Peter G. Casazza, Ole Christensen
We give sufficient conditions for translates and modulates of a function g in L^2(R) to be a frame for its closed linear span. Even in the case where this family spans all of L^2(R), wou conditions are significantly weaker than the previous known conditions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper we have generalized and studied the $K$-Weyl-Heisenberg frames, where $K$ is a bounded linear operator on $L^2(\mathbb{R}^d)$. We have obtained necessary and sufficient conditions for acertain system to be a $K$-Weyl-Heisenberg frame. We have also given the invariance property of these $K$-Weyl-Heisenberg frames.
We investigate the relevance of admissibility criteria based on Plancherel measure for the characterization of tight Weyl-Heisenberg frames with integer oversampling. For this purpose we observe that functions giving rise to such Weyl-Heisenberg frames are admissible with respect to the action of suitably defined type-I discrete group G. This allows to relate the construction of Weyl-Heisenberg frames to the Plancherel measure of G, which provides an alternative proof and a new interpretation of the well-known Zak transform based criterion for tight Weyl-Heisenberg frames with integer oversamp…
We examine the question of which characteristic functions yield Weyl-Heisenberg frames for various values of the parameters. We also give numerous applications of frames of characteristic functions to the general case (g,a,b).
We study the construction of Gabor frames and wavelet frames for Weyl-Heisenberg group and extended affine group by using contraction between the affine group and the Weyl-Heisenberg group due to Subag, Baruch, Birman and Mann. Firstly, we give construction of Gabor frames with several generators from a unitary irreducible representation associated to the Weyl-Heisenberg group. Wavelet frames with several generators associated with an extended affine group have been obtained. A relation between frames for Weyl-Heisenberg group and extended affine group is also discussed. Finally, we show that …
An explicit expression for the Kohn-Nirenberg symbol of a Weyl- Heisenberg frame operator on $L^2(\mathbb{R})$ is obtained directly from the Gabor atom coming from new classes of window functions. This new approach, using only elementary Fourier analysis, is independent of the theory of distributions and works strictly inside $L^2(\mathbb{R})$. Kohn-Nirenberg operators are introduced and are shown to be Weyl-Heisenberg frame operators in suitable cases.
We develop a usable perturbation theory for Weyl-Heisenberg frames. In particular, we prove that if $(E_{mb}T_{na}g)_{m,n\inmathbb Z}$ is a WH-frame and $h$ is a function which is close to $g$ in the Wiener Amalgam space norm, then $(E_{mb}T_{na}h)_{m,n\in \mathbb Z}$ is also a WH-frame.
Let $a$, $b$ be two fixed positive constants. A function $g\in L^2({\mathbb R})$ is called a \textit{mother Weyl-Heisenberg frame wavelet} for $(a,b)$ if $g$ generates a frame for $L^2({\mathbb R})$ under modulates by $b$ and translates by $a$, i.e., $\{e^{imbt}g(t-na\}_{m,n\in\mathbb{Z}}$ is a frame for $L^2(\mathbb{R})$. In this paper, we establish a connection between mother Weyl-Heisenberg frame wavelets of certain special forms and certain strongly positive definite quadratic forms of infinite dimension. Some examples of application in matrix algebra are provided.
Inspired by the work of A. Bonami and S. Poornima that a non-constant function which is homogeneous of degree $0$ cannot be a Fourier multiplier on homogeneous Sobolev spaces, we establish analogous results for Fourier multipliers on the Heisenberg group $ \mathbb{H}^n $ and Weyl multipliers on $ \mathbb{C}^n $ acting on Sobolev Spaces.
In this paper we prove weighted norm inequalities for Weyl multipliers satisfying Mauceri's condition. As applications of this we obtain some estimates for $L^p$ multipliers on the Heisenberg group and also show in the context of a theorem of Weis on operator valued Fourier multipliers that the R-boundedness of the derivative of the multiplier is not necessary for the boundedness of the multiplier transform.
In this article, we study two different types of operators, the localization operator and Weyl transform, on the reduced Heisenberg group with multidimensional center $\mathcal{G}$. The group $\mathcal{G}$ is a quotient group of non-isotropic Heisenberg group with multidimensional center $\mathcal{H}^m$ by its center subgroup. Firstly, we define the localization operator using a wavelet transform on $\mathcal{G}$ and obtain the product formula for the localization operators. Next, we define the Weyl transform associated to the Wigner transform on $\mathcal{G}$ with the operator-valued symbol. …
In this paper, we present necessary and sufficient conditions for some types of linear combination of frame elements (wave packet) to be a frame for $L^2(\mathbb{K})$, where $\mathbb{K}$ is a local field of positive characteristic.
A Weyl-Heisenberg frame for L^2(R) is a frame consisting of translates and modulates of a fixed function. In this paper we give necessary and sufficient conditions for this family to form a tight WH-frame. This allows us to write down explicitly all functions g for which all translates and modulates of g form an orthonormal basis for L^2(R). There are a number of consequences of this classification, including a simple direct classification of the alternate dual frames to a WH-frame (A result originally due to Janssen).
In this article, we define Weyl transform on second countable type - $I$ locally compact group $G,$ and as an operator on $L^2(G),$ we prove that the Weyl transform is compact when the symbol lies in $L^p(G\times \hat{G})$ with $1\leq p\leq 2.$ Further, for the Euclidean motion group and Heisenberg motion group, we prove that the Weyl transform can not be extended as a bounded operator for the symbol belongs to $L^p(G\times \hat{G})$ with $2<p<\infty.$ To carry out this, we construct positive, square integrable and compactly supported function, on the respective groups, such that $L^{p'}$ norm…
Let $a$, $b$ be two fixed non-zero constants. A measurable set $E\subset \mathbb{R}$ is called a Weyl-Heisenberg frame set for $(a, b)$ if the function $g=χ_{E}$ generates a Weyl-Heisenberg frame for $L^2(\mathbb{R})$ under modulates by $b$ and translates by $a$, i.e., $\{e^{imbt}g(t-na\}_{m,n\in\mathbb{Z}}$ is a frame for $L^2(\mathbb{R})$. It is an open question on how to characterize all frame sets for a given pair $(a,b)$ in general. In the case that $a=2π$ and $b=1$, a result due to Casazza and Kalton shows that the condition that the set $F=\bigcup_{j=1}^{k}([0,2π)+2n_{j}π)$ (where $\{n_…
$L^p$- Heisenberg--Pauli--Weyl uncertainty inequalities on certain two-step nilpotent Lie groups
本文在 Métivier 群上建立了新的 $L^p$-海森堡-泡利-韦伊尔不确定原理，通过群傅里叶变换证明非零函数与变换无法同时精确局部化，显著提升了原有不等式的精度。
A complexified Heisenberg matrix group $\mathrm{H}_\mathbb{C}$ with entries from an infinite-dimensional Hilbert space $H$ is investigated. The Weyl--Schrödinger type irreducible representations of $\mathrm{H}_\mathbb{C}$ on the space $L^2_χ$ of square-integrable scalar functions is described. The integrability is understood under the invariant probability measure $χ$ which satisfies an abstract Kolmogorov consistency conditions over the infinite-dimensional unitary group $U(\infty)$ irreducible acted on $H$. The space $L^2_χ$ is generated by Schur polynomials in variables on Paley--Wiener map…
Heisenberg-Pauli-Weyl uncertainty principles for the fractional Dunkl transform on the real line
本文针对实线上的分数阶Dunkl变换，给出了已有不确定性不等式取等号的显式函数形式，建立了$1 \leq p \leq 2$的$L^p$型Heisenberg-Pauli-Weyl不确定性原理，并在$p=2$时推导出更尖锐的结果及取等条件。
$L^p$-Heisenberg--Pauli--Weyl Uncertainty Inequalities on the Laguerre Hypergroup
该论文在拉盖尔超群上建立$1\leq p\leq2$的$L^p$-海森堡-泡利-外尔不确定性不等式，通过傅里叶 - 拉盖尔变换等进行分析，改进了$p = 2$时的结果，避免热核技术，提供统一框架，加强了不同调和分析间联系。
Nonstationary frames of translates and frames for the Weyl--Heisenberg group and the extended affine group
本文研究平移非平稳框架的存在条件，并据此构造 Weyl–Heisenberg 群的 Gabor 框架与扩展仿射群的小波框架，同时刻画其对偶、算子逆逼近及 Riesz 基性质。
We investigate the group $\mathcal{H}_\mathbb{C}$ of complexified Heisenberg matrices with entries from an infinite-dimensional complex Hilbert space $H$. Irreducible representations of the Weyl--Schr{ö}dinger type on the space $L^2_χ$ of quadratically integrable $\mathbb{C}$-valued functions are described. Integrability is understood with respect to the projective limit $χ=\varprojlimχ_i$ of probability Haar measures $χ_i$ defined on groups of unitary $i\times i$-matrices $U(i)$. The measure $χ$ is invariant under the infinite-dimensional group $U(\infty)=\bigcup U(i)$ and satisfies the abstr…
Sums of Frames from the Weyl--Heisenberg Group and Applications to Frame Algorithm
本文研究了Weyl-Heisenberg群生成的Gabor帧在$ L^2(\mathbb{R}) $空间中的帧界与帧求和标量的关系，给出了有限帧求和成为帧的充分条件，并探讨了帧求和在帧算法中的应用。
Since $(\mathbb{H}^n\rtimes U(n),U(n))$ is a Gelfand pair, an exact analogue of the Heisenberg group result due to Narayanan and Ratnakumar is not possible for the Heisenberg motion group. In this article, we prove that if the Weyl transform of a finitely supported integrable function on the Heisenberg motion group is non-zero only for finitely many Fourier-Wigner pieces and have finite rank, then the function must be zero. We also prove an analogue of the Heisenberg group result on the quaternion Heisenberg group. In the end, a quantitative interpretation of these results is described through…
In this paper we introduce and investigate the concept of reproducing pairs which generalizes continuous frames. We will introduce a concept that represents a unifying way to look at certain continuous frames (resp. reproducing pairs) on LCA groups, which can be described as continuous nonstationary Gabor systems and investigate conditions for these systems to form a continuous frame (resp. reproducing pair). As a byproduct we identify the structure of the frame operator (resp. resolution operator). Moreover, we ask the question, whether there always exist mutually dual systems with the same s…
This paper studies how differentiable representations of certain subsemigroups of the Weyl-Heisenberg group may be obtained in suitably constructed rigged Hilbert spaces. These semigroup representations are induced from a continuous unitary representation of the Weyl-Heisenberg group in a Hilbert space. Aspects of the rigged Hilbert space formulation of time asymmetric quantum mechanics are also investigated within the context of the results developed here.
We provide a detailed development of a function valued inner product known as the bracket product and used effectively by de Boor, Devore, Ron and Shen to study translation invariant systems. We develop a version of the bracket product specifically geared to Weyl-Heisenberg frames. This bracket product has all the properties of a standard inner product including Bessel's inequality, a Riesz Representation Theorem, and a Gram-Schmidt process which turns a sequence of functions $(g_{n})$ into a sequence $(e_{n})$ with the property that $(E_{mb}e_{n})_{m,n\in \Bbb Z}$ is orthonormal in $L^{2}(\Bb…
We present in this paper a construction for Gabor-type frames built out of generalized Weyl-Heisenberg groups. These latter are obtained via central extensions of groups which are direct products of locally compact abelian groups and their duals. Our results generalize many of the results, appearing in the literature, on frames built out of the Schrödinger representation of the standard Weyl-Heisenberg group. In particular, we obtain a generalization of the result in \cite{PO}, in which the product $ab$ determines whether it is possible for the Gabor system $\{E_{mb}T_{na}g \}_{m,n\in \mathbb …
The bracket map was originally considered for locally compact abelian groups. In this work we extend the study of bracket maps to the noncommutative setting, providing characterizations of bases and frames for cyclic subspaces of the Heisenberg group. We also indicate how to generalize these results to a class of non-abelian nilpotent Lie groups whose irreducible representations are square integrable modulo the center.
In a previous Note we established a necessary and sufficient condition for a multivariate Weyl--Heisenberg system G(ϕ,Λ) to be a frame when the window is a generalized Gaussian (squeezed coherent state) and Λ a rectangular lattice. In this Note we extend this result to lattices of the type $Λ=MZ^{2n}$ where M is a positive definite symmetric matrix.
In 1990, Daubechies proved a fundamental identity for Weyl-Heisenberg systems which is now called the Weyl-Heisenberg Frame Identity. WH-Frame Identity: If $g\in W(L^{\infty},L^{1})$, then for all continuous, compactly supported functions f we have: \[\sum_{m,n}|<f,E_{mb}T_{na}g>|^{2} = \frac{1}{b}\sum_{k}\int_{\Bbb R}\bar{f(t)}f(t-k/b)\sum_{n} g(t-na)\bar{g(t-na-k/b)} dt.\] It has been folklore that the identity will not hold universally. We make a detailed study of the WH-Frame Identity and show: (1) The identity does not require any assumptions on ab (such as the requirement that $ab\le 1$ …
We present a comprehensive analysis of the convergence properties of the frame operators of Weyl-Heisenberg systems and shift-invariant systems, and relate these to the convergence of the Walnut representation. We give a deep analysis of necessary conditions and sufficient conditions for convergence of the frame operator. We show that symmetric, norm and unconditional convergence of the Walnut series are all different, but that weak and norm convergence are the same, while there are WH-systems for which the Walnut representation has none of these convergence properties. We make a detailed stud…
Entanglement entropy and hyperuniformity of Ginibre and Weyl-Heisenberg ensembles
本文证明一类平面行列式点过程的纠缠熵增长速率等于其方差涨落速率，并据此说明Weyl-Heisenberg系综因I类超均匀性而满足面积律。
Let ϕ be an arbitrary generalized Gaussian (squeezed coherent state), Λ_{αβ}=(α_1 Z \times\cdot\cdot\cdot\times α_{n}\mathbb{Z)\times}(β_{1}\mathbb{Z}\times\cdot\cdot\cdot \timesβ_{n}\mathbb{Z)}$ a rectangular lattice. We show that there exists a positive definite symplectic matrix M (depending on ϕ) such that the multivariate Weyl--Heisenberg system G(ϕ,MΛ_{αβ}) is a frame. In a forthcoming Note we will prove a converse to this result.
A new 2-parameter quadratic deformation of the quantum oscillator algebra and its 1-parameter deformed Heisenberg subalgebra are considered. An infinite dimensional Fock module representation is presented which at roots of unity contains null vectors and so is reducible to a finite dimensional representation. The cyclic, nilpotent and unitary representations are discussed. Witten's deformation of $sl_2$ and some deformed infinite dimensional algebras are constructed from the $1d$ Heisenberg algebra generators. The deformation of the centreless Virasoro algebra at roots of unity is mentioned. F…
We outline an abstract approach to the pseudo-differential Weyl calculus for operators in function spaces in infinitely many variables. Our earlier approach to the Weyl calculus for Lie group representations is extended to the case of representations associated with infinite-dimensional coadjoint orbits. We illustrate the approach by the case of infinite-dimensional Heisenberg groups. The classical Weyl-Hörmander calculus is recovered for the Schrödinger representations of the finite-dimensional Heisenberg groups.
Translation complete subgroups of affine Weyl-Heisenberg groups and their generalized wavelet systems
本文研究仿射韦伊-海森堡群的特定子群形式，通过平方可积表示确定反演公式存在的条件，提出类似卡勒昂条件的适定性准则，并探讨其与Wigner函数基准则的比较，展示二维和三维中的新例子。
The paper deals with the necessary and sufficient conditions for obtaining reconstruction formulae and sampling theorems for every function belonging to the principal shift invariant subspace of $L^2(\mathbb{H}^n)$, both in the time domain and a transform domain, where $\mathbb{H}^n$ denotes the Heisenberg group.
Weyl-Heisenberg ensembles are translation-invariant determinantal point processes on $\mathbb{R}^{2d}$ associated with the Schrödinger representation of the Heisenberg group, and include as examples the Ginibre ensemble and the polyanalytic ensembles, which model the higher Landau levels in physics. We introduce finite versions of the Weyl-Heisenberg ensembles and show that they behave analogously to the finite Ginibre ensembles. More specifically, guided by the observation that the Ginibre ensemble with $N$ points is asymptotically close to the restriction of the infinite Ginibre ensemble to …
We introduce higher order (polynomial) extensions of the unique (up to isomorphisms) non trivial central extension of the Heisenberg algebra. Using the boson representation of the latter, we construct the corresponding polynomial analogue of the Weyl $C^*$-algebra and use this result to deduce the explicit form of the composition law of the associated generalization of the 1-dimensional Heisenberg group. These results are used to calculate the vacuum characteristic func- tions as well as the moments of the observables in the Galilei algebra. The continuous extensions of these objects gives a n…
We consider a non-canonical phase-space deformation of the Heisenberg-Weyl algebra that was recently introduced in the context of quantum cosmology. We prove the existence of minimal uncertainties for all pairs of non-commuting variables. We also show that the states which minimize each uncertainty inequality are ground states of certain positive operators. The algebra is shown to be stable and to violate the usual Heisenberg-Pauli-Weyl inequality for position and momentum. The techniques used are potentially interesting in the context of time-frequency analysis.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。