Paley-Wiener Theorem for Probabilistic Frames

This paper establishes Paley-Wiener perturbation theorems for probabilistic frames. The classical Paley-Wiener perturbation theorem shows that if a sequence is close to a basis in a Banach space, then this sequence is also a basis. Similar perturbation results have been established for frames in Hilbert spaces. In this work, we show that if a probability measure is sufficiently close to a probabilistic frame in an appropriate sense, then this probability measure is also a probabilistic frame. Moreover, we obtain explicit frame bounds for such probability measures that are close to a given probabilistic frame in the $2$-Wasserstein metric. This yields an alternative proof of the fact that the set of probabilistic frames is open in $\mathcal{P}_2(\mathbb{R}^n)$ under the $2$-Wasserstein topology.