# Continuous Schauder frames for Banach spaces

## TL;DR

This paper introduces continuous Schauder frames for Banach spaces, generalizing existing concepts and demonstrating their application to wavelets in Lp spaces, while extending fundamental Banach space theorems.

## Contribution

It defines continuous Schauder frames, connects them to wavelet systems, and extends classical Banach space properties to this new framework.

## Key findings

- Wavelets in Lp generate continuous Schauder frames.
- Many James theorems extend to continuous Schauder frames.
- Continuous Schauder frames unify concepts from Hilbert and Banach space theory.

## Abstract

We introduce the notion of a continuous Schauder frame for a Banach space. This is both a generalization of continuous frames and coherent states for Hilbert spaces and a generalization of unconditional Schauder frames for Banach spaces. As a natural example, we prove that any wavelet for $L_p(\R)$ with $1<p<\infty$ generates a continuous wavelet Schauder frame. Furthermore, we generalize the properties shrinking and boundedly complete to the continuous Schauder frame setting, and prove that many of the fundamental James theorems still hold in this general context.