on the Markov sequence problem for Jacobi polynomials will be presented.
It is based on the spectral analysis of a family of operators, the above mentioned extreme points, that arise in the study of a probabilistic model of colliding molecules introduced by Marc Kac.
This is joint work with Eric Carlen and Jeff Geronimo.