In this paper we examine the general theory of continuous frame multipliers in Hilbert space. These operators are a generalization of the widely used notion of (discrete) frame multipliers. Well-known examples include Anti-Wick operators, STFT multipliers or Calderón-Toeplitz operators. Due to the possible peculiarities of the underlying measure spaces, continuous frames do not behave quite as well as their discrete counterparts. Nonetheless, many results similar to the discrete case are proven for continuous frame multipliers as well, for instance compactness and Schatten class properties. Furthermore, the concepts of controlled and weighted frames are transferred to the continuous setting.2000 Mathematics Subject Classification. Primary 42C40; Secondary 41A58, 47A58.

Abstract. Time-frequency localization operators are a quantization procedure that maps symbols on R 2d to operators and depends on two window functions. We study the range of this quantization and its dependence on the window functions. If the short-time Fourier transform of the windows does not have any zero, then the range is dense in the Schatten p-classes. The main tool is new version of the Berezin transform associated to operators on L 2 (R d ). Although some results are analogous to results about Toeplitz operators on spaces of holomorphic functions, the absence of a complex structure requires the development of new methods that are based on time-frequency analysis.

The α-modulation transform is a time-frequency transform generated by square-integrable representations of the affine Weyl-Heisenberg group modulo suitable subgroups. In this paper we prove new conditions that guarantee the admissibility of a given window function. We also show that the generalized coorbit theory can be applied to this setting, assuming specific regularity of the windows. This then yields canonical constructions of Banach frames and atomic decompositions in α-modulation spaces. In particular, we prove the existence of compactly supported (in time domain) vectors that are admissible and satisfy all conditions within the coorbit machinery, which considerably go beyond known results.Math Subject Classification: 42C15, 42C40, 46E15 (primary), 46E35, 57S25 (secondary).