Gabor Frames over Irregular Lattices

We give necessary and sufficient conditions for gW(L ,¹) to generate a Gabor frame over certain irregular lattices.     

Non-separable Lattices,Gabor Orthonormal Bases and Tilings

Let \(K\subset {\mathbb {R}}^d\) be a bounded set with positive Lebesgue measure. Let \(\Lambda =M({\mathbb {Z}}^{2d})\) be a lattice in \({\mathbb {R}}^{2d}\) with density dens\((\Lambda )=1\). It is well-known that if M is a diagonal block matrix with diagonal matrices A and B, then \({\mathcal {G}}(|K|^{-1/2}\chi _K, \Lambda )\) is an orthonormal basis for \(L^2({\mathbb {R}}^d)\) if and only if K tiles both by \(A({\mathbb {Z}}^d)\) and \(B^{-t}({\mathbb {Z}}^d)\). However, there has not been any intensive study when M is not a diagonal matrix. We investigate this problem for a large class of important cases of M. In particular, if M is any lower block triangular matrix with diagonal matrices A and B, we prove that if \({\mathcal {G}}(|K|^{-1/2}\chi _K, \Lambda )\) is an orthonormal basis, then K can be written as a finite union of fundamental domains of \(A({{\mathbb {Z}}}^d)\) and at the same time, as a finite union of fundamental domains of \(B^{-t}({{\mathbb {Z}}}^d)\). If \(A^tB\) is an integer matrix, then there is only one common fundamental domain, which means K tiles by a lattice and is spectral. However, surprisingly, we will also illustrate by an example that a union of more than one fundamental domain is also possible. We also provide a constructive way for forming a Gabor window function for a given upper triangular lattice. Our study is related to a Fuglede’s type problem in Gabor setting and we give a partial answer to this problem in the case of lattices.

     

Quantum Harmonic Analysis on Lattices and Gabor Multipliers

Journal of Fourier Analysis and Applications - We calculate the norm of the Fourier operator from $$L^p(X)$$ to $$L^q({hat{X}})$$ when X is an infinite locally compact abelian group that is,...     

Representation of Operators in the Time-Frequency Domain and Generalized Gabor Multipliers

Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A characterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator’s best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations. Generalizations (multiple Gabor multipliers) are introduced for better approximation of overspread operators. The Riesz property of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator’s best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows.     

Tight Gabor Frames Associated with Non-separable Lattices and the Hyperbolic Secant

We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L ²(?) when the translations and modulations of the window are associated with certain non-separable lattices in ?² which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on ?²ⁿ with the property that the short-time Fourier transform defines an isometric embedding from L ²(? ⁿ ) to L _μ ² (?²ⁿ ) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point.     

拟半连续格和交半连续格

作为半连续格的推广, 引入了拟半连续格的概念. 讨论了拟半连续格的基本性质. 在拟半连续格上得到了类似于拟连续偏序集的一些主要结果. 同时探讨了半连续格、拟半连续格、交半连续格、交连续格、强连续格几种不同结构之间的关系. 最后,讨论了半连续函数空间仍是半连续格的条件.     

On the structure of Gabor and super Gabor spaces

We study the structure of Gabor and super Gabor spaces inside L²(\mathbbR^2d){L^{2}(\mathbb{R}^{2d})} and specialize the results to the case where the spaces are generated by vectors of Hermite functions. We then construct an isometric isomorphism between such spaces and Fock spaces of polyanalytic functions and use it in order to obtain structure theorems and orthogonal projections for both spaces at once, including explicit formulas for the reproducing kernels. In particular we recover a structure result obtained by N. Vasilevski using complex analysis and special functions. In contrast, our methods use only time-frequency analysis, exploring a link between time-frequency analysis and the theory of polyanalytic functions, provided by the polyanalytic part of the Gabor transform with a Hermite window, the polyanalytic Bargmann transform.     

On the structure of Gabor and super Gabor spaces

We study the structure of Gabor and super Gabor spaces inside ${L^{2}(\mathbb{R}^{2d})}$ and specialize the results to the case where the spaces are generated by vectors of Hermite functions. We then construct an isometric isomorphism between such spaces and Fock spaces of polyanalytic functions and use it in order to obtain structure theorems and orthogonal projections for both spaces at once, including explicit formulas for the reproducing kernels. In particular we recover a structure result obtained by N. Vasilevski using complex analysis and special functions. In contrast, our methods use only time-frequency analysis, exploring a link between time-frequency analysis and the theory of polyanalytic functions, provided by the polyanalytic part of the Gabor transform with a Hermite window, the polyanalytic Bargmann transform.     

Smooth格和强Smooth格

引入了强smooth格的概念,讨论了smooth格与强smooth格的一些基本性质,证明了强smooth格可用保任意交和Scott闭集之并的映射嵌入到某方体[0,1]X之中.     

真蕴涵格与优蕴涵格

研究蕴涵格的性质及其结构。借助数学软件给出8阶及以下蕴涵格的完全分类,说明了真蕴涵格与优蕴涵格的存在性,并纠正了多个文献中关于蕴涵格的一些错误结论。同时,引入局部有限蕴涵格和次结合蕴涵格的概念,证明了局部有限蕴涵格是优蕴涵格。     

半连续格和半代数格的映射性质

研究了半连续格及半代数格的一些映射性质,讨论了强连续格的函数空间,给出了强连续格到方体的嵌入定理.     

蕴涵格的同余关系及熵蕴涵格

在蕴涵格中引入了蕴涵滤子的概念 ,讨论了蕴涵滤子的一些基本性质 ,并由此建立了由素蕴涵滤子决定的同余关系及其商蕴涵格 ,以便为Fuzzy推理建立严格的逻辑基础作些必要的准备 .     

The existence of tight Gabor duals for Gabor frames and subspace Gabor frames

Let K and L be two full-rank lattices in Rd. We give a complete characterization for all the Gabor frames that admit tight dual of the same type. The characterization is given in terms of the center-valued trace of the von Neumann algebra generated by the left regular projective unitary representations associated with the time-frequency lattice K×L. Two applications of this characterization were obtained: (i) We are able to prove that every Gabor frame has a tight dual if and only if the volume of K×L is less than or equal to . (ii) We are able to obtain sufficient or necessary conditions for the existence of tight Gabor pseudo-duals for subspace Gabor frames in various cases. In particular, we prove that every subspace Gabor frame has a tight Gabor pseudo-dual if either the volume or v(K×L)?2. Moreover, if K=αZd, L=βZd with αβ=1, then a subspace Gabor frame G(g,L,K) has a tight Gabor pseudo-dual only when G(g,L,K) itself is already tight.     

分配格与模格的另一二条件等价定义

针对分配格与模格的格等式定义问题,得知了二条件是定义分配格与模格的最少条件,并进一步证明了Sholander's basis是定义分配格的最短最少变量格等式,最后又从分配格和模格的基本定义出发给出了新的分配格的二条件和三条件等价定义等式及模格的二条件与三条件等价定义等式.     

剩余格与正则剩余格的特征定理

本文进一步研究了具有广泛应用的一类模糊逻辑代数系统——剩余格，并引入了正则剩余格的概念，对剩余格与正则剩余格的定义进行了讨论，给出了剩余格与正则剩余格的特征定理，其中包含剩余格与正则剩余格的等式特征，从而这两个格类都构成簇．本文还讨论了剩余格与正则剩余格公理系统的独立性，以及它们与相近代数结构的关系．     

Involutive Residuated Lattices Based on Modular and Distributive Lattices

An involutive residuated lattice (IRL) is a lattice-ordered monoid possessing residual operations and a dualizing element. We show that a large class of self-dual lattices may be endowed with an IRL structure, and give examples of lattices which fail to admit IRLs with natural algebraic conditions. A classification of all IRLs based on the modular lattices M _n is provided.     

映射族诱导的邻近格与半一致格

我们在[1]与[2]中初步讨论了邻近格与半一致格,本文继续这一工作。本文保持[1]与[2]的记号及对格与映射所作的基本假定,但加*号的结论需用到以下附加条件: 1°所论的格如X上定义了分子集或X≤b。 2°所涉及的映射f:X→Y映X中的分子为Y中的分子。 本文通篇考虑映射族,X_t上定义了某种(同类的)结构。从拓扑结构理论通常的观点看来,以下定义是合适的:     

Lattices That are Embeddable in Suborder Lattices

Various types of lattices are embedded in suborder lattices of posets possessing certain properties. In particular, it is shown that the class of lattices isomorphic to sublattices of suborder lattices of posets of length at most n is a variety, for any n < ω.Supported by INTAS-YSF grant 2001/1-65, by RFBR-DFG grant No. 01-01-04003, by RF Ministry of Education grant No. E02-1.0-32, by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2112.2003.1, and by a grant from the Russian Science Support Foundation.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 483–511, July–August, 2005.     

Embedding Finite Lattices into Finite Biatomic Lattices

For a class C of finite lattices, the question arises whether any lattice in C can be embedded into some atomistic, biatomic lattice in C. We provide answers to the question above for C being, respectively,– the class of all finite lattices;– the class of all finite lower bounded lattices (solved by the first author's earlier work);– the class of all finite join-semidistributive lattices (this problem was, until now, open).We solve the latter problem by finding a quasi-identity valid in all finite, atomistic, biatomic, join-semidistributive lattices but not in all finite join-semidistributive lattices.     

Isometrical Embeddings of Lattices into Geometric Lattices

A lattice is said to be finite height generated if it is complete and every element is the join of some elements of finite height. Extending former results by Grätzer and Kiss (Order 2:351–365, 1986) on finite lattices, we prove that every finite height generated algebraic lattice that has a pseudorank function is isometrically embeddable into a geometric lattice.