New inequalities on sparse representation in pairs of bases

In this study, the authors investigated some new inequalities on sparse representation for pairs of bases and frames, which would enrich the theory ensemble. First, for fixed pairs of bases, frames and the signal to be represented, we presented the bounds (which can be used in practice directly) of l₀-norm of signal coefficients by the max/min cross-inner-products, as is of much importance to bases and frames selections in sparse representation. Also, the error bounds associated with the given min cross-inner-products are achieved. Moreover, the equivalence condition between l₀ solution and l₁ solution was achieved via relations of cross-inner-products, as can be applied directly for selection of optimal bases. Finally, the estimation of the parameters of cross-inner-products was shown.