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 $\{T^k\phi {\}}_{k=0}^{\mathrm{\infty }}$ is a Carleson frame, then every subsequence of the form $\{T^{Nk+j_k}\phi {\}}_{k=0}^{\mathrm{\infty }}$ where $N\in \mathbb{N}$ and $0\le j_k<N$ is also a frame.