5 research outputs found
Unconditional convergence and invertibility of multipliers
In the present paper the unconditional convergence and the invertibility of
multipliers is investigated. Multipliers are operators created by (frame-like)
analysis, multiplication by a fixed symbol, and resynthesis. Sufficient and/or
necessary conditions for unconditional convergence and invertibility are
determined depending on the properties of the analysis and synthesis sequences,
as well as the symbol. Examples which show that the given assertions cover
different classes of multipliers are given. If a multiplier is invertible, a
formula for the inverse operator is determined. The case when one of the
sequences is a Riesz basis is completely characterized.Comment: 31 pages; changes to previous version: 1.) the results from the
previous version are extended to the case of complex symbols m. 2.) new
statements about the unconditional convergence and boundedness are added
(3.1,3.2 and 3.3). 3.) the proof of a preliminary result (Prop. 2.2) was
moved to a conference proceedings [29]. 4.) Theorem 4.10. became more
detaile
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 Theory for Signal Processing in Psychoacoustics
This review chapter aims to strengthen the link between frame theory and
signal processing tasks in psychoacoustics. On the one side, the basic concepts
of frame theory are presented and some proofs are provided to explain those
concepts in some detail. The goal is to reveal to hearing scientists how this
mathematical theory could be relevant for their research. In particular, we
focus on frame theory in a filter bank approach, which is probably the most
relevant view-point for audio signal processing. On the other side, basic
psychoacoustic concepts are presented to stimulate mathematicians to apply
their knowledge in this field