Expansion of approximate Bessel sequences to approximate Schauder frames for Banach spaces

TL;DR

This paper investigates the conditions under which approximate Bessel sequences in Banach spaces can be expanded into approximate Schauder frames, revealing limitations and characterizations distinct from Hilbert space theory.

Contribution

It characterizes Banach spaces where approximate Bessel sequences can be expanded into approximate Schauder frames, highlighting differences from Hilbert space results.

Findings

Not all Banach spaces allow such expansions.

A characterization of Banach spaces enabling the expansion.

Counterexamples showing limitations in Banach spaces.

Abstract

It is known in Hilbert space frame theory that a Bessel sequence can be expanded to a frame. Contrary to Hilbert space situation, using a result of Casazza and Christensen, we show that there are Banach spaces and approximate Bessel sequences which can not be expanded to approximate Schauder frames. We characterize Banach spaces in which one can expand approximate Bessel sequences to approximate Schauder frames.