Related papers: Continuous Wavelet Frames on the Sphere: The Group…

The wavefunction in quantum field theory is an invaluable tool for tackling a variety of problems, including probing the interior of Minkowski spacetime and modelling boundary observables in de Sitter spacetime. Here we study the analytic…

We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D_4=SO(4,2)/(SO(4)\times SO(2)) of the conformal group SO(4,2) (locally isomorphic to SU(2,2)) in 1+3 dimensions. The manifold D_4 can be mapped…

Given a real, expansive dilation matrix we prove that any bandlimited function $\psi \in L^2(\mathbb{R}^n)$, for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame for certain translation…

In the first part of the paper, we study conformal groups that act properly discontinuously and cocompactly on simply connected, non-flat homogeneous plane waves. We show that proper cocompact similarity actions that are not isometric can…

The problem of quantizing a particle on a 2-sphere has been treated by numerous approaches, including Isham's global method based on unitary representations of a symplectic symmetry group that acts transitively on the phase space. Here we…

Let G be a stratified Lie group and L be the sub-Laplacian on G. Let 0 \neq f\in S(R^+). We show that Lf(L)\delta, the distribution kernel of the operator Lf(L), is an admissible function on G. We also show that, if \xi f(\xi) satisfies…

We reelaborate on the basic properties of lossless multilayers. We show that the transfer matrices for these multilayers have essentially the same algebraic properties as the Lorentz group SO(2,1) in a (2+1)-dimensional spacetime, as well…

The usual Helmholtz decomposition gives a decomposition of any vector valued function into a sum of gradient of a scalar function and rotation of a vector valued function under some mild condition. In this paper we show that the vector…

Group based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group based moving…

We investigate how a uniformly rotating frame is defined as the rest frame of an observer rotating with constant angular velocity $\Omega$ around the $z$ axis of an inertial frame. Assuming that this frame is a Lorentz one, we second…

The group theoretical approach to the relativistic wave equations on the real reducible spaces for spin~0, 1/2 and~1 massless particles is considered. The invariant wave equations which determine the appropriate irreducible representations…

The application of the continuous wavelet transform to study of a wide class of physical processes with oscillatory dynamics is restricted by large central frequencies due to the admissibility condition. We propose an alternative…

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 $g=\chi_{E}$ generates a Weyl-Heisenberg frame for $L^2(\mathbb{R})$ under modulates by…

In the present paper, a wavelet family over the $n$-dimensional sphere is constructed such that for each scale the wavelet is a polynomial and the inverse wavelet transform of a continuous function converges in the supremum norm.

We present a new proof of a theorem of Mallat which describes a construction of wavelets starting from a quadrature mirror filter. Our main innovation is to show how the scaling function associated to the filter can be used to identify a…

For plane frameworks with reflection or rotational symmetries, where the group action is not necessarily free on the vertex set, we introduce a phase-symmetric orbit rigidity matrix for each irreducible representation of the group. We then…

There are many two-by-two matrices in layer optics. It is shown that they can be formulated in terms of a three-parameter group whose algebraic property is the same as the group of Lorentz transformations in a space with two space-like and…

The quantum mechanical harmonic oscillator Hamiltonian generates a one-parameter unitary group W(\theta) in L^2(R) which rotates the time-frequency plane. In particular, W(\pi/2) is the Fourier transform. When W(\theta) is applied to any…

In recent years directional multiscale transformations like the curvelet- or shearlet transformation have gained considerable attention. The reason for this is that these transforms are - unlike more traditional transforms like wavelets -…

We present a framework that reformulates gravitational lensing as an optical phenomenon governed by an effective refractive index, enabling exploration of modified gravity theories using undergraduate-level mathematics and optics. After…