A necessary condition is given for general nonuniform Gabor frames, which generalizes Benedetto and Walnut＇s theorem. A sufficient and necessary condition for a class of nonuniform Gabor frames is proved.
In this work two aspects of theory of frames are presented: a side necessary condition on irregular wavelet frames is obtained, another perturbation of wavelet and Gabor frames is considered. Specifically, we prese...In this work two aspects of theory of frames are presented: a side necessary condition on irregular wavelet frames is obtained, another perturbation of wavelet and Gabor frames is considered. Specifically, we present the results obtained on frame stability when one disturbs the mother of wavelet frame, or the parameter of dilatation, and in Gabor frames when the generating function or the parameter of translation are perturbed. In all cases we work without demanding compactness of the support, neither on the generating function, nor on its Fourier transform.展开更多
In this paper, we give a method which allows us to construct a class of Parseval frames for L2(R) from Fourier frame for L2(I). The result shows that the function which generates a Gabor frame by translations and modu...In this paper, we give a method which allows us to construct a class of Parseval frames for L2(R) from Fourier frame for L2(I). The result shows that the function which generates a Gabor frame by translations and modulations has "good" properties, i.e., it is sufficiently smooth and compactly supported.展开更多
We consider Gabor localization operators ?defined by two parameters, the generating function ?of a tight Gabor frame , indexed by a lattice , and a domain ?whose boundary consists of line segments connecting certain p...We consider Gabor localization operators ?defined by two parameters, the generating function ?of a tight Gabor frame , indexed by a lattice , and a domain ?whose boundary consists of line segments connecting certain points of . We provide an explicit formula for the boundary form , the normalized limit of the projection functional , where ?are the eigenvalues of the localization operators ?applied to dilated domains , R is an integer and is the area of the fundamental domain. The boundary form expresses quantitatively the asymptotic interactions between the generating function ?and the oriented boundary ?from the point of view of the projection functional, which measures to what degree a given trace class operator fails to be an orthogonal projection. Keeping the area of the localization domain ?bounded above corresponds to controlling the relative dimensionality of the localization problem.展开更多
Due to its potential applications in multiplexing techniques, the study of superframes has interested some researchers. This paper addresses dual super wavelet and Gabor frames in the subspace setting. We obtain a bas...Due to its potential applications in multiplexing techniques, the study of superframes has interested some researchers. This paper addresses dual super wavelet and Gabor frames in the subspace setting. We obtain a basic-equation characterization for subspace dual super wavelet and Gabor frames. In addition, applying this characterization, we derive a procedure that allows for constructing subspace dual super wavelet frames based on certain subspace dual super Gabor frames, and vice versa. Our results are new even in L^2(R, CL) setting.展开更多
This paper addresses the theory of multi-window subspace Gabor frame with rational time-frequency parameter products.With the help of a suitable Zak transform matrix,we characterize multi-window subspace Gabor frames,...This paper addresses the theory of multi-window subspace Gabor frame with rational time-frequency parameter products.With the help of a suitable Zak transform matrix,we characterize multi-window subspace Gabor frames,Riesz bases,orthonormal bases and the uniqueness of Gabor duals of type I and type II.Using these characterizations we obtain a class of multi-window subspace Gabor frames,Riesz bases,orthonormal bases,and at the same time we derive an explicit expression of their Gabor duals of type I and type II.As an application of the above results,we obtain characterizations of multi-window Gabor frames,Riesz bases and orthonormal bases for L2(R),and derive a parametric expression of Gabor duals for multi-window Gabor frames in L2(R).展开更多
For a time-frequency lattice Λ = A Z d × B Z d , it is known that an orthonormal super Gabor frame of length L exists with respect to this lattice if and only if |det( AB) | = 1 L . The proof of this result invo...For a time-frequency lattice Λ = A Z d × B Z d , it is known that an orthonormal super Gabor frame of length L exists with respect to this lattice if and only if |det( AB) | = 1 L . The proof of this result involves various techniques from multi-lattice tiling and operator algebra theory, and it is far from constructive. In this paper we present a very general scheme for constructing super Gabor frames for the rational lattice case. Our method is based on partitioning an arbitrary fundamental domain of the lattice in the frequency domain such that each subset in the partition tiles R d by the lattice in the time domain. This approach not only provides us a simple algorithm of constructing various kinds of orthonormal super Gabor frames with flexible length and multiplicity, but also allows us to construct super Gabor (non-Riesz) frames with high order smoothness and regularity. Several examples are also presented.展开更多
Given L, N, M ∈ N and an NZ-periodic set S in Z, let l2(S) be the closed subspace of l2(Z) consisting of sequences vanishing outside S. For f = { fl : 0≤l≤L-1 }l2(Z), we denote by G(f, N, M) the Gabor system genera...Given L, N, M ∈ N and an NZ-periodic set S in Z, let l2(S) be the closed subspace of l2(Z) consisting of sequences vanishing outside S. For f = { fl : 0≤l≤L-1 }l2(Z), we denote by G(f, N, M) the Gabor system generated by f, and by L(f, N, M) the closed linear subspace generated by G(f, N, M). This paper addresses density results, frame conditions for a Gabor system G(g, N, M) in l2(S), and Gabor duals of the form G(a, N, M) in some L(h, N, M) for a frame G(g, N, M) in l2(S) (so-called oblique duals). We obtain a density theorem for a Gabor system G(g, N, M) in l2(S), and show that such condition is suficient for theexistence of g={XE1:0≤l≤L-1} with G(g,N,m) We characterize g with G(g,N,m) being respectively a frame for L(g,N,m) being a tight frame for l2(S).and G(h, N, M ) in L(h, N, M ), we establish a criterion for the existence of an oblique Gabor dual of g in L(h, N, M), study the uniqueness of oblique Gabor dual, and derive an explicit expression of a class of oblique Gabor duals (among which the one with the smallest norm is obtained). Some examples are also provided.展开更多
为 Gabor 的基本事实装裱 {E_(u (m) b )T_(u (n) 一) g )}_(m， n ∈ P ) 在本地人上，地被调查。精确地，正规框架双 {E_(u (m) b )T_(u (n) 一) g }_(m， n ∈ P ) 也， Gabor 结构被显示出；产品 ab 决定它是否是可能的为 {E_(u (m)...为 Gabor 的基本事实装裱 {E_(u (m) b )T_(u (n) 一) g )}_(m， n ∈ P ) 在本地人上，地被调查。精确地，正规框架双 {E_(u (m) b )T_(u (n) 一) g }_(m， n ∈ P ) 也， Gabor 结构被显示出；产品 ab 决定它是否是可能的为 {E_(u (m) b )T_(u (n) 一) g }_(m， n ∈ P ) 被讨论是为 L^2 (K) 的一个框架；一些必要条件和为 L^2 (K) 的 Gabor 框架的二个足够的条件被建立。一个例子最后被给。展开更多
This paper investigates the fine structure of the Gabor frame generated by the B-spline B<sub>3</sub>. In other words, one extends the known part of the Gabor frame set for the 3-spline with the constructi...This paper investigates the fine structure of the Gabor frame generated by the B-spline B<sub>3</sub>. In other words, one extends the known part of the Gabor frame set for the 3-spline with the construction of the compactly supported dual windows. The frame set of the function B<sub>3</sub> is the subset of all parameters (a,b) ∈R<sup>2</sup>+ </sub>for which the time-frequency shifts of B<sub>3</sub> along aZ × bZ form a Gabor frame for L<sup>2</sup>(R).展开更多
基金The project is partially supported by a grant from Beijing Educational Committee (KM200410005013)
文摘A necessary condition is given for general nonuniform Gabor frames, which generalizes Benedetto and Walnut＇s theorem. A sufficient and necessary condition for a class of nonuniform Gabor frames is proved.
基金This work was supported by CONICET and Universidad Nacional de San Luis
文摘In this work two aspects of theory of frames are presented: a side necessary condition on irregular wavelet frames is obtained, another perturbation of wavelet and Gabor frames is considered. Specifically, we present the results obtained on frame stability when one disturbs the mother of wavelet frame, or the parameter of dilatation, and in Gabor frames when the generating function or the parameter of translation are perturbed. In all cases we work without demanding compactness of the support, neither on the generating function, nor on its Fourier transform.
基金Supported by Henan Province Education Department Natural Science Foundation of China（2008B510001）
文摘In this paper, we give a method which allows us to construct a class of Parseval frames for L2(R) from Fourier frame for L2(I). The result shows that the function which generates a Gabor frame by translations and modulations has "good" properties, i.e., it is sufficiently smooth and compactly supported.
文摘We consider Gabor localization operators ?defined by two parameters, the generating function ?of a tight Gabor frame , indexed by a lattice , and a domain ?whose boundary consists of line segments connecting certain points of . We provide an explicit formula for the boundary form , the normalized limit of the projection functional , where ?are the eigenvalues of the localization operators ?applied to dilated domains , R is an integer and is the area of the fundamental domain. The boundary form expresses quantitatively the asymptotic interactions between the generating function ?and the oriented boundary ?from the point of view of the projection functional, which measures to what degree a given trace class operator fails to be an orthogonal projection. Keeping the area of the localization domain ?bounded above corresponds to controlling the relative dimensionality of the localization problem.
基金supported by National Natural Science Foundation of China (Grant No. 11271037)
文摘Due to its potential applications in multiplexing techniques, the study of superframes has interested some researchers. This paper addresses dual super wavelet and Gabor frames in the subspace setting. We obtain a basic-equation characterization for subspace dual super wavelet and Gabor frames. In addition, applying this characterization, we derive a procedure that allows for constructing subspace dual super wavelet frames based on certain subspace dual super Gabor frames, and vice versa. Our results are new even in L^2(R, CL) setting.
基金supported by National Natural Science Foundation of China (Grant No. 11271037)Beijing Natural Science Foundation (Grant No. 1122008)the Scientific Research Common Program of Beijing Municipal Commission of Education (Grant No. KM201110005030)
文摘This paper addresses the theory of multi-window subspace Gabor frame with rational time-frequency parameter products.With the help of a suitable Zak transform matrix,we characterize multi-window subspace Gabor frames,Riesz bases,orthonormal bases and the uniqueness of Gabor duals of type I and type II.Using these characterizations we obtain a class of multi-window subspace Gabor frames,Riesz bases,orthonormal bases,and at the same time we derive an explicit expression of their Gabor duals of type I and type II.As an application of the above results,we obtain characterizations of multi-window Gabor frames,Riesz bases and orthonormal bases for L2(R),and derive a parametric expression of Gabor duals for multi-window Gabor frames in L2(R).
基金supported by Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministrythe Fundamental Research Funds for the Central Universitiesthe grant of Young Teachers Study Abroad of China Scholarship Council (2005)
文摘For a time-frequency lattice Λ = A Z d × B Z d , it is known that an orthonormal super Gabor frame of length L exists with respect to this lattice if and only if |det( AB) | = 1 L . The proof of this result involves various techniques from multi-lattice tiling and operator algebra theory, and it is far from constructive. In this paper we present a very general scheme for constructing super Gabor frames for the rational lattice case. Our method is based on partitioning an arbitrary fundamental domain of the lattice in the frequency domain such that each subset in the partition tiles R d by the lattice in the time domain. This approach not only provides us a simple algorithm of constructing various kinds of orthonormal super Gabor frames with flexible length and multiplicity, but also allows us to construct super Gabor (non-Riesz) frames with high order smoothness and regularity. Several examples are also presented.
基金supported by National Natural Science Foundation of China (Grant Nos. 10901013, 10671008)Beijing Natural Science Foundation (Grant No. 1092001)+1 种基金the Scientific Research Common Program of Beijing Municipal Commission of Education (Grant No. KM201110005030)the Project Sponsored by SRF for ROCS, SEM of China
文摘Given L, N, M ∈ N and an NZ-periodic set S in Z, let l2(S) be the closed subspace of l2(Z) consisting of sequences vanishing outside S. For f = { fl : 0≤l≤L-1 }l2(Z), we denote by G(f, N, M) the Gabor system generated by f, and by L(f, N, M) the closed linear subspace generated by G(f, N, M). This paper addresses density results, frame conditions for a Gabor system G(g, N, M) in l2(S), and Gabor duals of the form G(a, N, M) in some L(h, N, M) for a frame G(g, N, M) in l2(S) (so-called oblique duals). We obtain a density theorem for a Gabor system G(g, N, M) in l2(S), and show that such condition is suficient for theexistence of g={XE1:0≤l≤L-1} with G(g,N,m) We characterize g with G(g,N,m) being respectively a frame for L(g,N,m) being a tight frame for l2(S).and G(h, N, M ) in L(h, N, M ), we establish a criterion for the existence of an oblique Gabor dual of g in L(h, N, M), study the uniqueness of oblique Gabor dual, and derive an explicit expression of a class of oblique Gabor duals (among which the one with the smallest norm is obtained). Some examples are also provided.
基金Project supported by the National Natural Science Foundation of China (No. 10571084, No. 10671062) the Postdoctoral Science Foundation of China (No. 2005037694)the Natural Science Foundation of the Education Department of Henan Province of China (No. 2006110001).
文摘为 Gabor 的基本事实装裱 {E_(u (m) b )T_(u (n) 一) g )}_(m， n ∈ P ) 在本地人上，地被调查。精确地，正规框架双 {E_(u (m) b )T_(u (n) 一) g }_(m， n ∈ P ) 也， Gabor 结构被显示出；产品 ab 决定它是否是可能的为 {E_(u (m) b )T_(u (n) 一) g }_(m， n ∈ P ) 被讨论是为 L^2 (K) 的一个框架；一些必要条件和为 L^2 (K) 的 Gabor 框架的二个足够的条件被建立。一个例子最后被给。
基金Supported by Development Program for Young Teacher in the Science Research Project for Colleges and Universities of Xinjiang Province of China (XJEDU2009S67)
文摘This paper investigates the fine structure of the Gabor frame generated by the B-spline B<sub>3</sub>. In other words, one extends the known part of the Gabor frame set for the 3-spline with the construction of the compactly supported dual windows. The frame set of the function B<sub>3</sub> is the subset of all parameters (a,b) ∈R<sup>2</sup>+ </sub>for which the time-frequency shifts of B<sub>3</sub> along aZ × bZ form a Gabor frame for L<sup>2</sup>(R).
