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Extensions of Szegö’s theory of orthogonal polynomials. II. (English) Zbl 0635.42023
[For part I see the second author, Lect. Notes Math. 1171, 230-238 (1985; Zbl 0591.42016)]. Let $$d\mu$$ be a finite positive measure on the unit circle in the complex plane and $$\{\phi_ n(D\mu)\}$$ be a system of orthonormal polynomials on the unit circle with respect to a measure $$d\mu$$. If supp(d$$\mu)$$ is an infinite set then $$d\mu$$ is a distribution. Szegö’s theory of orthogonal polynomials is concerned with the asymptotic behavior of $$\phi_ n(d\mu)$$ when log $$\mu$$ ’$$\in L^ 1.$$
The authors study the asymptotic behavior of the fraction $$\phi_ n(d\mu_ 1)/\phi_ n(d\mu_ 2)$$ outside the unit circle. The consequences for orthogonal polynomials on the real line are also discussed. A few printing mistakes are there in the text.
Reviewer: A.N.Srivastava

##### MSC:
 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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