Optimal dual frames and dual pairs for probability modelled erasures using weighted average of operator norm and spectral radius

TL;DR

This paper investigates the design of optimal dual frames and pairs in finite frame theory to minimize probabilistic reconstruction errors, using a weighted average of operator norm and spectral radius as the optimality criterion.

Contribution

It introduces a new optimality measure combining operator norm and spectral radius, and characterizes the existence, uniqueness, and properties of optimal dual frames and pairs under probabilistic erasures.

Findings

Existence and uniqueness of optimal dual frames established.

Topological properties of optimal duals analyzed.

Relations with other probabilistic duals discussed.

Abstract

The prime focus of this paper is the study of optimal duals of a given finite frame as well as optimal dual pairs, in the context of probability modelled erasures of frame coefficients. We characterize optimal dual frames (and dual pairs) which, among all dual frames (and dual pairs), minimize the maximum measure of the error operator obtained while considering all possible locations of probabilistic erasures of frame coefficients in the reconstruction with respect to each dual frame(dual pair). For a given weight number sequence associated with the probabilities, the measure of the probabilistic error operator is taken to be the weighted average of the operator norm and the spectral radius. Using this as an optimality measure, the existence and uniqueness of optimal dual frames (and optimal dual pairs) and their topological properties are studied. Also, their relations with probabilistic optimal dual frames as well as dual pairs in other contexts, such as those obtained using operator norm and spectral radius as the measure of the error operator, are analyzed.