Dual fusion frames in the sense of Kutyniok, Paternostro and Philipp with applications to invertible Bessel fusion multipliers

TL;DR

This paper surveys dual fusion frames in the sense of Kutyniok, Paternostro, and Philipp, introduces a new definition of Bessel fusion multipliers, and studies dual fusion frames induced by invertible Bessel fusion multipliers.

Contribution

It provides a comprehensive survey of dual fusion frames, proposes a modified definition of Bessel fusion multipliers, and analyzes dual fusion frames generated by invertible Bessel fusion multipliers.

Findings

Different fusion frames have distinct duals.

The new Bessel fusion multiplier behaves like ordinary Bessel multipliers in many cases.

Dual fusion frames can be induced by invertible Bessel fusion multipliers.

Abstract

To achieve our main research goal, first we survey the approaches towards dual fusion frames existing in the literature and agree on the notion of duality for fusion frames in the sense of Kutyniok, Paternostro and Philipp ({\it Oper. Matrices} {\bf11} (2017), no. 2, 301--336). As a main result we show that different fusion frames have different dual fusion frames. Moreover, this duality notion leads to a new definition of Bessel fusion multipliers which is a slightly modified version of the commonly used definitions. Particularly, we show that with this definition in many cases Bessel fusion multipliers behave similar to ordinary Bessel multipliers. Finally, special attention is devoted to the study of dual fusion frames induced by an invertible Bessel fusion multiplier.