A Geometric Construction of Tight Multivariate Gabor Frames with Compactly Supported Smooth Windows

Pfander, Götz E.; Rashkov, Peter; Yang, Wang

doi:10.1007/s00041-011-9198-x

Cited by 17 publications

(17 citation statements)

References 36 publications

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“…We will need to be familiar with the theory of frames (see [1], [9] and [6]). Given a countable sequence {f i } i∈I of vectors in a Hilbert space H, we say {f i } i∈I forms a frame if and only if there exist strictly positive real numbers A, B such that for any vector f ∈ H,…”

Section: Results On Frames and Orthonormal Basesmentioning

confidence: 99%

Sampling and interpolation on some nilpotent Lie groups

Oussa

2014

Forum Mathematicum

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Let N be a non-commutative, simply connected, connected, two-step nilpotent Lie group with Lie algebra n such that,j≤d is a non-vanishing homogeneous polynomial in the unknowns Z 1 , · · · , Z n−2d where {Z 1 , · · · , Z n−2d } is a basis for the center of the Lie algebra. Using well-known facts from time-frequency analysis, we provide some precise sufficient conditions for the existence of sampling spaces with the interpolation property, with respect to some discrete subset of N . The result obtained in this work can be seen as a direct application of time-frequency analysis to the theory of nilpotent Lie groups. Several explicit examples are computed. This work is a generalization of recent results obtained for the Heisenberg group by Currey, and Mayeli in [3].

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Section: Results On Frames and Orthonormal Basesmentioning

confidence: 99%

Sampling and interpolation on some nilpotent Lie groups

Oussa

2014

Forum Mathematicum

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“…calculated along the Hamiltonian trajectory leading from z 0 at time t 0 = 0 to z t at time t. One shows that under suitable conditions on the Hamiltonian H the approximate solution satisfies, for |t| ≤ T , an estimate of the type ||ψ(·, t) − ψ(·, t)|| ≤ C(z 0 , T ) √ |t| (45) where C(z 0 , T ) is a positive constant depending only on the initial point z 0 and the time interval [−T, T ] (Hagedorn [30,31]).…”

Section: The Semiclassical Approachmentioning

confidence: 99%

“…Strangely enough, the use of symplectic techniques in the theory of Gabor frames is very often ignored; one example (among many others) being Casazza's seminal paper [9] on modern tools for Weyl-Heisenberg frame theory, where the word "symplectic" does not appear a single time in the 127 pages of this paper! There are however exceptions: in Gröchenig's treatise [27] the metaplectic representation is used to study various symmetries; the same applies to the recent paper by Pfander et al [45], elaborating on earlier work [29] by Han and Wang, where symplectic transformations are exploited to study various properties of Gabor frames.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian deformations of Gabor frames: First steps

Gosson

2015

Applied and Computational Harmonic Analysis

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Gabor frames can advantageously be redefined using the Heisenberg–Weyl operators familiar from harmonic analysis and quantum mechanics. Not only does this redefinition allow us to recover in a very simple way known results of symplectic covariance, but it immediately leads to the consideration of a general deformation scheme by Hamiltonian isotopies (i.e. arbitrary paths of non-linear symplectic mappings passing through the identity). We will study in some detail an associated weak notion of Hamiltonian deformation of Gabor frames, using ideas from semiclassical physics involving coherent states and Gaussian approximations. We will thereafter discuss possible applications and extensions of our method, which can be viewed – as the title suggests – as the very first steps towards a general deformation theory for Gabor frames.

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“…Riesz sequences) π(Γ)g with arbitrary smooth window g ∈ L 2 (R d ) exist for any lattice Γ ≤ R 2d satisfying vol(R 2d /Γ) < 1 (resp. vol(R 2d /Γ) > 1), see also [58,96]. Only recently has there been a first contribution [62] to this existence problem for Gabor frames in higher dimensions, namely for so-called nonrational lattices Γ ≤ R 2d , by exploiting the structural results on (irrational) non-commutative tori [103] and its link with Gabor frames [76]; see Section 7.1 for a more detailed discussion.…”

Section: Introductionmentioning

confidence: 99%

Smooth lattice orbits of nilpotent groups and strict comparison of projections

Bédos,

Enstad,

van Velthoven

2021

Preprint

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This paper provides sufficient density conditions for the existence of smooth vectors generating a frame or Riesz sequence in the lattice orbit of a square-integrable projective representation of a nilpotent Lie group. The conditions involve the product of lattice co-volume and formal dimension, and complement Balian-Low type theorems for the nonexistence of smooth frames and Riesz sequences at the critical density. The proof hinges on a connection between smooth lattice orbits and generators for an explicitly constructed finitely generated Hilbert C * -module. An important ingredient in the approach is that twisted group C * -algebras associated to finitely generated nilpotent groups have finite decomposition rank, hence finite nuclear dimension, which allows us to deduce that any matrix algebra over such a simple C * -algebra has strict comparison of projections.

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A Geometric Construction of Tight Multivariate Gabor Frames with Compactly Supported Smooth Windows

Cited by 17 publications

References 36 publications

Sampling and interpolation on some nilpotent Lie groups

Sampling and interpolation on some nilpotent Lie groups

Hamiltonian deformations of Gabor frames: First steps

Smooth lattice orbits of nilpotent groups and strict comparison of projections

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