Redundancy of Probabilistic Frames and Approximately Dual Probabilistic Frames

This paper studies the redundancy of probabilistic frames and introduces approximately dual probabilistic frames. The redundancy of a probabilistic frame is defined as the dimension of the kernel of the associated synthesis operator, which turns out to be the semi-Fredholm index of the synthesis operator. For a given non-redundant probabilistic frame, we claim that the canonical dual probabilistic frame is the only dual frame of pushforward type. Furthermore, we show that probabilistic frames with finite redundancy are atomic. In the second part, we introduce approximately dual probabilistic frames and characterize all the approximate duals of pushforward type. We also study the approximately dual frames of perturbed probabilistic frames, and show that if a probability measure is close to a probabilistic dual pair in some sense, then this probability measure is an approximately dual probabilistic frame. In particular, this yields discrete approximate duals of any probabilistic frame.