Norms and Eigenvalues of Time-Frequency Localization Operators
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In this report we study and compare two types of time-frequency localization operators, the first is based on composition of projections in time and frequency, and the second is Daubechies' localization operator. We provide a review of several uncertainty principles in time-frequency analysis and formulate these principles in terms of the operator norm of the localization operators. Proceeding, the main focus is a particular kind of the Daubechies' localization operator. These operators are characterized by a window and a weight function, and with a Gaussian window and spherically symmetric weight we attain simple, explicit formulas for the eigenvalues. For such operators we consider the case when the weight takes the form of the characteristic function of some spherically symmetric subset of the time-frequency plane. Based on the measure of the subset in question, we determine simple upper and lower bound estimates for the operator norm. For some specific examples of subsets we provide more accurate estimates for the operator norm. Notably, we consider the spherically symmetric Cantor set and derive precise asymptotics for the operator norm of the associated localization operator.