Discretization in generalized coorbit spaces: extensions, annotations and errata for "Continuous Frames, Function Spaces and the Discretization Problem" by M. Fornasier and H. Rauhut

TL;DR

This paper revises and extends a key section of Fornasier and Rauhut's work on discretization in coorbit spaces, correcting errors, adding clarifications, and generalizing definitions for broader applicability.

Contribution

It provides a corrected, annotated, and generalized version of a crucial section from Fornasier and Rauhut's foundational paper on discretization in coorbit spaces.

Findings

Corrected typographical errors in the original section.

Added annotations for clarity and accessibility.

Generalized a central definition to support broader applications.

Abstract

During the process of writing the manuscript ["Continuous warped time-frequency representations - Coorbit spaces and discretization", N. Holighaus, C. Wiesmeyr and P. Balazs], the work ["Continuous Frames, Function Spaces and the Discretization Problem" by M. Fornasier and H. Rauhut - (1)] was one of the major foundations of our results and, naturally, we found ourselves going back to reading that contribution once and again. In particular in Section 5, which is concerned with the discretization problem, we have found some typographical errors, small inaccuracies and some parts that we just would have wished to be slightly more accessible. Finally, for our own theory, a generalization of a central definition required us to verify that all the derivations in Section 5 of (1) still hold after the necessary modifications. Considering the importance of the results in (1) for the community, this was reason enough to start this side project of re-writing that least accessible portion of Fornasier and Rauhut's manuscript, eliminating the errors we found, adding annotations where we consider them useful and modifying the results to take into account the generalization we require for our own work. What you see is the result of our endeavor, a one-to-one substitute for Section 5 in (1).