Weighted and controlled frames

TL;DR

This paper systematically develops the theory of weighted and controlled frames, showing their relationships, equivalences, and numerical advantages, especially in the context of Gabor frames and frame operator inversion.

Contribution

It introduces a systematic development of weighted and controlled frames, establishes their mutual relationships, and explores their numerical benefits and applications.

Findings

Controlled frames are equivalent to standard frames, aiding in checking the frame condition.

Weighted frames can be optimized to tighten frames by adjusting weights.

Numerical analysis shows effectiveness in Gabor frames and dual frame approximation.

Abstract

Weighted and controlled frames have been introduced recently to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper we develop systematically these notions, including their mutual relationship. We will show that controlled frames are equivalent to standard frames and so this concept gives a generalized way to check the frame condition, while offering a numerical advantage in the sense of preconditioning. Next, we investigate weighted frames, in particular their relation to controlled frames. We consider the special case of semi-normalized weights, where the concepts of weighted frames and standard frames are interchangeable. We also make the connection with frame multipliers. Finally we analyze weighted frames numerically. First we investigate three possibilities for finding weights in order to tighten a given frame, i.e., decrease the frame bound ratio. Then we examine Gabor frames and how well the canonical dual of a weighted frame is approximated by the inversely weighted dual frame.