A Functional Version of the 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. 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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