Gabor systems and almost periodic functions

Boggiatto, Paolo; Fernández, Carmen; Galbis, Antonio

doi:10.1016/j.acha.2015.07.004

Cited by 10 publications

(5 citation statements)

References 20 publications

(28 reference statements)

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“…The space of almost periodic function on R is a closed subspace of L ∞ (R) spanned by the set of functions of the form e iλt , λ ∈ R. EIn other words, we say that the space of almost periodic functions is the uniform closure of the trigonometric polynomials of the form ∑ m 1 h k e iλ k t , where h k ∈ C, λ k ∈ R and m ∈ N. All almost periodic functions are uniformly continuous and bounded in nature. These functions have received considerable attention across various disciplines of science and technology such as quantum mechanics, optics, geophysics, differential equations, harmonic analysis, and so on [15,16]. Keeping the exciting developments of almost periodic functions in hindsight, it is desirable to study the behaviour of almost periodic functions in the framework of quadratic-phase wave-packet transform.…”

Section: Introductionmentioning

confidence: 99%

Quadratic-Phase Wave-Packet Transform in L2(R)

2022

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Wavelet transform is a powerful tool for analysing the problems arising in harmonic analysis, signal and image processing, sampling, filtering, and so on. However, they seem to be inadequate for representing those signals whose energy is not well concentrated in the frequency domain. In pursuit of representations of such signals, we propose a novel time-frequency transform coined as quadratic-phase wave packet transform in L2(R). The proposed transform is aimed at rectifying the conventional wavelet transform by employing a quadratic-phase Fourier transform with extra degrees of freedom. Besides the formulation of all the fundamental results, including the orthogonality relation, reconstruction formula and the characterization of range, we also derive a direct relationship between the well-known Wigner-Ville distribution and the proposed transform. In addition, we study the quadratic-phase wave-packet transform in the framework of almost periodic functions. Finally, we extend the scope of the present work by investigating the composition of quadratic-phase wave packet transforms.

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Section: Introductionmentioning

confidence: 99%

Quadratic-Phase Wave-Packet Transform in L2(R)

2022

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“…Almost periodic functions were recently investigated in the connection with Gabor frames in [26,10,4]. As the space of almost periodic functions is non-separable, it can not admit countable frames, and the problem arises in which sense frame-type inequalities are still possible for norm estimation in this space [10,26,4]. In [4] the authors also provide Gabor frames for a suitable separable subspaces of the space of almost periodic functions.…”

Section: Introductionmentioning

confidence: 99%

“…As the space of almost periodic functions is non-separable, it can not admit countable frames, and the problem arises in which sense frame-type inequalities are still possible for norm estimation in this space [10,26,4]. In [4] the authors also provide Gabor frames for a suitable separable subspaces of the space of almost periodic functions. We, on the other hand, use almost periodic functions as a tool to develop existence results for irregular Gabor frames for the space of square integrable functions.…”

Section: Introductionmentioning

confidence: 99%

Gabor Frames for Model Sets

Matusiak

2019

J Fourier Anal Appl

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We generalize three main concepts of Gabor analysis for lattices to the setting of model sets: Fundamental Identity of Gabor Analysis, Janssen's representation of the frame operator and Wexler-Raz biorthogonality relations. Utilizing the connection between model sets and almost periodic functions, as well as Poisson's summations formula for model sets we develop a form of a bracket product that plays a central role in our approach. Furthermore, we show that, if a Gabor system for a model set admits a dual which is of Gabor type, then the density of the model set has to be greater than one.and denote the inverse Fourier transform of f byf . In the sequel we will distinguish between Fourier transform on L 2 (R d ) and Fourier transform on L 2 (R 2d ) by writing F f for the latter, f ∈ L 2 (R 2d ).The Fourier transform has a property of interchanging translation and modulation, that isThe translation and modulation operators obey the following commutation relationCombining the last two properties, we have thatGiven a non-zero function g ∈ L 2 (R d ), the short-time Fourier transform of f ∈ L 2 (R d ) with respect to the window g, is defined asWe define the modulation spaces as follows: fix a non-zero Schwartz function g ∈ S(R d ), and letwith the norm f M p = V g f p . Different choices for g give rise to equivalent norms on M p (R d ). For p = 2, we have M 2 (R d ) = L 2 (R d ). The space M 1 (R d ), known as Feichtinger's algebra, can also be characterized asProposition 2.1.[11] The space M 1 (R d ) has the following properties:is a Banach algebra under pointwise multiplication.ii) M 1 (R d ) is a Banach algebra under convolution.iii) M 1 (R d ) is invariant under time-frequency shifts.vi) M 1 (R d ) is invariant under Fourier transform.

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“…Almost periodic functions were recently investigated in the connection with Gabor frames in [33,15,6]. As the space of almost periodic functions is non-separable, it can not admit countable frames, and the problem arises in which sense frame-type inequalities are still possible for norm estimation in this space [15,33,6]. In [6] the authors also provide Gabor frames for a suitable separable subspaces of the space of almost periodic functions.…”

Section: Introductionmentioning

confidence: 99%

“…As the space of almost periodic functions is non-separable, it can not admit countable frames, and the problem arises in which sense frame-type inequalities are still possible for norm estimation in this space [15,33,6]. In [6] the authors also provide Gabor frames for a suitable separable subspaces of the space of almost periodic functions. We, on the other hand, use almost periodic functions as a tool to develop existence results for irregular frames for the space of square integrable functions.…”

Section: Introductionmentioning

confidence: 99%

Frames of translates for model sets

Matusiak¹

2018

Preprint

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We study spanning properties of a family of functions translated along simple model sets. We characterize tight frame and dual frame generators for such irregular translates and we apply the results to Gabor systems. We use the connection between model sets and almost periodic functions and rely strongly on a Poisson summations formula for model sets to introduce the so-called bracket product, which then plays a crucial role in our approach. As a corollary to our main results we obtain a density statement for semi-regular Gabor frames.

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Gabor systems and almost periodic functions

Cited by 10 publications

References 20 publications

Quadratic-Phase Wave-Packet Transform in L2(R)

Quadratic-Phase Wave-Packet Transform in L2(R)

Gabor Frames for Model Sets

Frames of translates for model sets

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