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. In addition, 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 results 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.
Recommended Citation
C. Berner, "Sequences that do Frame Reconstruction," Annals of Functional Analysis, vol. 17, no. 2, article no. 24, Springer, Apr 2026.
The definitive version is available at https://doi.org/10.1007/s43034-026-00506-z
Department(s)
Mathematics and Statistics
Publication Status
Open Access
Keywords and Phrases
Exponential systems; Frame operator; Frames; Schauder basis; Stability
International Standard Serial Number (ISSN)
2008-8752; 2639-7390
Document Type
Article - Journal
Document Version
Citation
File Type
text
Language(s)
English
Rights
© 2026 Springer, All rights reserved.
Publication Date
01 Apr 2026
