**ABSTRACT**
###
A First Survey of Gabor Multipliers

####
H.G. Feichtinger

##### Department of
Mathematics, University of Vienna, Strudlhofgasse 4, A-1090, Vienna,
Austria

####
K. Nowak

##### Department of
Mathematics and Computer Science, Drexel University,
3141 Chestnut Street, Philadelphia, PA 19104-2875, USA

We describe various basic facts about *Gabor
multipliers* and their continuous
analogue which we will call *STFT-multipliers*.
These operators are obtained by going from the signal domain
to some transform domain, and applying a pointwise
multiplication operator before resynthesis. Although
such operators have been in use implicitly for quite some
time, this paper appears to be the first systematic
mathematical treatment of Gabor multipliers. Indeed, typical
time-frequency localization operators, or thresholding
algorithms involve simple $0/1$-multipliers.
The main results of this chapter are of a qualitative nature
and describe how the properties
of the Gabor multiplier depend on
the decay of the multiplier sequence, the time-frequency (TF)
concentration properties of the Gabor atom in use,
and the time-frequency-lattice.
These properties will be described in terms of the mapping
properties of the corresponding Gabor multiplier between
modulation spaces, or membership in some operator ideal
(such as trace-class or
Hilbert--Schmidt operator).
It is also possible to give relatively precise estimates on
behaviour of the sequence of eigenvalues of such operators,
especially for the case of tight Gabor frames.
We shall also discuss the problem of injectivity of the
linear mapping from the multiplier symbol to the operator,
recovery of Gabor multipliers from lower symbols, and a
related question concerning best approximation of operators
(e.g., from the Hilbert--Schmidt class) by Gabor multipliers
of a certain type.