Refined support and entropic uncertainty inequalities

Generalized versions of the entropic (Hirschman-Beckner) and support (Elad-Bruckstein) uncertainty principle are presented for frames representations. Moreover, a sharpened version of the support inequality has been obtained by introducing a generalization of the coherence. In the finite dimensional case and under certain conditions, minimizers of this inequalities are given as constant functions on their support. In addition, ${\mathrm{\ell }}^p$-norms inequalities are introduced as byproducts of the entropic inequalities.