Approximately Dual and Pseudo-Dual Probabilistic Frames
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.
Cite
@article{arxiv.2505.13885,
title = {Approximately Dual and Pseudo-Dual Probabilistic Frames},
author = {Dongwei Chen and Emily J. King and Clayton Shonkwiler},
journal= {arXiv preprint arXiv:2505.13885},
year = {2026}
}
Comments
Revised version from a major revision. A stronger result about the redundancy of probabilistic frames is added at Theorem 3.6