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: $\inf_{\mathbb{T}} w > 0$. 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 $\sup_{n > \ell\ge 0} \bigl(\frac{1}{n - \ell}\sum_{k=\ell}^{n-1} e^{2s_k}\bigr)\bigl(\frac{1}{n - \ell}\sum_{k=\ell}^{n-1} e^{-2s_k}\bigr) < \infty$.