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

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

Frame expansions for Gabor multipliers

AbstractDiscrete Gabor multipliers are composed of rank one operators. We shall prove, in the case of rank one projection operators, that the generating operators for such multipliers are either Riesz bases (exact frames) or not frames for their closed linear spans. The same dichotomy conclusion is valid for general rank one operators under mild and natural conditions. This is relevant since discrete Gabor multipliers have an emerging role in communications, radar, and waveform design, where redundant frame decompositions are increasingly applicable