There are Poincare group representations on complex Hilbert spaces, like the Dirac spinor field, or real Hilbert spaces, like the electromagnetic field tensor. The Majorana spinor is an element of a 4 dimensional real vector space. The Majorana spinor field is a space-time dependent Majorana spinor, solution of the free Dirac equation. The Majorana-Fourier and Majorana-Hankel transforms of Majo...

We identify multiresolution subspaces giving rise via Hankel transforms to Bessel functions. They emerge as orthogonal systems derived from geometric Hilbert-space considerations, the same way the wavelet functions from a multiresolution scaling wavelet construction arise from a scale of Hilbert spaces. We study the theory of representations of the C ∗-algebra Oν+1 arising from this multiresolu...

We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Székely, Rizzo and Bakirov, to general metric spaces. We show that for testing independence, it is necessary and sufficient that the metric space be of strong negative type. In particular, we show that this holds for separable Hilbert spaces, which answers a question of Kosorok. Instead of t...

It is well-known that the Gaussian functions and, more generally, their modulations-translations (the Gabor functions) have the unique property of being optimally localized in space and frequency in the sense of Heisenberg’s uncertainty principle. In this thesis, we address the construction of complex wavelets modeled on the Gabor function, and smoothing kernels based on the Gaussian. We procee...

For unbounded operators t, s between Hilbert C*-modules which admit the polar decompositions V|t|, W|s|, respectively, we obtain an explicit upper bound estimate for the gap between t and s in terms of the norm of the bounded operators V − W , C|t| − C|s| and C|t∗| − C|s∗|, where C|t| and C|s| are the Cayley transforms of |t| and |s|. The result are used to drive a criterion for continuity of t...

The numerical evaluation of a class of Hilbert transforms using an eigenfunction approach is considered. Judicious selection of convergence accelerators allows for the efficient evaluation of the resulting series. The approach is particularly accurate for functions having a Gaussian-like asymptotic behavior. For more slowly decreasing functions, the accuracy of the evaluation decreases.

The analysis of electrical signals is a pressing requirement for the optimal design of power distribution. In this context, this paper illustrates how to use a variety of numerical tools, such as the Fourier, wavelet, and Hilbert-Huang transforms, to obtain information relating to the active and reactive power absorbed by different types of users. In particular, the Fourier spectrum gives the m...

The evaluation of the Hilbert transform of a function is a very important problem that appears widely in science and engineering, specially in signal analysis. In this work, the numerical evaluation of a class of Hilbert transforms using an expansion in terms of Hermite functions, eigenfunctions of the Fourier transform, is performed. Judicious selection of convergence accelerators allows for t...

Contents Abstract Sinc approximation methods excel for problems whose solutions may have singularities, or innnite domains, or boundary layers. This article summarizes results obtained to date, on Sinc numerical methods of computation. Sinc methods provide procedures for function approximation over bounded or unbounded regions, encompassing interpolation, approximation of derivatives, approxima...

The ability of the Rigged Hilbert Space formalism to deal with continuous spectrum is demonstrated within the example of the square barrier potential. The non-square integrable solutions of the time-independent Schrödinger equation are used to define Dirac kets, which are (generalized) eigenvectors of the Hamiltonian. These Dirac kets are antilinear functionals over the space of physical wave f...