On the dual frame induced by an invertible frame multiplier

TL;DR

This paper extends the theory of invertible frame multipliers by proving the uniqueness of a specific dual frame, generalizing to non-semi-normalized symbols, and characterizing when it coincides with the canonical dual.

Contribution

It generalizes previous results by removing the semi-normalized restriction and establishes the uniqueness of the specific dual frame among all sequences.

Findings

The specific dual is unique among all Bessel sequences.

The results hold even when the symbol is not semi-normalized.

Conditions are characterized when the canonical and specific dual frames coincide.

Abstract

Recently it has been established that given an invertible frame multiplier with semi-normalized symbol, a specific dual of any of the two involved frames can be determined for the inversion purpose. The inverse can be represented as a multiplier with the reciprocal symbol, this particular dual of one of the given frames, and any dual of the other frame. The specific dual is the only one having this property among all Bessel sequences. In this manuscript we extend the results showing that the specific dual with the above mentioned property is unique among all possible sequences. Furthermore, we allow the symbol to be not necessarily semi-normalized. Finally we characterize cases when the canonical dual frame and the new specific dual frame coincide.