Partial Gabor frames and dual frames

2021 ◽  
pp. 2150035
Author(s):  
Yu Tian ◽  
Hui-Fang Jia ◽  
Guo-Liang He
Keyword(s):  
Density Theorem ◽  
Gabor Frame ◽  
Gabor Frames ◽  
Dual Frame ◽  
Gabor System ◽  
Dual Frames ◽  
Gabor Systems

The theory of Gabor frames has been extensively investigated. This paper addresses partial Gabor systems. We introduce the concepts of partial Gabor system, frame and dual frame. We present some conditions for a partial Gabor system to be a partial Gabor frame, and using these conditions, we characterize partial dual frames. We also give some examples. It is noteworthy that the density theorem does not hold for general partial Gabor systems.

RATIONAL TIME-FREQUENCY GABOR FRAMES ASSOCIATED WITH PERIODIC SUBSETS OF THE REAL LINE

2014 ◽  
Vol 12 (02) ◽  
pp. 1450013 ◽  
Author(s):  
JEAN-PIERRE GABARDO ◽  
YUN-ZHANG LI
Keyword(s):  
Positive Measure ◽  
Measurable Subset ◽  
Gabor Frame ◽  
Gabor Frames ◽  
Type I ◽  
Type Ii ◽  
Zak Transform ◽  
Gabor System ◽  
Gabor Systems ◽  
Time Frequency

For a, b > 0 and g ∈ L2(ℝ), write 𝒢(g, a, b) for the Gabor system: [Formula: see text] Let S be an aℤ-periodic measurable subset of ℝ with positive measure. It is well-known that the projection 𝒢(gχS, a, b) of a frame 𝒢(g, a, b) in L2(ℝ) onto L2(S) is a frame for L2(S). However, when ab > 1 and S ≠ ℝ, 𝒢(g, a, b) cannot be a frame in L2(ℝ) for any g ∈ L2(ℝ), while it is possible that there exists some g such that 𝒢(g, a, b) is a frame for L2(S). So the projections of Gabor frames in L2(ℝ) onto L2(S) cannot cover all Gabor frames in L2(S). This paper considers Gabor systems in L2(S). In order to use the Zak transform, we only consider the case where the product ab is a rational number. With the help of a suitable Zak transform matrix, we characterize Gabor frames for L2(S) of the form 𝒢(g, a, b), and obtain an expression for the canonical dual of a Gabor frame. We also characterize the uniqueness of Gabor duals of type I (respectively, type II).

scholarly journals A SHORT NOTE ON THE FRAME SET OF ODD FUNCTIONS

2018 ◽  
Vol 98 (3) ◽  
pp. 481-493 ◽  
Author(s):  
MARKUS FAULHUBER
Keyword(s):  
Wigner Distribution ◽  
Short Note ◽  
Gabor Frame ◽  
Gabor Frames ◽  
Simple Argument ◽  
Gabor Systems ◽  
One Dimensional ◽  
Separable Case ◽  
The One ◽  
Frame Set

We give a simple argument which shows that Gabor systems consisting of odd functions of$d$variables and symplectic lattices of density$2^{d}$cannot constitute a Gabor frame. In the one-dimensional, separable case, this follows from a more general result of Lyubarskii and Nes [‘Gabor frames with rational density’,Appl. Comput. Harmon. Anal.34(3) (2013), 488–494]. We use a different approach exploiting the algebraic relation between the ambiguity function and the Wigner distribution as well as their relation given by the (symplectic) Fourier transform. Also, we do not need the assumption that the lattice is separable and, hence, new restrictions are added to the full frame set of odd functions.

Gabor frames on non-Archimedean fields

2018 ◽  
Vol 15 (05) ◽  
pp. 1850079 ◽  
Author(s):  
Owais Ahmad ◽  
Firdous A. Shah ◽  
Neyaz A. Sheikh
Keyword(s):  
Time Domain ◽  
Positive Characteristic ◽  
Gabor Frame ◽  
Gabor Frames ◽  
Necessary And Sufficient Condition ◽  
Gabor System ◽  
Potential Applications ◽  
The Time Domain ◽  
Necessary And Sufficient ◽  
Fundamental Equations

In this paper, we introduce the concept of periodic Gabor frames on non-Archimedean fields of positive characteristic. We first establish a necessary and sufficient condition for a periodic Gabor system to be a Gabor frame for [Formula: see text]. Then, we present some equivalent characterizations of Parseval Gabor frames on non-Archimedean fields by means of some fundamental equations in the time domain. Finally, potential applications of Gabor frames on non-Archimedean fields are also discussed.

DISCRETE-TIME WILSON FRAMES WITH GENERAL LATTICES

2012 ◽  
Vol 10 (05) ◽  
pp. 1250044 ◽  
Author(s):  
QIAOFANG LIAN ◽  
YUNFANG LIAN ◽  
MINGHOU YOU
Keyword(s):  
Discrete Time ◽  
Tight Frame ◽  
Gabor Frame ◽  
Dual Frame ◽  
Necessary And Sufficient Condition ◽  
Dual Frames ◽  
Gabor Dual ◽  
Bessel Sequences ◽  
Necessary And Sufficient

In this paper, we focus on the construction of Wilson frames and their dual frames for general lattices of volume [Formula: see text] (K even) in the discrete-time setting. We obtain a necessary and sufficient condition for two Bessel sequences having Wilson structure to be dual frames for l2(ℤ). When the window function satisfies some symmetry property, we obtain a characterization of a Wilson system to be a tight frame for l2(ℤ), show that a Wilson frame for l2(ℤ) can be derived from the underlying Gabor frame, and that the dual frame having Wilson structure can also be derived from the canonical Gabor dual of the underlying Gabor frame.

Wilson frames for ℂL with general lattices

2016 ◽  
Vol 14 (06) ◽  
pp. 1650055
Author(s):  
Minghou You ◽  
Junqiao Yang ◽  
Qiaofang Lian
Keyword(s):  
Digital Signal ◽  
Tight Frame ◽  
Gabor Frame ◽  
Dual Frame ◽  
Dual Frames ◽  
Volume Formula ◽  
Discrete Signals ◽  
Gabor Dual ◽  
Necessary And Sufficient

In digital signal and image processing one can only process discrete signals of finite length, and the space [Formula: see text] is the preferred setting. Recently, Kutyniok and Strohmer constructed orthonormal Wilson bases for [Formula: see text] with general lattices of volume [Formula: see text] ([Formula: see text] even). In this paper, we extend this construction to Wilson frames for [Formula: see text] with general lattices of volume [Formula: see text], where [Formula: see text] and [Formula: see text]. We obtain a necessary and sufficient condition for two sequences having Wilson structure to be dual frames for [Formula: see text]. When the window function satisfies some symmetry property, we obtain a characterization of a Wilson system to be a tight frame for [Formula: see text], show that a Wilson frame for [Formula: see text] can be derived from the underlying Gabor frame, and that the dual frame having Wilson structure can also be derived from the canonical Gabor dual of the underlying Gabor frame.

scholarly journals Multigenerator Gabor Frames on Local Fields

2018 ◽  
Vol 33 (2) ◽  
pp. 307
Author(s):  
Owais Ahmad ◽  
Neyaz Ahmad Sheikh
Keyword(s):  
Sufficient Conditions ◽  
Complete Characterization ◽  
Necessary And Sufficient Conditions ◽  
Local Fields ◽  
Gabor Frames ◽  
Gabor System ◽  
Necessary And Sufficient

The main objective of this paper is to provide complete characterization of multigenerator Gabor frames on a periodic set $\Omega$ in $K$. In particular, we provide some necessary and sufficient conditions for the multigenerator Gabor system to be a frame for $L^2(\Omega)$. Furthermore, we establish the complete characterizations of multigenerator Parseval Gabor frames.

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 The Feichtinger Conjecture for Wavelet Frames, Gabor Frames and Frames of Translates

2006 ◽  
Vol 58 (6) ◽  
pp. 1121-1143 ◽  
Author(s):  
Marcin Bownik ◽  
Darrin Speegle
Keyword(s):  
Regularity Condition ◽  
Finite Union ◽  
Gabor Frame ◽  
Gabor Frames ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Wavelet System ◽  
Weak Regularity

AbstractThe Feichtinger conjecture is considered for three special families of frames. It is shown that if a wavelet frame satisfies a certain weak regularity condition, then it can be written as the finite union of Riesz basic sequences each of which is a wavelet system. Moreover, the above is not true for general wavelet frames. It is also shown that a sup-adjoint Gabor frame can be written as the finite union of Riesz basic sequences. Finally, we show how existing techniques can be applied to determine whether frames of translates can be written as the finite union of Riesz basic sequences. We end by giving an example of a frame of translates such that any Riesz basic subsequence must consist of highly irregular translates.

Sign in / Sign up

Export Citation Format

Share Document