The Homogeneous Approximation Property and the Comparison Theorem for Coherent Frames

TL;DR

This paper demonstrates that the homogeneous approximation property and the comparison theorem are valid for all coherent frames, extending existing theories and addressing previously unresolved questions about frame density.

Contribution

It establishes the validity of key properties for arbitrary coherent frames, broadening the scope of frame theory beyond prior limitations.

Findings

Homogeneous approximation property holds for all coherent frames

Comparison theorem applies universally to coherent frames

Addresses open questions about frame density in the literature

Abstract

We show that the homogeneous approximation property and the comparison theorem hold for arbitrary coherent frames. This observation answers some questions about the density of frames that are not covered by the theory of Balan, Casazza, Heil, and Landau. The proofs are a variation of the method developed by Ramanathan and Steger.