Additive Stability of Frames

TL;DR

This paper presents an algorithm for transforming frames into tight frames via additive perturbations, providing bounds for perturbation stability and exploring extensions to infinite-dimensional spaces.

Contribution

It introduces a novel iterative perturbation method to produce tight frames and analyzes stability bounds and extension conditions.

Findings

Algorithm successfully produces tight frames in finite steps

Sharp bounds established for additive perturbations preserving frames

Conditions identified for extending results to infinite-dimensional spaces

Abstract

Given a frame in a finite dimensional Hilbert space we construct additive perturbations which decrease the condition number of the frame. By iterating this perturbation, we introduce an algorithm that produces a tight frame in a finite number of steps. Additionally, we give sharp bounds on additive perturbations which preserve frames and we study the effect of appending and erasing vectors to a given tight frame. We also discuss under which conditions our finite-dimensional results are extendable to infinite-dimensional Hilbert spaces.