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A result of Widom from the 60's shows that a measure $\mu$ on the unit disc $\mathbb D$ concentrated on $(-1,1)$ is a Carleson measure if and only if $$ \int_{(-1,1)} t^k\,d\mu(t) = O(1/k)\quad\text{ as }k\to +\infty. $$ Now, let $\mu$ be a finite positive measure on $\mathbb D$. Then it defines the sequence and the matrix $$ a_k = \int_{\mathbb D}z^k\,d\mu(z)\quad\text{ and }\quad b_{jk} = \int_{\mathbb D}z^k\overline{z}^j\,d\mu(z). $$ On the other hand, as a consequence of Stone-Weierstraß, $\mu$ is uniquely determined by the sequence $(a_k)_{k\in\mathbb N}$ (the matrix $(b_{jk})_{j,k\in\mathbb N}$). Is there any property of $(a_k)_{k\in\mathbb N}$ or $(b_{jk})_{j,k\in\mathbb N}$ that determines whether $\mu$ is a Carleson measure (for example $b_{jk} = O((j+k)^{-1})$)?

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