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

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.