Multilevel Hermite--Padé approximations

The talk is devoted to a multilevel Hermite--Pad\'e interpolation problem for a Nikishin system of Markov   functions. First to all we show that the Nikishin systems are perfect with respect to this problem. The polynomials which solve this problem have nice algebraic properties. They satisfy the nearest neighbor recurrence relations and the zero-interlacing property. They also define some simultaneous rational approximants for the system of Markov functions. Using the Gonchar–Rakhmanov vector equilibrium potential method we prove the convergence of ray sequences of the approximants and also estimate the rate of convergence. We briefly discuss some applications in integrable systems (the Degasperis–Procesi equation), random matrices (a two-matrix model) and operator theory (Jacobi matrices on trees).