On recurrence coefficients of Steklov measures

• Autores: Roman V. Bessonov
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 69, Fasc. 2, 2018, págs. 237-248
• Idioma: inglés
• DOI: 10.1007/s13348-017-0203-9
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• Resumen

  • 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.