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

This article is cited in 8 scientific papers (total in 8 papers)

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.


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

Bibliographic databases:

UDC: 517.5
MSC: Primary 42C05; Secondary 30F35
Received: 25.12.2001 and 09.02.2004

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
\by A.~L.~Lukashov
\paper Circular parameters of polynomials orthogonal on several arcs of the unit circle
\jour Mat. Sb.
\yr 2004
\vol 195
\issue 11
\pages 95--118
\jour Sb. Math.
\yr 2004
\vol 195
\issue 11
\pages 1639--1663

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    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  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib
    4. F. Peherstorfer, P. Yuditskii, “Almost periodic Verblunsky coefficients and reproducing kernels on Riemann surfaces”, Journal of Approximation Theory, 139:1-2 (2006), 91  crossref  mathscinet  zmath
    5. Peherstorfer F., Steinbauer R., “Note on “Rational compacts and exposed quadratic irrationalities” by Sergey Khrushchev”, J. Approx. Theory, 159:2 (2009), 290–292  crossref  mathscinet  zmath  isi  elib
    6. S. P. Suetin, “Strong asymptotics of polynomials orthogonal with respect to a complex weight”, Sb. Math., 200:1 (2009), 77–93  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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  crossref  mathscinet  zmath  isi  elib
    8. F. Peherstorfer, “Orthogonal polynomials on several intervals: accumulation points of recurrence coefficients and of zeros”, Journal of Approximation Theory, 2011  crossref  mathscinet  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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