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学者姓名：李云章
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Abstract ：
Quaternion algebra is a noncommutative associative algebra. Noncommutativity limits the flexibility of computation and makes analysis related to quaternions nontrivial and challenging. Due to its applications in signal analysis and image processing, quaternionic Fourier analysis has received increasing attention in recent years. This paper addresses phase retrievability in quaternion Euclidean spaces HM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}}<^>{M}$$\end{document}. We obtain a sufficient condition on phase retrieval frames for quaternionic left Hilbert module (HM,(center dot,center dot))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ({\mathbb {H}}<^>{M},\,(\cdot ,\,\cdot )\big )$$\end{document} of the form {emTng}m,n is an element of NM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{e_{m}T_{n}g\}_{m,\,n\in {\mathbb {N}}_{M}}$$\end{document}, where {em}m is an element of NM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{e_{m}\}_{m\in {\mathbb {N}}_{M}}$$\end{document} is an orthonormal basis for (HM,(center dot,center dot))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ({\mathbb {H}}<^>{M},\,(\cdot ,\,\cdot )\big )$$\end{document} and (center dot,center dot)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\cdot ,\,\cdot )$$\end{document} is the Euclidean inner product on HM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}}<^>{M}$$\end{document}. It is worth noting that {em}m is an element of NM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{e_{m}\}_{m\in {\mathbb {N}}_{M}}$$\end{document} is not necessarily 1Me2 pi im center dot Mm is an element of NM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \frac{1}{\sqrt{M}}e<^>{\frac{2\pi im\cdot }{M}}\right\} _{m\in {\mathbb {N}}_{M}}$$\end{document}, and that our method also applies to phase retrievability in CM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}<^>{M}$$\end{document}. For the real Hilbert space (HM,⟨center dot,center dot⟩)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ({\mathbb {H}}<^>{M},\,\langle \cdot ,\,\cdot \rangle \big )$$\end{document} induced by (HM,(center dot,center dot))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ({\mathbb {H}}<^>{M},\,(\cdot ,\,\cdot )\big )$$\end{document}, we present a sufficient condition on phase retrieval frames {emTng}m is an element of N4M,n is an element of NM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{e_{m}T_{n}g\}_{m\in {\mathbb {N}}_{4M},\,n\in {\mathbb {N}}_{M}}$$\end{document}, where {em}m is an element of N4M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{e_{m}\}_{m\in {\mathbb {N}}_{4M}}$$\end{document} is an orthonormal basis for (HM,⟨center dot,center dot⟩)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ({\mathbb {H}}<^>{M},\,\langle \cdot ,\,\cdot \rangle \big )$$\end{document}. We also give a method to construct and verify general phase retrieval frames for (HM,⟨center dot,center dot⟩)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ({\mathbb {H}}<^>{M},\,\langle \cdot ,\,\cdot \rangle \big )$$\end{document}. Finally, some examples are provided to illustrate the generality of our theory.
Keyword ：
Frame Frame Quaternion Quaternion Phase retrieval Phase retrieval
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| GB/T 7714 | Yang, Ming , Li, Yun-Zhang . Phase Retrieval in Quaternion Euclidean Spaces [J]. | BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY , 2024 , 47 (2) . |
| MLA | Yang, Ming 等. "Phase Retrieval in Quaternion Euclidean Spaces" . | BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY 47 . 2 (2024) . |
| APA | Yang, Ming , Li, Yun-Zhang . Phase Retrieval in Quaternion Euclidean Spaces . | BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY , 2024 , 47 (2) . |
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Abstract ：
Recently, Gabor analysis on locally compact abelian (LCA) groups has become the focus of an active research. In practice, the time variable cannot be negative. The half real line R+ = (0,infinity) is an LCA group under multiplication and the usual topology, with the Haar measure d mu = dx/x. This paper addresses Gabor frame multipliers and Parseval duals for L-2(R+, d mu). We introduce and characterize Gabor frame multipliers and Parseval Gabor frame multipliers based on Zak transform matrices. Our Zak transform matrix is essentially different from the conventional Zibulski-Zeevi matrix. It allows us to define Gabor frame generators by designing suitable matrix-valued functions of finite size. We also prove that an arbitrary Gabor frame g(g, a, b) admits a Parseval dual frame/tight dual frame whenever ln a. In b are rational numbers not greater than 1/2.
Keyword ：
Gabor frame Gabor frame Parseval Gabor frame multiplier Parseval Gabor frame multiplier Frame Frame Parseval dual Parseval dual Gabor frame multiplier Gabor frame multiplier
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| GB/T 7714 | Yang, Ming , Li, Yun-Zhang . Gabor frame multipliers and Parseval duals on the half real line [J]. | INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING , 2024 , 22 (04) . |
| MLA | Yang, Ming 等. "Gabor frame multipliers and Parseval duals on the half real line" . | INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING 22 . 04 (2024) . |
| APA | Yang, Ming , Li, Yun-Zhang . Gabor frame multipliers and Parseval duals on the half real line . | INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING , 2024 , 22 (04) . |
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Abstract ：
A conjugation on a Hilbert space H means an antilinear bounded operator that the squares to the identity, which generalizes the traditional conjugation on complex Euclidean spaces. In this paper, with the help of normalized conjugation we introduce the notion of conjugate phase retrieval on general Hilbert spaces. We characterize frames that do conjugate phase retrieval; prove that every conjugate phase retrieval frame for H consisting of all real vectors has the complement property in H, and the converse is true if dim(H) <= 2; and also prove that a small perturbation of conjugate phase retrieval frame still gives a conjugate phase retrieval frame if dim(H) < infinity, but it is false if dim(H) = infinity.
Keyword ：
complement property complement property Frame Frame conjugate phase retrieval conjugate phase retrieval conjugation conjugation phase retrieval phase retrieval
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| GB/T 7714 | Li, Ya-Nan , Li, Yun-Zhang . Conjugate phase retrieval on general Hilbert spaces [J]. | LINEAR & MULTILINEAR ALGEBRA , 2024 , 72 (17) : 2845-2878 . |
| MLA | Li, Ya-Nan 等. "Conjugate phase retrieval on general Hilbert spaces" . | LINEAR & MULTILINEAR ALGEBRA 72 . 17 (2024) : 2845-2878 . |
| APA | Li, Ya-Nan , Li, Yun-Zhang . Conjugate phase retrieval on general Hilbert spaces . | LINEAR & MULTILINEAR ALGEBRA , 2024 , 72 (17) , 2845-2878 . |
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Abstract ：
In last decades, the operator theory of Krein spaces, Krein space approaches, and various generalizations of frames have interested many mathematicians due to their potential applications in mathematics and engineering. This paper addresses the frame theory for Krein spaces. We present some properties of J-orthonormal bases, Parseval frames and frames for Krein spaces, and a parametric expression of all duals of an arbitrarily given frame in Krein spaces. This study shows that the frame theory for Krein spaces is not a direct generalization of the frame theory for Hilbert spaces.
Keyword ：
Frame Frame J-orthonormal basis J-orthonormal basis Parseval frame Parseval frame Krein space Krein space dual frame dual frame
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| GB/T 7714 | Li, Yun-Zhang , Dong, Rui-Qi . Frames and Dual Frames for Krein Spaces [J]. | NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION , 2024 , 45 (4-6) : 355-372 . |
| MLA | Li, Yun-Zhang 等. "Frames and Dual Frames for Krein Spaces" . | NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 45 . 4-6 (2024) : 355-372 . |
| APA | Li, Yun-Zhang , Dong, Rui-Qi . Frames and Dual Frames for Krein Spaces . | NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION , 2024 , 45 (4-6) , 355-372 . |
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Abstract ：
Quaternion algebra H is a noncommutative associative algebra. In recent years, quaternionic Fourier analysis has received increasing attention due to its applications in signal analysis and image processing. This paper addresses conjugate phase retrieval problem in the quaternion Euclidean space H-M with M >= 2. Write C-eta = {xi : xi = xi(0) + beta eta, xi(0), beta is an element of R} for eta is an element of {i, j, k}. We remark that not only C-eta(M)-vectors cannot allow traditional conjugate phase retrieval in H-M, but also C-i(i)M boolean OR C-j(M)-complex vectors cannot allow phase retrieval in H-M . We are devoted to conjugate phase retrieval of C-i(M) boolean OR C-j(M) -complex vectors in H-M, where "conjugate" is not the traditional conjugate. We introduce the notions of conjugation, maximal commutative subset and conjugate phase retrieval. Using the phase lifting techniques, we present some sufficient conditions on complex vectors allowing conjugate phase retrieval. And some examples are also provided to illustrate the generality of our theory.
Keyword ：
phaselift phaselift quaternion quaternion conjugation conjugation frame frame Conjugate phase retrieval Conjugate phase retrieval
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| GB/T 7714 | Li, Yun-Zhang , Yang, Ming . What conjugate phase retrieval complex vectors can do in quaternion Euclidean spaces [J]. | FORUM MATHEMATICUM , 2024 , 36 (6) : 1585-1601 . |
| MLA | Li, Yun-Zhang 等. "What conjugate phase retrieval complex vectors can do in quaternion Euclidean spaces" . | FORUM MATHEMATICUM 36 . 6 (2024) : 1585-1601 . |
| APA | Li, Yun-Zhang , Yang, Ming . What conjugate phase retrieval complex vectors can do in quaternion Euclidean spaces . | FORUM MATHEMATICUM , 2024 , 36 (6) , 1585-1601 . |
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Abstract ：
The concept of approximate oblique dual frame was introduced by Diaz, Heineken and Morillas. It is more general than traditional dual frame, oblique dual frame, and approximate dual frame. This paper addresses constructing more approximate oblique dual frame pairs starting from one given oblique dual frame pair. Using "analysis and synthesis operator", "portrait", and "gap" perturbation techniques, we present several sufficient conditions for constructing approximate oblique dual frame pairs under the general Hilbert space setting. As an application, we then focus on constructing approximate oblique dual frame pairs in shift-invariant subspaces of L2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>{2}(\mathbb R)$$\end{document}.
Keyword ：
Frame Frame Approximate dual frame Approximate dual frame Approximate oblique dual frame Approximate oblique dual frame Oblique dual frame Oblique dual frame Dual frame Dual frame
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| GB/T 7714 | Li, Yun-Zhang , Wu, Li-Juan . Making more approximate oblique dual frame pairs [J]. | ANNALS OF FUNCTIONAL ANALYSIS , 2024 , 15 (2) . |
| MLA | Li, Yun-Zhang 等. "Making more approximate oblique dual frame pairs" . | ANNALS OF FUNCTIONAL ANALYSIS 15 . 2 (2024) . |
| APA | Li, Yun-Zhang , Wu, Li-Juan . Making more approximate oblique dual frame pairs . | ANNALS OF FUNCTIONAL ANALYSIS , 2024 , 15 (2) . |
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Abstract ：
This paper addresses the Hilbert-Schmidt frame (HS-frame) theory. We introduce the concept of generalized dual HS-frame (g-dual HS-frame) which generalizes that of g-dual frame. We prove that two equivalent HS-frames form a g-dual HS-frame pair, characterize operators on l2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\ell <^>2$$\end{document} that transform a pair of HS-Riesz bases into a g-dual HS-frame pair, and present a parametric expression of all g-dual HS-frames of an arbitrarily given HS-frame. Also the perturbation-stability and topological properties of g-dual HS-frames are investigated. Finally, applying our results, we not only recover some known results but also derive some new results in the classical Hilbert space frame setting.
Keyword ：
Frame Frame HS-frame HS-frame topological property topological property perturbation perturbation g-dual g-dual
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| GB/T 7714 | Dong, Rui-Qi , Li, Yun-Zhang . Generalized Dual Hilbert-Schmidt Frames and Their Topological Properties [J]. | RESULTS IN MATHEMATICS , 2024 , 79 (2) . |
| MLA | Dong, Rui-Qi 等. "Generalized Dual Hilbert-Schmidt Frames and Their Topological Properties" . | RESULTS IN MATHEMATICS 79 . 2 (2024) . |
| APA | Dong, Rui-Qi , Li, Yun-Zhang . Generalized Dual Hilbert-Schmidt Frames and Their Topological Properties . | RESULTS IN MATHEMATICS , 2024 , 79 (2) . |
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Abstract ：
Due to R+ not being a group under addition, L2(R+) admits no traditional Gabor system as L2(R). Observing that R+ is a group under a new addition "circle plus", we in this paper introduce and characterize a class of weak Gabor dual frames in L2(R+) based on this new group structure. Some examples are also provided.
Keyword ：
Gabor frame Gabor frame Gabor dual Gabor dual Half real line Half real line Frame Frame Weak Gabor dual Weak Gabor dual
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| GB/T 7714 | Zhang, Yan , Li, Yun-Zhang . A time domain characterization of weak Gabor dual frames on the half real line [J]. | FILOMAT , 2023 , 37 (7) : 2237-2249 . |
| MLA | Zhang, Yan 等. "A time domain characterization of weak Gabor dual frames on the half real line" . | FILOMAT 37 . 7 (2023) : 2237-2249 . |
| APA | Zhang, Yan , Li, Yun-Zhang . A time domain characterization of weak Gabor dual frames on the half real line . | FILOMAT , 2023 , 37 (7) , 2237-2249 . |
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| GB/T 7714 | Zhang, Xiao-Li , Li, Yun-Zhang . Quaternionic Gabor frame characterization and the density theorem (vol 17, 64, 2023) [J]. | BANACH JOURNAL OF MATHEMATICAL ANALYSIS , 2023 , 17 (4) . |
| MLA | Zhang, Xiao-Li 等. "Quaternionic Gabor frame characterization and the density theorem (vol 17, 64, 2023)" . | BANACH JOURNAL OF MATHEMATICAL ANALYSIS 17 . 4 (2023) . |
| APA | Zhang, Xiao-Li , Li, Yun-Zhang . Quaternionic Gabor frame characterization and the density theorem (vol 17, 64, 2023) . | BANACH JOURNAL OF MATHEMATICAL ANALYSIS , 2023 , 17 (4) . |
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Abstract ：
Reproducing systems in L-2(R) such as wavelet and Gabor dual frames have been extensively studied, but reducing systems in L-2(R+) with R+ = (0, 8) have not. In practice, L-2(R+) models the causal space since the time variable cannot be negative. Due to R+ not being a group under addition,L-2(R+) admits no nontrivial shift invariant system and thus admits no traditional wavelet or Gabor analysis. However, L-2(R+) admits nontrivial dilation systems due to R+ being a group under multiplication. This paper addresses the frame theory of a class of dilation-and-modulation (MD) systems generated by a finite family in L-2(R+). We obtain a parametric expression of MD-frames, and a density theorem for such MD-systems which is parallel to that of traditional Gabor systems in L-2(R). It is well known that an arbitrary Gabor frame must admit dual frames with the same structure. Interestingly, it is not the case for MD-frames. We prove that an MD-frame admits MD-dual frames if and only if log(b) a is an integer, where a and b are dilation and modulation parameters, respectively. And in this case, we characterize and express all MD-dual generators for an arbitrarily given MD-frame. Some examples are also provided.
Keyword ：
Riesz basis Riesz basis MD - system MD - system MD-dual frame MD-dual frame MD-frame MD-frame Frame Frame
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| GB/T 7714 | Li, Ya-Nan , Li, Yun-Zhang . A class of reproducing systems generated by a finite family in L-2(R+) [J]. | BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY , 2023 , 46 (3) . |
| MLA | Li, Ya-Nan 等. "A class of reproducing systems generated by a finite family in L-2(R+)" . | BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY 46 . 3 (2023) . |
| APA | Li, Ya-Nan , Li, Yun-Zhang . A class of reproducing systems generated by a finite family in L-2(R+) . | BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY , 2023 , 46 (3) . |
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