Demystifying Carleson Frames
Ilya Krishtal, Brendan Miller
TL;DR
The paper addresses the redundancy and spanning properties of Carleson frames arising from operator orbits by proving that, when the Carleson spectrum is real and in $(0,1)$ with increasing entries, any subsequence of the form $\\{T^{Nk+j_k}\\varphi\\}_{k\ge0}$ remains a frame for any $N\in\\mathbb{N}$ and choice $j_k\in\{0,1,\dots,N-1\}$, extending to fractional shifts. The main technique blends perturbation arguments for the synthesis operator with a generalized Müntz–Szász framework to establish surjectivity for these subsequences, and is complemented by analysis-operator arguments yielding completeness results for denser sampling sets via density measures $L(\\Lambda)$. Additional contributions include completeness results for certain complex Carleson spectra, explicit lower and upper frame bounds for discrete and continuous Carleson frames, and a detailed treatment of continuous frames $\\{D^t g\\}_{t\in[0,\\infty)}$ with concrete bounds in terms of the Carleson data. Overall, the work clarifies the structure of Carleson frames, provides robust subsequence-frame stability results, and offers concrete frame bounds and completeness results that advance the understanding of highly redundant dynamical frames in Hilbert spaces.
Abstract
We study spanning properties of Carleson systems and prove a recent conjecture on frame subsequences of Carleson frames. In particular, we show that if $\{T^k\varphi\}_{k=0}^\infty$ is a Carleson frame, then every subsequence of the form $\{T^{Nk+j_k}\varphi\}_{k=0}^\infty$ where $N\in\mathbb{N}$ and $0 \leq j_k < N$ is also a frame.