Designing optimal dual frames for $\ell^p-$average error optimization

Abstract: In this paper, we investigates the problem of optimal dual frame selection for signal reconstruction in the presence of erasures. Unlike traditional approaches relying on left inverses, we evaluate performance through the norms of error operators, using the Frobenius norm, spectral radius, and numerical radius as measures. Our central focus is the characterization of dual frames that minimize the $\ell^p-$average under these error operator measurements over all possible erasure patterns. We provide conditions under which the canonical dual frame is uniquely optimal and extend our results to multiple erasures. In the Frobenius norm case, we offer a complete characterization for any number of erasures in uniform tight frames. The paper also examines interconnections between optimality criteria across different norm measures and gives sufficient conditions ensuring uniqueness of the optimal dual.