We obtain exact solution of the Dirac equation with the Coulomb potential as an infinite series of square integrable functions. This solution is for all energies, the discrete as well as the continuous. The spinor basis elements are written in terms of the confluent hypergeometric functions and chosen such that the matrix representation of the DiracCoulomb operator is tridiagonal. The wave equa...

This note is concerned with the generalization of the continuous shearlet transform to higher dimensions. Similar to the twodimensional case, our approach is based on translations, anisotropic dilations and specific shear matrices. We show that the associated integral transform again originates from a square-integrable representation of a specific group, the full n-variate shearlet group. Moreo...

This paper is concerned with the uncertainty principle in the context of the affine-Weyl-Heisenberg group in one and two dimensions. As the representation of this group fails to be square integrable, we explore various admissible sections of this group, and calculate the resulting uncertainty principles as well as its minimizers with respect to these sections. Previous studies have shown that t...

A stochastic calculus similar to Malliavin’s calculus is worked out for Brownian excursions. The analogue of the Malliavin derivative in this calculus is not a differential operator, but its adjoint is (like the Skorohod integral) an extension of the Itô integral. As an application, we obtain an expression for the integrand in the stochastic integral representation of square integrable Wiener f...

We consider a Poisson process η on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of η. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener-Itô chaos expansion. We apply these results to extend well-known variance inequal...

This paper is concerned with the uncertainty principle in the context of the affine-Weyl-Heisenberg group in one and two dimensions. As the representation of this group fails to be square integrable, we explore various admissible sections of this group, and calculate the resulting uncertainty principles as well as its minimizers with respect to these sections. Previous studies have shown that t...

Adopting a purely group-theoretical point of view, we consider the star product of functions which is associated, in a natural way, with a square integrable (in general, projective) representation of a locally compact group. Next, we show that for this (implicitly defined) star product explicit formulae can be provided. Two significant examples are studied in detail: the group of translations o...

Continuous wavelet analysis and Gabor analysis have proven to be very useful tools for the analysis of signals in which local frequencies can be extracted. Both techniques can be described in the same footing using the theory of square-integrable group representations or derived theories. It is shown here how the same kind of techniques can be developed to construct phase-space representation t...

The minimal representation π of the indefinite orthogonal group O(m+ 1, 2) is realized on the Hilbert space of square integrable functions on R with respect to the measure |x|dx1 · · ·dxm. This article gives an explicit integral formula for the holomorphic extension of π to a holomorphic semigroup of O(m + 3,C) by means of the Bessel function. Taking its ‘boundary value’, we also find the integ...

This note is concerned with the generalization of the continuous shearlet transform to higher dimensions. Similar to the two-dimensional case, our approach is based on translations, anisotropic dilations and specific shear matrices. We show that the associated integral transform again originates from a square-integrable representation of a specific group, the full n-variate shearlet group. More...