The mystery of Carleson frames

In 2016 Aldroubi et al. constructed the first class of frames having the form ${\left\{T^k\phi \right\}}_{k=0}^{\infty }$ for a bounded linear operator on the underlying Hilbert space. In this paper we show that a subclass of these frames has a number of additional remarkable features that have not been identified for any other frames in the literature. Most importantly, the subfamily obtained by selecting each Nth element from the frame is itself a frame, regardless of the choice of $N\in N$. Furthermore, the frame property is kept upon removal of an arbitrarily finite number of elements.