Norm retrieval algorithms: A new frame theory approach

In this paper, we introduce a new approach based on the excess components of a frame for recognizing norm retrieval frames (NRFs) in finite-dimensional Hilbert spaces. This method leads to a complete characterization of NRFs in Double-struck capital R2$$ {\mathrm{\mathbb{R}}} circumflex 2 $$ and Double-struck capital R3$$ {\mathrm{\mathbb{R}}} circumflex 3 $$ and then to 1-excess NRFs in Double-struck capital Rn$$ {\mathrm{\mathbb{R}}} circumflex n $$. This approach is not only more effective in higher dimensions for detecting NRF than the previous approaches but is also applicable to constructing NRF from a given Riesz basis. In addition, we establish a relationship between the norm retrievability of a frame and its duals. We also show that, unlike the phase retrieval property, the set of all norm retrievable dual frames is generally not dense in the set of all dual frames. Then, we present numerical results to illustrate the potentiality of the NRF approach in signal recovery with respect to the classical framework in frame theory, and finally, we raise some perspectives and problems concerning NRF.