August 6, 2025Open Access

Functional Donoho-Elad-Gribonval-Nielsen-Fuchs Sparsity Theorem

Key Points

MAIN FINDING: Extends the renowned sparsity theorem to abstract Banach spaces through the use of Schauder frames.
KEY EVIDENCE: Unique sparse solutions can be derived from P-type l1-minimization, similar to earlier results in Hilbert spaces.
APPROACH: Used 1-approximate Schauder frames to generalize the normalized condition of previous findings.
SIGNIFICANCE: This work broadens the applicability of the sparsity theorem, impacting fields that rely on Banach space theory.

Abstract

Celebrated breakthrough sparsity theorem obtained independently by Donoho and Elad Proc. Natl. Acad. Sci. USA, 2003 and Gribonval and Nielson IEEE Trans. Inform. Theory, 2003 and Fuchs IEEE Trans. Inform. Theory, 2004 says that unique sparse solution to NP-Hard ₀-minimization problem can be obtained using unique solution to P-Type 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.

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Cite This Study

K. Mahesh Krishna (Tue,) studied this question.

synapsesocial.com/papers/689523d29f4f1c896c42a07a https://doi.org/https://doi.org/10.20944/preprints202508.0241.v1

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