arXiv subjects

Ram Narayan Mohapatra

Publications and source records attributed to Ram Narayan Mohapatra.

4 recordsLinked to original sources

arXiv2026

Refined upper bounds for the numerical radius via weighted operator means

We establish a parameterized family of upper bounds for the numerical radius of bounded linear operators on a complex Hilbert space, based on weighted expressions involving the modulus of an operator and its adjoint. The proposed estimates encompass a known numerical-radius inequality as a particular symmetric case, while optimization over the weight parameter provides additional flexibility and can lead to sharper bounds. Under suitable structural assumptions on the associated auxiliary operators, we further derive corresponding spectral-radius estimates. The weighted approach is also extended to $2\times2$ off-diagonal operator matrices. Finally, we examine sharpness and equality cases and provide explicit finite-dimensional examples illustrating the obtained estimates.

math.FA↗

arXiv2025

Optimal Dual Frame Pairs: A Synergy with Graph Theory

This paper investigates the optimization of dual frame pairs in the context of erasure problems in data transmission, using a graph theoretical approach. Frames are essential for mitigating errors and signal loss due to their redundancy properties. We address the use of spectral radius and operator norm for error measurements, presenting conditions for the optimality of dual pairs for one and two erasures. Our study shows that a tight frame generated by connected graphs and its canonical dual pair is optimal for one-erasure scenarios. Additionally, we compute the spectral radius of the error operator for one and two erasures in graph-generated frames, establishing necessary conditions for dual pair optimality.

math.FA↗

arXiv2025

On the paper Optimal dual frames of probabilistic erasures

In the paper Optimal Dual Frames for Probabilistic Erasures, the authors have given conditions under which the canonical dual is claimed to be the unique probability optimal dual for 1-erasure reconstruction. In this paper, we demonstrate via counterexamples that the conditions provided are not sufficient to guarantee uniqueness. We also noticed a mistake in the proof of the theorem and proved the correct version of the theorem with a stronger but valid condition. Furthermore, we show that the corollary asserting uniqueness for a tight frame assumption is also incorrect. Our results refine the understanding of probability optimal dual frame constructions and offer a more complete characterization of the 1-erasure probability optimal duals.

math.FA↗

arXiv2025

Robustness of infinite frames and Besselian structures

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.

math.FA↗