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Time-frequency partitions and characterizations of modulation spaces with localization operators. (English) Zbl 1210.42049
A time–frequency representation transforms a function \(f\) in \(\mathbb{R}^d\) into a function on the time–frequency space \(\mathbb{R}^d \times \mathbb{R}^d\). The goal is to obtain a description of \(f\) that is local both in time and in frequency. However, the pointwise interpretation of such time–frequency representation encounters difficulties because of the Heisenberg Uncertainty Principle. Therefore the question arises in which sense the short–time Fourier transform describes the local properties of a function and its Fourier transform.
Following I. Daubechies [IEEE Trans. Inf. Theory 34, No.4, 605–612 (1988; Zbl 0672.42007)] the authors use time–frequency operators to give meaning to the time–frequency content. By investigating a whole family of such operators, the authors obtain a new characterization of modulation spaces. This characterization relates the size of the localization operators to the global time–frequency distribution. As a by–product, they obtain a new proof of the existence of multi–window Gabor frames and extend the structure theory of Gabor frames,

MSC:
42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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