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学者姓名:李云章
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Abstract :
The notion of Hilbert-Schmidt frame (HS-frame) is more general than that of g-frame. This paper addresses frame properties of HS-operator sequences. From the literature, a g-frame is a HS-frame in some sense. Interestingly, in this paper we prove that a g-Riesz basis is not a HS-Riesz basis whenever the cardinality of its index set is greater than 1. Also we present some operator parametric expressions of HS-Bessel sequences, HS-orthonormal bases, HS-orthonormal systems, HS-frames, HS-frame sequences, HS-Riesz bases and HS-Riesz sequences; characterize HS-Riesz bases and Riesz sequences using minimality; and obtain a representation of orthogonal projection operators in terms of subspace HS-frames.
Keyword :
HS-frame sequence HS-frame sequence HS-frame HS-frame HS-Riesz basis HS-Riesz basis Frame Frame HS-Riesz sequence HS-Riesz sequence
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GB/T 7714 | Li, Yun-Zhang , Zhang, Xiao-Li . Frame Properties of Hilbert-Schmidt Operator Sequences [J]. | MEDITERRANEAN JOURNAL OF MATHEMATICS , 2023 , 20 (1) . |
MLA | Li, Yun-Zhang 等. "Frame Properties of Hilbert-Schmidt Operator Sequences" . | MEDITERRANEAN JOURNAL OF MATHEMATICS 20 . 1 (2023) . |
APA | Li, Yun-Zhang , Zhang, Xiao-Li . Frame Properties of Hilbert-Schmidt Operator Sequences . | MEDITERRANEAN JOURNAL OF MATHEMATICS , 2023 , 20 (1) . |
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Abstract :
Due to its applications in signal analysis and image processing, the quaternionic Fourier analysis has received increasing attention. In particular, quaternionic Gabor frames (QGFs) attracted some mathematicians' interest. From the literatures, some results on QGFs are based on quaternionic Fourier orthonormal bases. But those used so-called quaternionic Fourier orthonormal bases have a gap that they are all incomplete. In this paper, we present a class of quaternionic Fourier orthonormal bases, and using them derive the corresponding Gabor orthonormal bases.
Keyword :
quaternionic Gabor orthonormal basis quaternionic Gabor orthonormal basis Quaternion Quaternion quaternionic Fourier orthonormal basis quaternionic Fourier orthonormal basis
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GB/T 7714 | Li, Yun-Zhang , Zhang, Xiao-Li . A class of quaternionic Fourier orthonormal bases [J]. | FORUM MATHEMATICUM , 2023 . |
MLA | Li, Yun-Zhang 等. "A class of quaternionic Fourier orthonormal bases" . | FORUM MATHEMATICUM (2023) . |
APA | Li, Yun-Zhang , Zhang, Xiao-Li . A class of quaternionic Fourier orthonormal bases . | FORUM MATHEMATICUM , 2023 . |
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Abstract :
Recently, Gabor analysis on locally compact abelian (LCA) groups has interested some mathematicians. The half real line Double-struck capital R+=(0,infinity)$$ {\mathbb{R}}_{+}=\left(0,\infty \right) $$ is an LCA group under multiplication and the usual topology. This paper addresses spline Gabor frames for L2(Double-struck capital R+,d mu)$$ {L}<^>2\left({\mathbb{R}}_{+}, d\mu \right) $$, where mu$$ \mu $$ is the corresponding Haar measure. We introduce the concept of spline functions on Double-struck capital R+$$ {\mathbb{R}}_{+} $$ by mu$$ \mu $$-convolution and estimate their Gabor frame sets, that is, lattice sets such that spline generating Gabor systems are frames for L2(Double-struck capital R+,d mu)$$ {L}<^>2\left({\mathbb{R}}_{+}, d\mu \right) $$. For an arbitrary spline Gabor frame with special lattices, we present its one dual Gabor frame window, which has the same smoothness as the initial window function. For a class of special spline Gabor Bessel sequences, we prove that they can be extended to a tight Gabor frame by adding a new window function, which has compact support and same smoothness as the initial windows. And we also demonstrate that two spline Gabor Bessel sequences can always be extended to a pair of dual Gabor frames with the adding window functions being compactly supported and having the same smoothness as the initial windows.
Keyword :
Gabor frame set Gabor frame set dual frame dual frame frame frame Gabor system Gabor system spline function spline function
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GB/T 7714 | Yang, Ming , Li, Yun-Zhang . Spline function generating Gabor systems on the half real line [J]. | MATHEMATICAL METHODS IN THE APPLIED SCIENCES , 2023 , 46 (8) : 9415-9441 . |
MLA | Yang, Ming 等. "Spline function generating Gabor systems on the half real line" . | MATHEMATICAL METHODS IN THE APPLIED SCIENCES 46 . 8 (2023) : 9415-9441 . |
APA | Yang, Ming , Li, Yun-Zhang . Spline function generating Gabor systems on the half real line . | MATHEMATICAL METHODS IN THE APPLIED SCIENCES , 2023 , 46 (8) , 9415-9441 . |
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Abstract :
Recently, Gabor analysis on locally compact abelian (LCA) groups has interested some mathematicians. The half real line R+ = (0, infinity) is an LCA group under multiplication and the usual topology, with the Haar measure d mu = dxx. This paper addresses rationally sampled Gabor frames for L2(R+, d mu). Given a function in L2(R+, d mu), we introduce a new Zak transform matrix associated with it, which is different from the conventional Zibulski-Zeevi matrix. It allows us to define a function by designing its Zak transform matrix. Using our Zak transform matrix method, we characterize and express complete Gabor systems, Bessel sequences, Gabor frames, Riesz bases and Gabor duals of an arbitrarily given Gabor frame for L2(R+, d mu), and prove the minimality of the canonical dual frames in some sense. Some examples are also provided to illustrate the generality of our theory.(c) 2023 Elsevier Inc. All rights reserved.
Keyword :
Zak transform matrix Zak transform matrix Riesz basis Riesz basis Gabor duals Gabor duals Zak transform Zak transform Gabor frame Gabor frame
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GB/T 7714 | Li, Yun-Zhang , Yang, Ming . Rationally sampled Gabor frames on the half real line [J]. | JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS , 2023 , 532 (1) . |
MLA | Li, Yun-Zhang 等. "Rationally sampled Gabor frames on the half real line" . | JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 532 . 1 (2023) . |
APA | Li, Yun-Zhang , Yang, Ming . Rationally sampled Gabor frames on the half real line . | JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS , 2023 , 532 (1) . |
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Abstract :
This paper addresses quaternionic dual Gabor frames under the condition that the products of time-frequency shift parameters are rational numbers. For a general overcomplete quaternionic Gabor frame with the product of time-frequency shift parameters not equal to 12$$ \frac{1}{2} $$, we show that its corresponding frame and translation operators do not commute, which leads to its canonical dual frame not having the Gabor structure, but it may have other dual frames with Gabor structure. We characterize when two quaternionic Gabor Bessel sequences form a pair of dual frames, and present a class of quaternionic dual Gabor frames. We also characterize quaternionic Gabor Riesz bases and prove that their canonical dual frames have Gabor structure.
Keyword :
quaternionic Gabor Riesz basis quaternionic Gabor Riesz basis Gabor frame Gabor frame quaternionic dual Gabor frame quaternionic dual Gabor frame
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GB/T 7714 | Li, Yun-Zhang , Zhang, Xiao-Li . Characterization of rationally sampled quaternionic dual Gabor frames [J]. | MATHEMATICAL METHODS IN THE APPLIED SCIENCES , 2023 , 47 (2) : 1095-1112 . |
MLA | Li, Yun-Zhang 等. "Characterization of rationally sampled quaternionic dual Gabor frames" . | MATHEMATICAL METHODS IN THE APPLIED SCIENCES 47 . 2 (2023) : 1095-1112 . |
APA | Li, Yun-Zhang , Zhang, Xiao-Li . Characterization of rationally sampled quaternionic dual Gabor frames . | MATHEMATICAL METHODS IN THE APPLIED SCIENCES , 2023 , 47 (2) , 1095-1112 . |
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Abstract :
The study of quaternionic Gabor systems has interested some mathematicians in recent years. From the literature, we found that most existing results on quaternionic Gabor frames focus on the case of the product of time-frequency shift parameters being equal to 1, and have a gap that the involved quaternionic Gabor systems are all incomplete according to the symmetric real scalar inner product. In this paper, we introduce quaternionic Zak transformation and a class of quaternionic Gabor systems. Under the condition that the products of time-frequency shift parameters are rational numbers, we characterize completeness and frame property of quaternionic Gabor systems in terms of Zak transformation matrices. From this, we derive the density theorem for quaternionic Gabor systems.
Keyword :
Density theorem Density theorem Quaternionic Gabor frame Quaternionic Gabor frame Frame Frame
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GB/T 7714 | Zhang, Xiao-Li , Li, Yun-Zhang . Quaternionic Gabor frame characterization and the density theorem [J]. | BANACH JOURNAL OF MATHEMATICAL ANALYSIS , 2023 , 17 (4) . |
MLA | Zhang, Xiao-Li 等. "Quaternionic Gabor frame characterization and the density theorem" . | BANACH JOURNAL OF MATHEMATICAL ANALYSIS 17 . 4 (2023) . |
APA | Zhang, Xiao-Li , Li, Yun-Zhang . Quaternionic Gabor frame characterization and the density theorem . | BANACH JOURNAL OF MATHEMATICAL ANALYSIS , 2023 , 17 (4) . |
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Abstract :
Due to R+=(0,infinity)R+=(0,infinity) not being a group under addition, L2(R+)L2(R+) admits no traditional wavelet or Gabor frames. This paper addresses a class of modulation-dilation frames (MDMD-frames) for L2(R+)L2(R+). We obtain a theta-transform matrix-based expression of adding generators to generate MD-tight frames from a MDMD-Bessel sequences in L2(R+)L2(R+); and present criteria on Phi Phi with MD(Psi boolean OR Phi,a,b)MD(Psi boolean OR Phi,a,b) being a Parseval frame (an orthonormal basis) for an arbitrary Parseval frame sequence (an orthonormal sequence) MD(Psi,a,b)MD(Psi,a,b) in L2(R+)L2(R+). Some examples are also presented.
Keyword :
Frame Frame Parseval frame Parseval frame Modulation-dilation frame Modulation-dilation frame
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GB/T 7714 | Li, Ya-Nan , Li, Yun-Zhang . Extensions of modulation-dilation Bessel Systems in L-2(R+) [J]. | COLLECTANEA MATHEMATICA , 2022 . |
MLA | Li, Ya-Nan 等. "Extensions of modulation-dilation Bessel Systems in L-2(R+)" . | COLLECTANEA MATHEMATICA (2022) . |
APA | Li, Ya-Nan , Li, Yun-Zhang . Extensions of modulation-dilation Bessel Systems in L-2(R+) . | COLLECTANEA MATHEMATICA , 2022 . |
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Abstract :
Due to their potential in pure and applied mathematics, the notions of basis and frame have various generalizations. Recently, g-frame and g-basis were introduced and studied by some mathematicians. It is well known that a frame-based series expansion of a vector is unconditionally convergent, while a basis-based one need not be. In applications unconditionality is more favourable than conditionality. In this paper, we introduce the notion of g-unconditional basis which leads to unconditional convergence, and establish a characterization of g-unconditional bases.
Keyword :
g-unconditional basis g-unconditional basis g-basis g-basis g-linearly independence g-linearly independence g-frame g-frame
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GB/T 7714 | Zhang, Yan , Li, Yun-Zhang . SOME CHARACTERIZATIONS OF g-UNCONDITIONAL BASES [J]. | MATHEMATICAL REPORTS , 2022 , 24 (4) : 751-770 . |
MLA | Zhang, Yan 等. "SOME CHARACTERIZATIONS OF g-UNCONDITIONAL BASES" . | MATHEMATICAL REPORTS 24 . 4 (2022) : 751-770 . |
APA | Zhang, Yan , Li, Yun-Zhang . SOME CHARACTERIZATIONS OF g-UNCONDITIONAL BASES . | MATHEMATICAL REPORTS , 2022 , 24 (4) , 751-770 . |
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Abstract :
This paper addresses the dilation problem on (dual) Hilbert-Schmidt frames (HS-frames). We present a dilation theorem from an HS-frame (a Parseval HS-frame, a dual HS-frame pair) to an HS-Riesz basis (an HS-orthonormal basis, a dual HS-Riesz basis pair); and prove that the corresponding orthogonal complementary HS-frame (joint complementary HS-frame) is unique up to equivalence (unitary equivalence, joint equivalence). A remark is also provided. It demonstrates that our results and approach can recover some existing dilation results on frames and g-frames.
Keyword :
HS-frame HS-frame Frame Frame Dilation theorem Dilation theorem HS-Riesz basis HS-Riesz basis Dual HS-frame Dual HS-frame
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GB/T 7714 | Li, Yun-Zhang , Zhang, Xiao-Li . Dilations of (dual) Hilbert-Schmidt frames [J]. | ANNALS OF FUNCTIONAL ANALYSIS , 2022 , 13 (3) . |
MLA | Li, Yun-Zhang 等. "Dilations of (dual) Hilbert-Schmidt frames" . | ANNALS OF FUNCTIONAL ANALYSIS 13 . 3 (2022) . |
APA | Li, Yun-Zhang , Zhang, Xiao-Li . Dilations of (dual) Hilbert-Schmidt frames . | ANNALS OF FUNCTIONAL ANALYSIS , 2022 , 13 (3) . |
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Abstract :
It is well known that there are two approaches applicable in constructing frames starting from one fixed frame. One is based on l(2)-operator portraits by which, using a suitable bounded linear operator on l(2), one can construct an arbitrary frame from one fixed frame. The other is based on perturbation that allows suitable perturbing a frame leaving a frame. The study of Hilbert-Schmidt frames (HS-frames) has interested some mathematicians in recent years. This paper addresses l(2)-operator portraits and perturbations in the setting of HS-frames. We prove that the portrait of a HS-frame under a bounded invertible operator on l(2) is still a HS-frame; present a sufficient condition on bounded operators on l(2) which transform an l(2)-decomposable HS-frame into another HS-frame (HS-Riesz basis, HS-frame sequence and HS-Riesz sequence); and prove that suitable perturbing a HS-frame sequence (HS-Riesz sequence) leaves a HS-frame sequence (HS-Riesz sequence). Finally, using these results we recover some conclusions on frames.
Keyword :
Frame Frame HS-Riesz sequence HS-Riesz sequence HS-frame HS-frame HS-frame sequence HS-frame sequence HS-Riesz basis HS-Riesz basis
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GB/T 7714 | Zhang, Xiao-Li , Li, Yun-Zhang . Portraits and Perturbations of Hilbert-Schmidt Frame Sequences [J]. | BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY , 2022 , 45 (6) : 3197-3223 . |
MLA | Zhang, Xiao-Li 等. "Portraits and Perturbations of Hilbert-Schmidt Frame Sequences" . | BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY 45 . 6 (2022) : 3197-3223 . |
APA | Zhang, Xiao-Li , Li, Yun-Zhang . Portraits and Perturbations of Hilbert-Schmidt Frame Sequences . | BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY , 2022 , 45 (6) , 3197-3223 . |
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