Some Results for the Two-Sided Quaternionic Gabor Fourier Transform and Quaternionic Gabor Frame Operator

2020 ◽  
Vol 31 (1) ◽  
Author(s):  
Jinxia Li ◽  
Jianxun He
Keyword(s):  
Fourier Transform ◽  
Gabor Frame ◽  
Frame Operator

scholarly journals Invertibility of the Gabor frame operator on the Wiener amalgam space

2008 ◽  
Vol 153 (2) ◽  
pp. 212-224 ◽  
Author(s):  
Ilya A. Krishtal ◽  
Kasso A. Okoudjou
Keyword(s):  
Gabor Frame ◽  
Frame Operator ◽  
Wiener Amalgam Space

scholarly journals Hilbert space valued Gabor frames in weighted amalgam spaces

2019 ◽  
Vol 10 (4) ◽  
pp. 377-394
Author(s):  
Anirudha Poria ◽  
Jitendriya Swain
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Gabor Frame ◽  
Window Function ◽  
Gabor Frames ◽  
Frame Operator ◽  
Wiener Amalgam Space ◽  
Amalgam Spaces ◽  
Special Case

AbstractLet {\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the {\mathbb{H}}-valued Gabor frame operator on {\mathbb{H}}-valued weighted amalgam spaces {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space {W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space {\mathbb{H}}.

scholarly journals Behavior of Gabor frame operators on Wiener amalgam spaces

2016 ◽  
Vol 14 (04) ◽  
pp. 1650028 ◽  
Author(s):  
Anirudha Poria
Keyword(s):  
Weak Sense ◽  
Gabor Frame ◽  
Operator Norm ◽  
Identity Operator ◽  
Frame Operator ◽  
Wiener Amalgam Spaces ◽  
Sampling Density ◽  
Amalgam Spaces

It is well-known that the Gabor expansions converge to identity operator in weak* sense on the Wiener amalgam spaces as sampling density tends to infinity. In this paper, we prove the convergence of Gabor expansions to identity operator in the operator norm as well as weak* sense on [Formula: see text] as the sampling density tends to infinity. Also we show the validity of the Janssen’s representation and the Wexler–Raz biorthogonality condition for Gabor frame operator on [Formula: see text].

ON HILBERT TRANSFORM OF GABOR AND WILSON SYSTEMS

2014 ◽  
Vol 12 (02) ◽  
pp. 1450012 ◽  
Author(s):  
A. M. JARRAH ◽  
SUMAN PANWAR
Keyword(s):  
Hilbert Transform ◽  
Gabor Frame ◽  
Sufficient Condition ◽  
Gabor System ◽  
Frame Operator ◽  
Bessel Sequence ◽  
The Hilbert Transform

Given that the Gabor system {EmbTnag}m,n∈ℤ is a Gabor frame for L2(ℝ), a sufficient condition is obtained for the Gabor system {EmbTnaHg}m,n∈ℤ to be a Gabor frame, where Hg denotes the Hilbert transform of g ∈ L2(ℝ). It is proved that the Hilbert transform operator and the frame operator for the Gabor Bessel sequence {EmbTnag}m,n∈ℤ commute with each other under certain conditions. Also, a sufficient condition is obtained for the Wilson system [Formula: see text] to be a Wilson frame given that [Formula: see text] is a Wilson frame. Finally, we obtain conditions under which the Hilbert transform operator and the frame operator for the Wilson Bessel sequence [Formula: see text] commute with each other.

Comparison of Calculated and Recorded Electron Energy-Loss Spectra of Aluminium and Carbon

1990 ◽  
Vol 48 (2) ◽  
pp. 64-65
Author(s):  
L. Reimer ◽  
R. Oelgeklaus
Keyword(s):  
Fourier Transform ◽  
Energy Loss ◽  
Electron Energy ◽  
Rotational Symmetry ◽  
Mean Free Path ◽  
Single Scattering ◽  
Specimen Thickness ◽  
Two Dimensional ◽  
Image Position ◽  
Electron Energy Loss

Quantitative electron energy-loss spectroscopy (EELS) needs a correction for the limited collection aperture α and a deconvolution of recorded spectra for eliminating the influence of multiple inelastic scattering. Reversely, it is of interest to calculate the influence of multiple scattering on EELS. The distribution f(w,θ,z) of scattered electrons as a function of energy loss w, scattering angle θ and reduced specimen thickness z=t/Λ (Λ=total mean-free-path) can either be recorded by angular-resolved EELS or calculated by a convolution of a normalized single-scattering function ϕ(w,θ). For rotational symmetry in angle (amorphous or polycrystalline specimens) this can be realised by the following sequence of operations :(1)where the two-dimensional distribution in angle is reduced to a one-dimensional function by a projection P, T is a two-dimensional Fourier transform in angle θ and energy loss w and the exponent -1 indicates a deprojection and inverse Fourier transform, respectively.

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