Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda lattices. (English. Russian original) Zbl 0608.42016

Aptekarev, A. I.

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.

Reviewer: A.G.Baskakov

MSC:

42C05	Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
30E15	Asymptotic representations in the complex plane

Keywords:

Jordan curves; Riemann theta-function; weight; discrete Sturm-Liouville operators