{"title":"Sequences that do frame reconstruction","authors":"Chad Berner","doi":"10.1007/s43034-026-00506-z","DOIUrl":null,"url":null,"abstract":"<div><p>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 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.</p></div>","PeriodicalId":48858,"journal":{"name":"Annals of Functional Analysis","volume":"17 2","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2026-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43034-026-00506-z.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s43034-026-00506-z","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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 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.
期刊介绍：
Annals of Functional Analysis is published by Birkhäuser on behalf of the Tusi Mathematical Research Group.
Ann. Funct. Anal. is a peer-reviewed electronic journal publishing papers of high standards with deep results, new ideas, profound impact, and significant implications in all areas of functional analysis and all modern related topics (e.g., operator theory). Ann. Funct. Anal. normally publishes original research papers numbering 18 or fewer pages in the journal’s style. Longer papers may be submitted to the Banach Journal of Mathematical Analysis or Advances in Operator Theory.
Ann. Funct. Anal. presents the best paper award yearly. The award in the year n is given to the best paper published in the years n-1 and n-2. The referee committee consists of selected editors of the journal.
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