Demystifying Carleson frames

We study spanning properties of Carleson systems and prove a recent conjecture on frame subsequences of Carleson frames. In particular, we show that if ${\left\{T^k\phi \right\}}_{k=0}^{\infty }$ is a Carleson frame, then every subsequence of the form ${\left\{T^{Nk+j_k}\phi \right\}}_{k=0}^{\infty }$ where $N\in N$ and $0\le j_k<N$ is also a frame.