Collectanea Mathematica

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

On recurrence coefficients of Steklov measures

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Abstract

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 \(\sup _{n > \ell \geqslant 0} (\frac{1}{n - \ell }\sum _{k=\ell }^{n-1} e^{2s_k})(\frac{1}{n - \ell }\sum _{k=\ell }^{n-1} e^{-2s_k}) < \infty \).

Keywords

Orthogonal polynomials Steklov conjecture Muckenhoupt class Bounded mean oscillation 

Mathematics Subject Classification

Primary 42C05 Secondary 33D45 

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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

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