Canonical forms of unconditionally convergent multipliers

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 (called multipliers in canonical form).

Hamiltonians defined by biorthogonal sets

“…Once we put D φ := span{φ n } and D ψ : Remark 12 It is worth to notice that the operators H α φ,ψ and H α ψ,φ introduced above looks quite similar to the so called multipliers, see [32,33], which are operators of the form…”

“…where {Φ n } and {Ψ n } are fixed sequences in the Hilbert space, while {m n } is a sequence of scalars. The interest in [32,33] was on mathematical aspects of these multipliers, like, for instance, their unconditional convergence. On the other hand, we are more interested in the role of G-quasi bases in the definition of our (physically-motivated) multipliers.…”

Frame Theory for Signal Processing in Psychoacoustics

“…However, in contrast to frame operators, multipliers in general loose the bijectivity (as well as self-adjointness and positivity). For some applications it might be necessary to invert multipliers, which brings the interest to bijective multipliers and formulas for their inverses -for interested readers, we refer to [10,[82][83][84] for some investigation in this direction.…”

The dual frame induced by an invertible frame multiplier

“…), the multiplier M m,Φ,Ψ will be denoted by M (c),Φ,Ψ . For works oriented to invertibility of multipliers refer to [6,22,24,25]. Notice that the assumptions of all assertions in the next sections lead to welldefindedness of the multipliers M m,Φ,Ψ and we will not mention this explicitly in the statements.…”