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

Celebrated breakthrough sparsity theorem obtained independently by Donoho and Elad $\mathit{\text{[Proc.}}\mathit{\text{Natl.}}\mathit{\text{Acad.}}\mathit{\text{Sci.}}\mathit{\text{USA,}}\mathit{\text{2003]}}$ and Gribonval and Nielsen $\mathit{\text{[IEEE}}\mathit{\text{Trans.}}\mathit{\text{Inform.}}\mathit{\text{Theory,}}\mathit{\text{2003]}}$ and Fuchs $\mathit{\text{[IEEE}}\mathit{\text{Trans.}}\mathit{\text{Inform.}}\mathit{\text{Theory,}}\mathit{\text{2004]}}$ says that unique sparse solution to NP-Hard ${\mathrm{\ell }}_0$-minimization problem can be obtained using unique solution to P-Type ${\mathrm{\ell }}_1$-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.