Localized Bounded Below Approximate Schauder Frames are Finite Unions of Approximate Riesz Sequences

TL;DR

This paper introduces the concept of localization for approximate Schauder frames and Riesz sequences, demonstrating that localized bounded below ASFs can be decomposed into a finite union of ARSs, building on the Feichtinger conjecture.

Contribution

It extends the Feichtinger conjecture to localized approximate Schauder frames, showing they can be expressed as finite unions of approximate Riesz sequences.

Findings

Localized bounded below ASFs are finite unions of ARSs

Built on the Feichtinger conjecture and its localized version

Provides a new structural understanding of ASFs and ARSs

Abstract

Based on the truth of Feichtinger conjecture by Marcus, Spielman and Srivastava \textit{[Ann. of Math. (2), 2015]} and from the localized version by Gr\"{o}chenig \textit{[Adv. Comput. Math., 2003]}, we introduce the notion of localization of approximate Schauder frames (ASFs) and approximate Riesz sequences (ARSs). We show that localized bounded below ASFs are finite unions of ARSs.