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 $s_k={\alpha }_0+\dots +{\alpha }_k$ satisfy the discrete Muckenhoupt condition ${\sup }_{n>\ell \ge 0}\left(\frac{1}{n-\ell }{\sum }_{k=\ell }^{n-1}{e}^{2s_k}\right)\left(\frac{1}{n-\ell }{\sum }_{k=\ell }^{n-1}{e}^{-2s_k}\right)<\infty$.

12 pages

References in corpus (1)

• Sampling measures, Muckenhoupt Hamiltonians, and triangular factorization