Optimal properties of the canonical tight probabilistic frame

A probabilistic frame is a Borel probability measure with finite second moment whose support spans R. A Parseval probabilistic frame is one for which the associated matrix of second moment is the identity matrix in R. Each probabilistic frame is canonically associated to a Parseval probabilistic frame. In this paper, we show that this canonical Parseval probabilistic frame is the closest Parseval probabilistic frame to a given probabilistic frame in the 2−Wasserstein distance. Our proof is based on two main ingredients. On the one hand, we show that a probabilistic frame can be approximated in the 2−Wasserstein metric with (compactly supported) finite frames whose bounds can be controlled. On the other hand we establish some fine continuity properties of the function that maps a probabilistic frame to its canonical Parseval probabilistic frame. Our results generalize similar ones for finite frames and their associated Parseval frames.