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 Mat. Sb., 2004, Volume 195, Number 11, Pages 95–118 (Mi msb860)

Circular parameters of polynomials orthogonal on several arcs of the unit circle

A. L. Lukashov

Saratov State University named after N. G. Chernyshevsky

Abstract: The asymptotic behaviour of the circular parameters $(a_n)$ of the polynomials orthogonal on the unit circle with respect to Geronimus measures is analysed. It is shown that only when the harmonic measures of the arcs making up the support of the orthogonality measure are rational do the corresponding parameters form a pseudoperiodic sequence starting from some index (that is, after a suitable rotation of the circle and the corresponding modification of the orthogonality measures they form a periodic sequence). In addition it is demonstrated that if the harmonic measures of these arcs are linearly independent over the field of rational numbers, then the sets of limit points of the sequences of absolute values of the circular parameters $|a_n|$ and of their ratios $(a_{n+k}/a_n)_{n=1}^\infty$ are a closed interval on the real line and a continuum in the complex plane, respectively.

DOI: https://doi.org/10.4213/sm860

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English version:
Sbornik: Mathematics, 2004, 195:11, 1639–1663

Bibliographic databases:

UDC: 517.5
MSC: Primary 42C05; Secondary 30F35

Citation: A. L. Lukashov, “Circular parameters of polynomials orthogonal on several arcs of the unit circle”, Mat. Sb., 195:11 (2004), 95–118; Sb. Math., 195:11 (2004), 1639–1663

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Lukashov A.L., Peherstorfer F., “Zeros of polynomials orthogonal on two arcs of the unit circle”, J. Approx. Theory, 132:1 (2005), 42–71
2. Gesztesy F., Zinchenko M., “Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle”, J. Approx. Theory, 139:1-2 (2006), 172–213
3. Gesztesy F., Zinchenko M., “A Borg-type theorem associated with orthogonal polynomials on the unit circle”, J. London Math. Soc. (2), 74:3 (2006), 757–777
4. F. Peherstorfer, P. Yuditskii, “Almost periodic Verblunsky coefficients and reproducing kernels on Riemann surfaces”, Journal of Approximation Theory, 139:1-2 (2006), 91
5. Peherstorfer F., Steinbauer R., “Note on “Rational compacts and exposed quadratic irrationalities” by Sergey Khrushchev”, J. Approx. Theory, 159:2 (2009), 290–292
6. S. P. Suetin, “Strong asymptotics of polynomials orthogonal with respect to a complex weight”, Sb. Math., 200:1 (2009), 77–93
7. Zinchenko M., “Trace formulas and a Borg-type theorem for CMV operators with matrix-valued coefficients”, Math. Nachr., 283:2, Special Issue: Erhard Schmidt Memorial Issue, Part II (2010), 312–329
8. F. Peherstorfer, “Orthogonal polynomials on several intervals: accumulation points of recurrence coefficients and of zeros”, Journal of Approximation Theory, 2011
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