We give an introduction to irregular Weyl-Heisenberg frames showing the latest developments and open problems. We provide several new results for semiirregular WH-frames as well as giving new and more accessable proofs for several results from the literature.

We have applied the method of integration of the Heisenberg equation of motion proposed by Bender and Dunne, and M. Kamella and M. Razavy to the potential V(q) = v q - µ q with linear and nonlinear dissipation. We concentrate our calculations on the evolution of basis set of Weyl Ordered Operators and calculate the mean position , velocity , the commutation relation [q, p], and the energ...

The goal of this paper is to characterise certain probability laws on a class of quantum groups or braided groups that we will call nilpotent. First we introduce a braided analogue of the Heisenberg-Weyl group, which shall serve as standard example. We introduce Gaussian functionals on quantum groups or braided groups as functionals that satisfy an analogue of the Bernstein property, i.e. that ...

The well-known second order moment Heisenberg-Weyl inequality (or uncertainty relation) in Fourier Analysis states: Assume that f : R → C is a complex valued function of a random real variable x such that f ∈ L(R). Then the product of the second moment of the random real x for |f | and the second moment of the random real ξ for ∣∣∣f̂ ∣∣∣2 is at least E|f |2 / 4π, where f̂ is the Fourier transform...

Time-frequency transforms represent a signal as a mixture of its time domain representation and its frequency domain representation. We present efficient algorithms for the quantum Zak transform and quantum Weyl-Heisenberg transform.

In a first part, we are concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg–Weyl, its Bargmann–Fock representation with differential operators and the associated one-parameter group. Upon this basis, the paper is then devoted to the groups of Riordan matrices associated to the related transformations of matrices (i.e. substitutions with prefun...

In a first part, we are concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg–Weyl, its Bargmann–Fock representation with differential operators and the associated one-parameter group. Upon this basis, the paper is then devoted to the groups of Riordan matrices associated to the related transformations of matrices (i.e., substitutions with prefu...

Applying the methods developed in [3], in [4] we have constructed generalized CS attached to the Jacobi group G1 = H1 oSU(1, 1), based on the homogeneous Kähler manifold D 1 = H1/R × SU(1, 1)/U(1) = C 1 × D1. Here D1 denotes the unit disk D1 = {w ∈ C ; |w| < 1}, and Hn is the (2n + 1)-dimensional real Heisenberg-Weyl group with Lie algebra hn. In [4] we have also emphasized that, when expressed...