Weaving Schauder frames

TL;DR

This paper extends the concept of weaving frames from Hilbert spaces to Banach spaces, establishing conditions under which different types of Schauder bases and frames are woven and stable under perturbations.

Contribution

It introduces the notion of weaving Schauder frames in Banach spaces, explores conditions for their equivalence, and proves perturbation theorems for approximate Schauder frames.

Findings

Every weaving of two approximate Schauder frames is an approximate Schauder frame iff uniformly bounded.

Two notions of weaving Schauder bases coincide when all weavings are unconditional.

Perturbation theorems ensure stability of approximate Schauder frames under small changes.

Abstract

We extend the concept of weaving Hilbert space frames to the Banach space setting. Similar to frames in a Hilbert space, we show that for any two approximate Schauder frames for a Banach space, every weaving is an approximate Schauder frame if and only if there is a uniform constant $C\geq 1$ such that every weaving is a $C$-approximate Schauder frame. We also study weaving Schauder bases, where it is necessary to introduce two notions of weaving. On one hand, we can ask if two Schauder bases are woven when considered as Schauder frames with their biorthogonal functionals, and alternatively, we can ask if each weaving of two Schauder bases remains a Schauder basis. We will prove that these two notions coincide when all weavings are unconditional, but otherwise they can be different. Lastly, we prove two perturbation theorems for approximate Schauder frames.