Sequences that do frame reconstruction
Chad Berner
TL;DR
This work extends frame theory by introducing and thoroughly analyzing sequences that do frame reconstruction with a bounded, not necessarily invertible operator $B$, i.e., $f= obreak \\sum_{n} ig\langle f, Bf_n\big\rangle f_n$. It provides a complete classification of such sequences via a frame operator, presents concrete nonframe examples that still enable reconstruction, and proves a Paley–Wiener type stability result showing reconstruction persists under small $\ell^1$ perturbations. The paper further connects these sequences to weighted frames and classifies unconditional and exponential Schauder bases that satisfy the relaxed reconstruction property, including a full characterization for exponential bases tied to measures on the torus. Overall, it broadens stable reconstruction beyond invertible frame operators, linking frame-like decompositions with Kaczmarz-type methods, weighted frames, and Schauder-basis frameworks.
Abstract
Frames allow all elements of a Hilbert space to be reconstructed by inner product data in a stable manner. Recently, there is interest in relaxing the definition of frames to understand the implications for stable signal recovery. In this paper, we relax the definition of a frame by allowing the operator in the frame decomposition formula to not be invertible. We provide a complete classification of sequences that allow this decomposition via a type of frame operator. Additionally, we provide several examples of sequences that allow this reconstruction property that are not frames and illustrate in which ways they fail to be frames. Furthermore, we provide a Paley-Wiener type stability result for sequences that do this frame-like reconstruction, which is also stable under the non-frame property. Finally, we classify certain Schauder bases-such as unconditional and exponential bases-that satisfy this relaxed frame reconstruction condition.