Robustness of infinite frames and Besselian structures
Shankhadeep Mondal, Geetika Verma, Ram Narayan Mohapatra
TL;DR
The paper addresses robustness and erasure resilience of infinite frames in Hilbert spaces by developing a comprehensive Minimal Redundancy Condition (MRC) framework and extending robustness concepts to Besselian frames. It proves that MRC for a frame $F$ is equivalent to MRC for its canonical dual $S_F^{-1}F$, and provides operator-based criteria for robustness when duals are formed as $G=\{S_F^{-1}f_i + h_i\}$. It also shows that $m$-erasure robustness is preserved under unitary transformations and orthogonal projections, with a link to frame excess via a matrix $\Gamma_F$, whose first $m$ columns are independent, and extends these ideas to Besselian frames through 1-erasure robustness criteria based on bridging sets and projections. Altogether, the results offer a cohesive view of redundancy, stability, and erasure recovery for infinite frames, with practical implications for reliable signal reconstruction under data loss.
Abstract
This paper extends the concepts of Minimal Redundancy Condition (MRC) and robustness of erasures for infinite frames in Hilbert spaces. We begin by establishing a comprehensive framework for the MRC, emphasizing its importance in ensuring the stability and resilience of frames under finite erasures. Furthermore, we discussed the robustness of erasures, which generalizes the ability of a frame to withstand information loss. The relationship between robustness, MRC, and excess of a frame is carefully examined, providing new insights into the interplay between these properties. The robustness of Besselian frames, highlighting their potential in applications where erasure resilience is critical. Our results contribute to a deeper understanding of frame theory and its role in addressing challenges posed by erasure recovery.