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学者姓名：李云章
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Abstract ：
Due to its potential application in signal analysis and image processing, quaternionic Fourier analysis has received increasing attention. This paper addresses quaternionic subspace Gabor frames under the condition that the products of time-frequency shift parameters are rational numbers. We characterize subspace quaternionic Gabor frames in terms of quaternionic Zak transformation matrices. For an arbitrary subspace Gabor frame, we give a parametric expression of its Gabor duals of type I and type II, and characterize the uniqueness Gabor duals of type I and type II. And as an application, given a Gabor frame for the whole space L2(R2,H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>{2}({\mathbb {R}}<^>{2},\,{\mathbb {H}})$$\end{document}, we give a parametric expression of its all Gabor duals, and derive its unique Gabor dual of type II. Some examples are also provided.
Keyword ：
quaternionic subspace Gabor frame quaternionic subspace Gabor frame Quaternion Quaternion quaternionic dual Gabor frame quaternionic dual Gabor frame
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| GB/T 7714 | Li, Yun-Zhang , Zhang, Xiao-Li . Quaternionic Subspace Gabor Frames and Their Duals [J]. | ADVANCES IN APPLIED CLIFFORD ALGEBRAS , 2024 , 34 (4) . |
| MLA | Li, Yun-Zhang 等. "Quaternionic Subspace Gabor Frames and Their Duals" . | ADVANCES IN APPLIED CLIFFORD ALGEBRAS 34 . 4 (2024) . |
| APA | Li, Yun-Zhang , Zhang, Xiao-Li . Quaternionic Subspace Gabor Frames and Their Duals . | ADVANCES IN APPLIED CLIFFORD ALGEBRAS , 2024 , 34 (4) . |
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Abstract ：
四元数Hilbert空间在应用物理科学特别是量子物理中占有重要地位.本文讨论四元数Hilbert空间的框架理论,在四元数Hilbert空间中引入了Riesz基的概念,在此基础上刻画了Riesz基,给出了它们的一些等价条件;特别地,得到了四元数Hilbert空间中的一个序列是Riesz基的充要条件是它是一个具有双正交序列的完备Bessel序列,且它的双正交序列也是一个完备Bessel序列;并进一步证明了双正交序列中一个序列的完备性可以从特征刻画中去除.文中举例说明了双正交性、完备性和Bessel性质之间的关系.
Keyword ：
Riesz基 Riesz基 完备性 完备性 框架 框架 四元数Hilbert空间 四元数Hilbert空间
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| GB/T 7714 | 张伟 , 李云章 . 四元数Hilbert空间中Riesz基的刻画 [J]. | 数学年刊A辑(中文版) , 2023 , 44 (01) : 97-112 . |
| MLA | 张伟 等. "四元数Hilbert空间中Riesz基的刻画" . | 数学年刊A辑(中文版) 44 . 01 (2023) : 97-112 . |
| APA | 张伟 , 李云章 . 四元数Hilbert空间中Riesz基的刻画 . | 数学年刊A辑(中文版) , 2023 , 44 (01) , 97-112 . |
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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 ：
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 ：
四元数Hilbert空间在应用物理科学特别是量子物理中占有重要地位.本文讨论四元数Hilbert空间的框架理论,在四元数Hilbert空间中引入了Riesz基的概念,在此基础上刻画了Riesz基,给出了它们的一些等价条件;特别地,得到了四元数Hilbert空间中的一个序列是Riesz基的充要条件是它是一个具有双正交序列的完备Bessel序列,且它的双正交序列也是一个完备Bessel序列;并进一步证明了双正交序列中一个序列的完备性可以从特征刻画中去除.文中举例说明了双正交性、完备性和Bessel性质之间的关系.
Keyword ：
完备性 完备性 四元数Hilbert空间 四元数Hilbert空间 Riesz基 Riesz基 框架 框架
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| GB/T 7714 | 张伟 , 李云章 . 四元数Hilbert空间中Riesz基的刻画 [J]. | 数学年刊A辑 , 2023 , 44 (1) : 97-112 . |
| MLA | 张伟 等. "四元数Hilbert空间中Riesz基的刻画" . | 数学年刊A辑 44 . 1 (2023) : 97-112 . |
| APA | 张伟 , 李云章 . 四元数Hilbert空间中Riesz基的刻画 . | 数学年刊A辑 , 2023 , 44 (1) , 97-112 . |
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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 ：
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 ：
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 , 36 (3) : 825-834 . |
| MLA | Li, Yun-Zhang 等. "A class of quaternionic Fourier orthonormal bases" . | FORUM MATHEMATICUM 36 . 3 (2023) : 825-834 . |
| APA | Li, Yun-Zhang , Zhang, Xiao-Li . A class of quaternionic Fourier orthonormal bases . | FORUM MATHEMATICUM , 2023 , 36 (3) , 825-834 . |
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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 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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