Can any unconditionally convergent multiplier be transformed to have the symbol (1) and Bessel sequences by shifting weights?

TL;DR

This paper investigates the conditions under which unconditionally convergent multipliers can be transformed into a standard form with symbol (1) and Bessel sequences by shifting weights, with implications for signal processing.

Contribution

It establishes necessary and sufficient conditions for unconditional convergence of multipliers and shows they can be transformed into a canonical form with symbol (1) and Bessel sequences.

Findings

Unconditionally convergent multipliers can be transformed to have symbol (1) and Bessel sequences.

Necessary and sufficient conditions for unconditional convergence are identified.

Shifting weights between symbol and sequence achieves the transformation.

Abstract

Multipliers are operators that combine (frame-like) analysis, a multiplication with a fixed sequence, called the symbol, and synthesis. They are very interesting mathematical objects that also have a lot of applications for example in acoustical signal processing. It is known that bounded symbols and Bessel sequences guarantee unconditional convergence. In this paper we investigate necessary and equivalent conditions for the unconditional convergence of multipliers. In particular we show that, under mild conditions, unconditionally convergent multipliers can be transformed by shifting weights between symbol and sequence, into multipliers with symbol (1) and Bessel sequences.