Global symmetric approximation of frames

TL;DR

This paper addresses the problem of finding the best symmetric approximation of arbitrary frames by Parseval frames in infinite-dimensional Hilbert spaces, providing explicit solutions and conditions for uniqueness.

Contribution

It introduces a comprehensive solution to the symmetric approximation problem, utilizing geometric structures and index theory to characterize all optimal Parseval frame approximations.

Findings

Explicit description of all best Parseval frame approximations.

A criterion for the uniqueness of the best approximation.

Parametrization of connected components using projection indices.

Abstract

We solve the problem of best approximation by Parseval frames to an arbitrary frame in a subspace of an infinite dimensional Hilbert space. We explicitly describe all the solutions and we give a criterion for uniqueness. This best approximation problem was previously solved under an additional assumption on the set of Parseval frames in M. Frank, V. Paulsen, T. Tiballi, Symmetric approximation of frames and bases in Hilbert spaces, Trans. Amer. Math. Soc. 354 (2002), 777-793. Our proof relies on the geometric structure of the set of all Parseval frames quadratically close to a given frame. In the process we show that its connected components can be parametrized by using the notion of index of a pair of projections, and we prove existence and uniqueness results of best approximation by Parseval frames restricted to these connected components.