Properties of balanced frames

HEINEKEN S. B.; MORILLAS P. M.; TARAZAGA PABLO

Congreso; Mathematical Congress of the Americas 2021; 2021

In this work we consider balanced frames, i.e. those frames which sum is zero, and in particular balanced unit norm tight frames, in finite dimensional Hilbert spaces. So far there has not been paid attention to frames that are balanced. Here we present several advantages of balanced unit norm tight frames in signal processing. They provide an exact reconstruction in the presence of systematic errors in the transmitted coefficients, and are optimal when these coefficients are corrupted with noises that can have non-zero mean. Moreover, using balanced frames we can know that the transmitted coefficients were perturbed, and we also have an indication of the source of the error. We analyze various properties of these types of frames. We define an equivalence relation in the set of the dual frames of a balanced frame. This allows to show that we can obtain all the duals from the balanced ones. We investigate the problem of finding the nearest balanced frame to a given frame, characterizing completely its existence and giving its expression. We introduce and study a new concept of complement for balanced frames. Finally, we present examples and methods for constructing balanced unit norm tight frames.