{"ID":2875718,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.09609","arxiv_id":"2510.09609","title":"Functional Donoho-Elad-Gribonval-Nielsen-Fuchs Sparsity Theorem","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 using 1-approximate Schauder frames. We notice that the `normalized' condition for Hilbert spaces can be generalized to a larger extent when we consider Banach spaces.","short_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 pro...","url_abs":"https://arxiv.org/abs/2510.09609","url_pdf":"https://arxiv.org/pdf/2510.09609v1","authors":"[\"K. Mahesh Krishna\"]","published":"2025-09-01T04:08:09Z","proceeding":"math.FA","tasks":"[\"math.FA\",\"cs.IT\",\"math.OC\"]","methods":"[]","has_code":false}