In this talk we plan to present a survey on results concerning so-called STFT-multipliers, alternatively called Anti-Wick operators, realized by multiplying the short-time Fourier transform of a distribution by some function or distribution. In the case this is a discrete measures sitting on a lattice we are talking about Gabor multipliers.

We are going to recall the concept of Banach Gelfand triples, based on the Segal algebra SO(Rd) (also known as modulation space M1(Rd)) which allows to describe the mapping properties of such operators under quite natural and general assumptions on the ingredients. From the computational point of view the question is how to find realizable variants of these operators, i.e. to describe finite-dimensional approximations which can be carried out on a computer, while at the same time approximating the output of a given operator to a given degree. Different choices of the Gabor atoms with corresponding adapted lattices in phase spaces are considered.