A Functional Version of the Sparsity Theorem

[2510.09609] A Functional Version of the Sparsity Theorem Skip to main content Search Submit Donate Log in Mathematics > Functional Analysis arXiv:2510.09609 (math) [Submitted on 1 Sep 2025 ( v1 ), last revised 25 Aug 2026 (this version, v2)] Title: A Functional Version of the Sparsity Theorem Authors: K. Mahesh Krishna View PDF HTML (experimental) Abstract: Celebrated breakthrough sparsity theorem obtained independently by Donoho and Elad \textit{[Proc. Natl. Acad. Sci. USA, 2003]} and Gribonval and Nielsen \textit{[IEEE Trans. Inform. Theory, 2003]} and Fuchs \textit{[IEEE Trans. Inform. Theory, 2004]} says that unique sparse solution to NP-Hard $\ell_0$-minimization problem can be obtained using unique solution to P-Type $\ell_1$-minimization problem. In this paper, we extend their result to abstract Banach spaces. We notice that the `normalized' condition for Hilbert spaces can be generalized to a larger extent when we consider Banach spaces. Comments: Title changed and major improvement. 8 Pages, 0 Figures Subjects: Functional Analysis (math.FA) ; Information Theory (cs.IT); Optimization and Control (math.OC) MSC classes: 42C15, 94A12, 46B45 Cite as: arXiv:2510.09609 [math.FA] (or arXiv:2510.09609v2 [math.FA] for this version) https://doi.org/10.48550/arXiv.2510.09609 Focus to learn more arXiv-issued DOI via DataCite Submission history From: K Mahesh Krishna [ view email ] [v1] Mon, 1 Sep 2025 04:08:09 UTC (8 KB) [v2] Tue, 25 Aug 2026 04:00:46 UTC (8 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.FA < prev | next > new | recent | 2025-10 Change to browse by: cs cs.IT math math.IT math.OC References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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