4,614 research outputs found
Recent Progress in Shearlet Theory: Systematic Construction of Shearlet Dilation Groups, Characterization of Wavefront Sets, and New Embeddings
The class of generalized shearlet dilation groups has recently been developed
to allow the unified treatment of various shearlet groups and associated
shearlet transforms that had previously been studied on a case-by-case basis.
We consider several aspects of these groups: First, their systematic
construction from associative algebras, secondly, their suitability for the
characterization of wavefront sets, and finally, the question of constructing
embeddings into the symplectic group in a way that intertwines the
quasi-regular representation with the metaplectic one. For all questions, it is
possible to treat the full class of generalized shearlet groups in a
comprehensive and unified way, thus generalizing known results to an infinity
of new cases. Our presentation emphasizes the interplay between the algebraic
structure underlying the construction of the shearlet dilation groups, the
geometric properties of the dual action, and the analytic properties of the
associated shearlet transforms.Comment: 28 page
Continuous Frames, Function Spaces, and the Discretization Problem
A continuous frame is a family of vectors in a Hilbert space which allows
reproductions of arbitrary elements by continuous superpositions. Associated to
a given continuous frame we construct certain Banach spaces. Many classical
function spaces can be identified as such spaces. We provide a general method
to derive Banach frames and atomic decompositions for these Banach spaces by
sampling the continuous frame. This is done by generalizing the coorbit space
theory developed by Feichtinger and Groechenig. As an important tool the
concept of localization of frames is extended to continuous frames. As a
byproduct we give a partial answer to the question raised by Ali, Antoine and
Gazeau whether any continuous frame admits a corresponding discrete realization
generated by sampling.Comment: 44 page
Wavelet frames, Bergman spaces and Fourier transforms of Laguerre functions
The Fourier transforms of Laguerre functions play the same canonical role in
wavelet analysis as do the Hermite functions in Gabor analysis. We will use
them as analyzing wavelets in a similar way the Hermite functions were recently
by K. Groechenig and Y. Lyubarskii in "Gabor frames with Hermite functions, C.
R. Acad. Sci. Paris, Ser. I 344 157-162 (2007)". Building on the work of K.
Seip, "Beurling type density theorems in the unit disc, Invent. Math., 113,
21-39 (1993)", concerning sampling sequences on weighted Bergman spaces, we
find a sufficient density condition for constructing frames by translations and
dilations of the Fourier transform of the nth Laguerre function. As in
Groechenig-Lyubarskii theorem, the density increases with n, and in the special
case of the hyperbolic lattice in the upper half plane it is given by b\log
a<\frac{4\pi}{2n+\alpha}, where alpha is the parameter of the Laguerre
function.Comment: 15 page
Time-frequency transforms of white noises and Gaussian analytic functions
A family of Gaussian analytic functions (GAFs) has recently been linked to
the Gabor transform of white Gaussian noise [Bardenet et al., 2017]. This
answered pioneering work by Flandrin [2015], who observed that the zeros of the
Gabor transform of white noise had a very regular distribution and proposed
filtering algorithms based on the zeros of a spectrogram. The mathematical link
with GAFs provides a wealth of probabilistic results to inform the design of
such signal processing procedures. In this paper, we study in a systematic way
the link between GAFs and a class of time-frequency transforms of Gaussian
white noises on Hilbert spaces of signals. Our main observation is a conceptual
correspondence between pairs (transform, GAF) and generating functions for
classical orthogonal polynomials. This correspondence covers some classical
time-frequency transforms, such as the Gabor transform and the Daubechies-Paul
analytic wavelet transform. It also unveils new windowed discrete Fourier
transforms, which map white noises to fundamental GAFs. All these transforms
may thus be of interest to the research program `filtering with zeros'. We also
identify the GAF whose zeros are the extrema of the Gabor transform of the
white noise and derive their first intensity. Moreover, we discuss important
subtleties in defining a white noise and its transform on infinite dimensional
Hilbert spaces. Finally, we provide quantitative estimates concerning the
finite-dimensional approximations of these white noises, which is of practical
interest when it comes to implementing signal processing algorithms based on
GAFs.Comment: to appear in Applied and Computational Harmonic Analysi
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