Scattering theory and polynomials orthogonal on the unit circle

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Abstract
The techniques of scattering theory are used to investigate polynomials orthogonal on the unit circle. The discrete analog of the Jost function, which has been shown to play an important role in the theory of polynomials orthogonal on a segment of the real line, is defined for this system and its properties are investigated. The relation between the Jost function and the weight function is discussed. The techniques of inverse scattering theory are developed and used to obtain new asymptotic formulas satisfied by the polynomials. A set of sum rules satisfied by the coefficients in the recurrence relaxation is exhibited. Finally, Szegö’s theorem on Toeplitz determinants is proved using the recurrence formulas and the Jost function. The techniques of inverse scattering theory are used to find the correction terms.

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