Wave packets and fourier integral operators

Continuous Frames, Function Spaces, and the Discretization Problem

“…The α-modulation spaces are known to have nice analysis properties. For instance, the mapping properties of pseudodifferential operators in Hörmander classes on α-modulation spaces as by Holschneider, Nazaret [42] and Borup [7] generalize classical results of Cordoba and Fefferman [10]. Moreover, we expect that such spaces have a key role in the study of pseudodifferential operators modeling the transmission of (digital) signals in wireless communications and in corresponding numerical methods [11].…”

“…So generalized concepts of coherent states have been introduced, where the continuous frame is indexed by a homogeneous space G/H [1]. Important examples can be described in this setting, such as continuous wavelet and Gabor frames on spheres [3,52] and continuous mixed Gabor / wavelet frames, i.e., continuous frames associated to the affine Weyl Heisenberg group [10,42,50,51,40,20,26]. As a matter of fact, the theory of Feichtinger and Gröchenig is no longer applicable in this setting.…”

Localization Operators Via Time-Frequency Analysis

“…Here we end up recalling an important application of formula (23). Indeed, formula (23) (24) represents a generalization of the classical result for Shubin-Sobolev spaces [28], recaptured in the framework of modulation spaces by means of (9).…”

“…Depending on the field of application, these operators are known under the names of Wick, anti-Wick or Toeplitz operators, as well as wave packets, Gabor or short time Fourier transform multipliers. There is a host of articles on the subject written by mathematicians, physicists and engineers; we refer, for instance, to [2,9,10,11,12,16,24,26,30,34].…”

Time-frequency partitions for the Gelfand triple $(S_0,L^2,S_0')$

“…They occur in signal analysis, in quantization procedures in physics [2], or in the approximation of pseudodifferential operators [5].…”