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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

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