Piecewise scalable frames

TL;DR

This paper introduces piecewise scalable frames, a new scaling method that transforms many frames into Parseval frames, with specific conditions and properties, especially in low-dimensional spaces.

Contribution

It defines the concept of piecewise scalable frames, establishes necessary and sufficient conditions, and explores their properties and invariance under transformations.

Findings

All frames in R^2 and R^3 are piecewise scalable.

Piecewise scalability is preserved under unitary transformations.

Frames with closely spaced vectors may not be piecewise scalable.

Abstract

In this paper we define "piecewise scalable frames". This new scaling process allows us to alter many frames to Parseval frames which is impossible by the previous standard scaling. We give necessary and sufficient conditions for a frame to be piecewise scalable. We show that piecewise scalability is preserved under unitary transformations. Unlike standard scaling, we show that all frames in $\RR^2$ and $\RR^3$ are piecewise scalable. We also show that if the frame vectors are close to each other, then they might not be piecewise scalable. Several properties of scaling constants are also presented.