Localization Operators Via Time-Frequency Analysis

Abstract

Abstract. A systematic overview of localization operators using a time-frequency approach is given. Sufficient and necessary regularity results for localization operators with symbols and windows living in various function spaces (such as L p or modulation spaces) are discussed. Finally, an exact and an asymptotic product formulae are presented.

On the Duality of Regular and Local Functions

“…Detailed studies of the interaction of both regularizations and localizations, can be found for example in [49,53,54,77] and in engineering literature, we encounter these interactions in terms of the interplay between "windowing" on one hand and "interpolation" on the other. An equivalent is the so-called "zero-padding" technique (e.g.…”

“…It culminated, however, in the term "localization operator". It appears in 1988 for the first time (see [53], p. 133 in [54]) in Daubechies' article [55] and later in Daubechies' 1992 standard textbook [52]. Meanwhile, "localizations" occur in many publications [2,52,[54][55][56][57][58][59][60][61][62][63], amongst others as "localized trigonometric functions" or "localized sine basis" [52,57,64], as "localized frames" [65], "local trigonometric bases", as "local representations" [6] or simply in terms of "locally integrable" functions.…”

Multiplier Theorems for the Short-Time Fourier Transform

“…If g(t) = γ(t) = e −πt 2 , then M g,γ,λ is the classical Anti-Wick operator. M g,γ,λ was investigated in many papers, such as Berezin [1], Shubin [25], Wong [29], Feichtinger and Nowak [13], Boggiato, Cordero, Gröchenig, Tabacco [2,4,5,6,7,8].…”