Approximately Dual and Pseudo-Dual Probabilistic Frames

TL;DR

This paper explores dual probabilistic frames, introduces approximate and pseudo-dual frames, and establishes structural properties and relationships with redundancy and atomicity.

Contribution

It introduces the concepts of approximately dual and pseudo-dual probabilistic frames and proves key structural results about their properties.

Findings

The canonical dual is the only dual of pushforward type with zero redundancy.

Probabilistic frames with finite redundancy are atomic and finite.

Every probabilistic frame has a finite discrete approximate dual.

Abstract

This paper studies properties of dual probabilistic frames -- in particular in relation to redundancy -- and introduces both approximately dual probabilistic frames and pseudo-dual probabilistic frames. We show that the canonical dual probabilistic frame is the only dual frame of pushforward type of a probabilistic frame with zero redundancy. Furthermore, we show that probabilistic frames with finite redundancy are atomic and finite. Approximately dual probabilistic frames generalize duality, with pseudo-duality being a further generalization. We introduce these concepts and prove certain structural results. In particular, every probabilistic frame has a discrete finite frame as an approximate dual.