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.
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.
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.
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.
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_…
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…
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…
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 …
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…
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.
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…
Two measurable sets $S, Λ\subseteq \mathcal{R}^d$ form a Heisenberg uniqueness pair, if every bounded measure $μ$ with support in S whose Fourier transform vanishes on Λ must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in $\mathcal{R}^d$. As a corollary we obtain a new, surprising version of the classical Cramér-Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of…
