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学者姓名：李云章
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Abstract ：
Due to its significance in mathematics and engineering, the operator theory of Krein spaces and Krein space approaches have been being attracted attention of many mathematicians. Recently, the concept of frame has been introduced in Krein spaces. This paper addresses the approximate oblique dual frames for Krein spaces. We present some parametric expressions and constructions of approximate oblique dual frames of a given frame sequence and prove that there exists a bijection between the approximate oblique dual frames of a given frame sequence and its portrait under a bounded invertible operator on & ell;2$$ {\ell}<^>2 $$ and between the approximate oblique dual frames of original and perturbed frame sequences. Also, we estimate the deviation of the canonical approximate oblique dual frames of original and perturbed frame sequences and give an explicit characterization for the best approximation of the approximate oblique dual frame of original frame sequence using that of perturbed frame sequence. Finally, applying our results to shift-invariant systems, we derive some new results in shift-invariant spaces.
Keyword ：
perturbation perturbation best approximation best approximation approximate oblique dual approximate oblique dual Krein space Krein space frame frame
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| GB/T 7714 | Li, Yun-Zhang , Dong, Rui-Qi . Approximate Oblique Dual Frames for Krein Spaces [J]. | MATHEMATICAL METHODS IN THE APPLIED SCIENCES , 2025 . |
| MLA | Li, Yun-Zhang 等. "Approximate Oblique Dual Frames for Krein Spaces" . | MATHEMATICAL METHODS IN THE APPLIED SCIENCES (2025) . |
| APA | Li, Yun-Zhang , Dong, Rui-Qi . Approximate Oblique Dual Frames for Krein Spaces . | MATHEMATICAL METHODS IN THE APPLIED SCIENCES , 2025 . |
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Abstract ：
This paper addresses the weaving theory of operator-valued frames (OPV-frames). We give a rigorous proof of the equivalence between "weakly woven" and "woven" of OPV-frames; estimate the optimal universal OPV-frame bounds of all weavings; and prove that OPV-frame and its dual OPV-frame are woven. Also, using examples, we show that "woven property" does not have transmissibility, and that a collection of pairwise weaving frames need not be woven. Finally, we give a sufficient condition for a collection of adjacent weaving OPV-frames to be woven.
Keyword ：
woven operator-valued frames woven operator-valued frames weaving weaving operator-valued frame operator-valued frame woven frames woven frames Frame Frame
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| GB/T 7714 | Li, Ya-Nan , Li, Yun-Zhang , Yan, Zhi-Chao . On weaving operator-valued frames [J]. | INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING , 2025 . |
| MLA | Li, Ya-Nan 等. "On weaving operator-valued frames" . | INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING (2025) . |
| APA | Li, Ya-Nan , Li, Yun-Zhang , Yan, Zhi-Chao . On weaving operator-valued frames . | INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING , 2025 . |
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Abstract ：
Due to their potential applications in video images, color images, and multi-spectral images, matrix-valued frames have attracted the attention of scholars from mathematics and engineering communities. A Hilbert-Schmidt frame (HS-frame) can be considered to be square matrix-valued. In this paper, we introduce the concepts of generalized Hilbert-Schmidt frame (GHS-frame) and oblique dual GHS-frame (ODGHS-frame). The former is more general than HS-frame and includes general matrix-valued frames (not necessarily square matrix-valued frames) as a special case. We give a parametric expression of all ODGHS-frames of an arbitrarily given GHS-frame, and prove that the portraits of a given ODGHS-frame pair (GHS-orthonormal basis, dual GHS-frame pair) under suitable bounded operators give other ODGHS-frame pairs. Finally, as an application of the above results, we derive an existing result and some new results on usual frames.
Keyword ：
Oblique dual GHS-frame pair Oblique dual GHS-frame pair Oblique dual GHS-frame Oblique dual GHS-frame GHS-frame GHS-frame Oblique dual Oblique dual Frame Frame
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| GB/T 7714 | Li, Yun-Zhang , Wu, Li-Juan . Oblique Duals of Generalized Hilbert-Schmidt Frames [J]. | BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY , 2025 , 48 (3) . |
| MLA | Li, Yun-Zhang 等. "Oblique Duals of Generalized Hilbert-Schmidt Frames" . | BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY 48 . 3 (2025) . |
| APA | Li, Yun-Zhang , Wu, Li-Juan . Oblique Duals of Generalized Hilbert-Schmidt Frames . | BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY , 2025 , 48 (3) . |
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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 ：
This paper addresses the dilation problem on (dual) frames for Krein spaces. We characterize Riesz bases for Krein spaces and equivalence ((J1,J2)-unitary equivalence) between frames for Krein spaces; prove that every frame (dual frame pair) for a Krein space can be dilated to a Riesz basis (dual Riesz basis pair) for a larger Krein space, and that the corresponding J-orthogonal complementary frame (J-joint complementary frame) is unique up to equivalence ((J1,J2)-joint equivalence). Also we illustrate that two equivalent Parseval frames for Krein spaces need not be (J1,J2)-unitarily equivalent and that not every Parseval frame can be dilated to a J-orthonormal basis for a larger Krein space, and derive a result on matrices of finite size as application.
Keyword ：
dual frame dual frame Riesz basis Riesz basis Frame Frame Krein space Krein space dilation dilation
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| GB/T 7714 | Dong, Rui-Qi , Li, Yun-Zhang . Dilations of (dual) frames for Krein spaces [J]. | INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING , 2024 . |
| MLA | Dong, Rui-Qi 等. "Dilations of (dual) frames for Krein spaces" . | INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING (2024) . |
| APA | Dong, Rui-Qi , Li, Yun-Zhang . Dilations of (dual) frames for Krein spaces . | INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING , 2024 . |
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Abstract ：
Due to its significance in reconstructing signals, the theory of dual frames has attracted the interest of many mathematicians. In this paper, we introduce the concept of approximate Q-oblique dual g-frames. We characterize the approximate Q-oblique dual g-frames of an arbitrarily given g-frame; present a method for constructing approximate Q-oblique dual g-frame pairs that have better approximation from a given approximate Q-oblique dual g-frame pair; and investigate the stability of approximate Q-oblique dual g-frame pairs under the analysis, synthesis and frame operator-perturbation, the gap-perturbation, and the l2 operator portrait-perturbation. Finally, as an application of the above results, we obtain some new results on usual frames which cover some existing results.
Keyword ：
Approximate Q-oblique dual g-frame Approximate Q-oblique dual g-frame g-frame g-frame frame frame dual frame dual frame perturbation perturbation
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| GB/T 7714 | Wu, Li-Juan , Li, Yun-Zhang . Characterization and Construction of Approximate Q-Oblique Dual g-Frames [J]. | NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION , 2024 , 46 (3) : 194-225 . |
| MLA | Wu, Li-Juan 等. "Characterization and Construction of Approximate Q-Oblique Dual g-Frames" . | NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 46 . 3 (2024) : 194-225 . |
| APA | Wu, Li-Juan , Li, Yun-Zhang . Characterization and Construction of Approximate Q-Oblique Dual g-Frames . | NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION , 2024 , 46 (3) , 194-225 . |
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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 ：
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 ：
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 ：
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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