Representation of the inverse of a frame multiplier

TL;DR

This paper explores the inverse of frame multipliers, revealing that for semi-normalized symbols, the inverse can be expressed using reciprocal symbols and dual frames, with implications for Gabor multipliers and applications.

Contribution

It provides a new representation of the inverse of invertible frame multipliers using dual frames and reciprocal symbols, and investigates conditions for canonical duals and Gabor multipliers.

Findings

Inverse of invertible frame multipliers can be represented with reciprocal symbols and dual frames.

One dual frame in the inverse representation is uniquely determined.

The set of dual frames uniquely determines the original frame.

Abstract

Certain mathematical objects appear in a lot of scientific disciplines, like physics, signal processing and, naturally, mathematics. In a general setting they can be described as frame multipliers, consisting of analysis, multiplication by a fixed sequence (called the symbol), and synthesis. They are not only interesting mathematical objects, but also important for applications, for example for the realization of time-varying filters. In this paper we show a surprising result about the inverse of such operators, if existing, as well as new results about a core concept of frame theory, dual frames. We show that for semi-normalized symbols, the inverse of any invertible frame multiplier can always be represented as a frame multiplier with the reciprocal symbol and dual frames of the given ones. Furthermore, one of those dual frames is uniquely determined and the other one can be arbitrarily chosen. We investigate sufficient conditions for the special case, when both dual frames can be chosen to be the canonical duals. In connection to the above, we show that the set of dual frames determines a frame uniquely. Furthermore, for a given frame, the union of all coefficients of its dual frames is dense in $\ell^2$. We also investigate invertible Gabor multipliers. Finally, we give a numerical example for the invertibility of multipliers in the Gabor case.