Frames for Metric Spaces

TL;DR

This paper systematically studies frames in metric spaces, establishing their existence, connections to Banach space frames, and stability properties, thereby advancing the understanding of metric space analysis.

Contribution

It introduces the concept of metric frames, proves their existence in separable metric spaces, and links them to Banach space frames through Lipschitz-free spaces.

Findings

Every separable metric space admits a metric -frame.

Established a correspondence between metric frames and Banach space frames.

Derived stability results for metric frames.

Abstract

We make a systematic study of frames for metric spaces. We prove that every separable metric space admits a metric $\mathcal{M}_d$-frame. Through Lipschitz-free Banach spaces we show that there is a correspondence between frames for metric spaces and frames for subsets of Banach spaces. We derive some characterizations of metric frames. We also derive stability results for metric frames.