The aim of the lecture is to offer a brief introduction to time-frequency localization operators. To motivate the talk we first discuss Fourier multiplier operators. Such operators are used for example in the design of frequency filters in signal analysis. However, in some situations it is of interest to treat time-frequency plane as one geometric whole rather than as two separate spaces. Operators which perform simultaneous localization in time and in frequency arise as a natural extension of Fourier multipliers. Historically, such operators were first observed by Felix Berezin in the context of quantization problem in quantum mechanics in early 1970s. 35 years ago Ingrid Daubechies published an influential paper on localization operators and their applications in optics and signal analysis. A convenient framework for the study of localization operators as short-time Fourier transform multipliers is given by Elena Cordero and Karlheinz Gröchenig in 2003. Feichtingers modulation spaces are used when considering continuity properties of such operators. The central part of the talk is devoted to some basic properties of localization operators and their connection to pseudodifferential operators. We will also briefly discuss bilinear localization operators. In the last part of the talk we will reconsider localization operators as continuous frame multipliers defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. Finally, we consider the tensor product setting for continuous frame multipliers. A specific feature in such context is the notion of partial trace. By using the partial trace theorem we will offer an interpretation of tensor product continuous frame multipliers as density operators for bipartite quantum systems in quantum mechanics.
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