Tight projections of frames on infinite dimensional Hilbert spaces

TL;DR

This paper characterizes which frames in infinite dimensional Hilbert spaces can be projected onto tight frames for subspaces, revealing that many frames cannot be projected onto infinite dimensional tight frames, contrary to finite-dimensional cases.

Contribution

It provides a characterization of frames in infinite dimensional spaces that can be projected onto tight frames, highlighting differences from finite-dimensional results.

Findings

Certain frames cannot be projected onto infinite dimensional tight frames.

Finite-dimensional results do not extend straightforwardly to infinite dimensions.

A large class of infinite-dimensional frames cannot produce tight frames through projection.

Abstract

We characterize the frames on an infinite dimensional separable Hilbert space that can be projected to a tight frame for an infinite dimensional subspace. A result of Casazza and Leon states that an arbitrary frame for a 2N- or (2N-1)-dimensional Hilbert space can be projected to a tight frame for an N-dimensional subspace. Surprisingly, we demonstrate a large class of frames for infinite dimensional Hilbert spaces which cannot be projected to a tight frame for any infinite dimensional subspace.