Tight Weyl–Heisenberg frames in `(Z) are the tool for short-time Fourier analysis in discrete time. They are closely related to paraunitary modulated filter banks and are studied here using techniques of the filter bank theory. Good resolution of short-time Fourier analysis in the joint time–frequency plane is not attainable unless some redundancy is introduced. That is the reason for consideri...

We use the functional representation of Heisenberg-Weyl group and obtain equation for the spectrum of the model, which is more complicated than Bethes ones, but can be written explicitly through theta functions.

We examine the Schrödinger algebra in the framework of Berezin quantization. First, the Heisenberg-Weyl and sl(2) algebras are studied. Then the Berezin representation of the Schrödinger algebra is computed. In fact, the sl(2) piece of the Schrödinger algebra can be decoupled from the Heisenberg component. This is accomplished using a special realization of the sl(2) component that is built fro...

The parentage between Weyl pairs, generalized Pauli group and unitary group is investigated in detail. We start from an abstract definition of the Heisenberg-Weyl group on the field R and then switch to the discrete Heisenberg-Weyl group or generalized Pauli group on a finite ring Zd. The main characteristics of the latter group, an abstract group of order d noted Pd, are given (conjugacy class...

We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partiall...

The well-known Poisson Summation Formula is analysed from the perspective of the coherent state systems associated with the Heisenberg–Weyl group. In particular, it is shown that the Poisson Summation Formula may be viewed abstractly as a relation between two sets of bases (Zak bases) arising as simultaneous eigenvectors of two commuting unitary operators in which geometric phase plays a key ro...

Two standard tools for signal analysis are the short{time Fourier transform and the continuous wavelet transform. These tools arise as matrix coeecients of square integrable representations of the Heisenberg and aane groups respectively, and discrete frame decompositions of L 2 arise from approximations of corresponding reproducing formulae. Here we study two groups, the so-called aane Weyl-Hei...

The unitary representations of the Canonical group that is defined to be the semidirect product of the unitary group with the Weyl-Heisenberg group, !!1, 3" ! "!1, 3""s #!1, 3" are studied. The Canonical group is equivalently written as the semi-direct product of the special unitary group with the Oscillator group !!1, 3" ! $"!1, 3""s %&!1, 3" . The non-abelian Weyl-Heisenberg group represents ...

The aim of this paper is to show that any continuous ∗-homomorphism of L(C)(with twisted convolution as multiplication) into B(L(R)) is essentially a Weyl transform. From this we deduce a similar characterisation for the group Fourier transform on the Heisenberg group, in terms of convolution.