Title: Generalized notions of sparsity and restricted isometry property. Part II: Applications,
Abstract: The restricted isometry property (RIP) is a universal tool for data recovery.
We explore the implication of the RIP in the framework of generalized sparsity
and group measurements introduced in the Part I paper. It turns out that for a
given measurement instrument the number of measurements for RIP can be improved
by optimizing over families of Banach spaces. Second, we investigate the
preservation of difference of two sparse vectors, which is not trivial in
generalized models. Third, we extend the RIP of partial Fourier measurements at
optimal scaling of number of measurements with random sign to far more general
group structured measurements. Lastly, we also obtain RIP in infinite dimension
in the context of Fourier measurement concepts with sparsity naturally replaced
by smoothness assumptions.