Nikishin systems on the unit circle

Abstract

We introduce Nikishin system of $r$ probability measures on the unit circle. We show that such systems satisfy the AT property and therefore normality, introduced in~\cite{KVMLOPUC}, for any multi-index $(n_1,\ldots,n_r)\in\mathbb{N}^r$ with same-parity components satisfying $n_1 \ge n_2 \ge\ldots\ge n_r$. In the case of $r=2$, we demonstrate that the same property holds without requiring $n_1 \ge n_2 \ge\ldots\ge n_r$. The analogous simple proof works for Nikishin systems on the real line for indices satisfying $n_j\ge \max\{n_{j+1},\ldots,n_r\}-1$, $j=1,\ldots,r-1$. This is related to the proof by Cousseement and Van Assche for $r=2$.