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 $\sup_{ n > \ell\ge 0} \bigl(\frac{ 1}{ n - \ell}\sum_{ k=\ell}^{ n-1} e^{ 2s_k}\bigr)\bigl(\frac{ 1}{ n - \ell}\sum_{ k=\ell}^{ n-1} e^{ -2s_k}\bigr) < \infty$.
Categories: math.SP math.CA