Matrix Riemann–Hilbert analysis for the case of higher genus — asymptotics of polynomials orthogonal on a system of intervals

The method of the matrix Riemann–Hilbert problem is adapted for obtaining the strong asymptotics of polynomials orthogonal on a system of intervals on the real axis. The use of the Riemann theta-functions for deriving the asymptotical formulas is the main ingredient of the approach. An extension of the technique under consideration to Boundary Values Problems for analytic matrix functions of higher dimensions (greater than $2\times 2$) is the main motivation of the work. Precisely this type of problem arise under asymptotical analysis of the Hermite–Padé approximants. The paper is continuation of the series of the lecture notes devoted to exposition of the “Riemann–Hilbert matrix problem” asymptotical techniques.