Collectanea Mathematica

, Volume 69, Issue 2, pp 237–248 | Cite as

On recurrence coefficients of Steklov measures

  • R. V. Bessonov


A measure \(\mu \) on the unit circle \(\mathbb {T}\) belongs to Steklov class \({\mathcal {S}}\) if its density w with respect to the Lebesgue measure on \(\mathbb {T}\) is strictly positive: \(\mathop {\mathrm {ess\,inf}}\nolimits _{\mathbb {T}} w > 0\). Let \(\mu \), \(\mu _{-1}\) be measures on the unit circle \({\mathbb {T}}\) with real recurrence coefficients \(\{\alpha _k\}\), \(\{-\alpha _k\}\), correspondingly. If \(\mu \in {\mathcal {S}}\) and \(\mu _{-1} \in {\mathcal {S}}\), then partial sums \(s_k=\alpha _0+ \ldots + \alpha _k\) satisfy the discrete Muckenhoupt condition \(\sup _{n > \ell \geqslant 0} (\frac{1}{n - \ell }\sum _{k=\ell }^{n-1} e^{2s_k})(\frac{1}{n - \ell }\sum _{k=\ell }^{n-1} e^{-2s_k}) < \infty \).


Orthogonal polynomials Steklov conjecture Muckenhoupt class Bounded mean oscillation 

Mathematics Subject Classification

Primary 42C05 Secondary 33D45 


  1. 1.
    Aptekarev, A., Denisov, S., Tulyakov, D.: On a problem by Steklov. J. Am. Math. Soc. 29(4), 1117–1165 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bessonov, R.V.: Sampling measures, Muckenhoupt Hamiltonians, and triangular factorization. preprint arXiv:1603.07533, accepted in IMRN, 2016
  3. 3.
    Denisov, S.: Remark on the formula by Rakhmanov and Steklovs conjecture. J. Approx. Theor 205, 102–113 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Denisov, S., Kupin, S.: On the growth of the polynomial entropy integrals for measures in the Szegö class. Adv. Math. 241, 18–32 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Denisov, S., Rush, K.: On Schur parameters in Steklovs problem. J. Approx. Theor. 215, 68–91 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Garnett, J.B.: Bounded analytic functions. Pure and Applied Mathematics, vol. 96. Academic Press Inc., Harcourt Brace Jovanovich Publishers, New York (1981)Google Scholar
  7. 7.
    Rahmanov, E.A.: On Steklov’s conjecture in the theory of orthogonal polynomials. Matem. Sb. 108, 581–608 (1979)MathSciNetGoogle Scholar
  8. 8.
    Rahmanov, E.A.: Estimates of the growth of orthogonal polynomials whose weight is bounded away from zero. Mat. Sb. (N.S.), 114(156)(2):269–298, 335, 1981Google Scholar
  9. 9.
    Barry, S.: Orthogonal polynomials on the unit circle. Part 1. Classical theory. American Mathematical Society Colloquium Publications, vol. 54, pp. xxvi+466 American Mathematical Society, Providence, (2005). ISBN: 0-8218-3446-0 MR2105088Google Scholar
  10. 10.
    Pavel Kondrat’evich S.V.A.: Steklov’s problem in the theory of orthogonal polynomials. Itogi Nauki i Tekhniki. Seriya “Matematicheskii Analiz”, 15:5–82, 1977Google Scholar

Copyright information

© Universitat de Barcelona 2017

Authors and Affiliations

  1. 1.St.Petersburg State UniversitySt.PetersburgRussia
  2. 2.St.Petersburg Department of Steklov Mathematical Institute of Russian Academy of ScienceSt.PetersburgRussia

Personalised recommendations