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.