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Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda lattices. (English. Russian original) Zbl 0608.42016
Math. USSR, Sb. 53, 233-260 (1986); translation from Mat. Sb., Nov. Ser. 125(167), No. 2, 231-258 (1984).
Let E be a set of points of $${\mathbb{C}}$$ consisting of the union of finitely many Jordan curves and arcs. The author obtains an expression in the form of a Riemann theta-function for the asymptotic behavior of the polynomials $$Q_ n(z)=z^ n+..$$. that are orthogonal on E with respect to the measure $$\rho (\xi)| d\xi |: \int_{E}Q_ n(\xi)\overline{\xi^ k}\rho (\xi)d\xi =0,$$ $$k=0,1,...,n-1$$, where $$\rho$$ is the weight nonnegative function. In the present article the approach connected with the theory of orthogonal polynomials is extended to periodic and ”finite-zone” nonlinear system connected with discrete Sturm-Liouville operators.