On recurrence coefficients of Steklov measures

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