On the Hilbert transform in the framework of generalized functions

In this paper, a definition of the Hilbert transform operating on Colombeau's temperated generalized functions is given. Similar results to some theorems that hold in the classical theory, or in certain subspaces of Schwartz distributions, have been obtained in this framework.

Differential calculus and integration of generalized functions over membranes

In this paper we continue the development of the differential calculus started in Aragona et al. (Monatsh. Math. 144: 13-29, 2005). Guided by the so-called sharp topology and the interpretation of Colombeau generalized functions as point functions on generalized point sets, we introduce the notion of membranes and extend the definition of integrals, given in Aragona et al. (Monatsh. Math. 144: 13-29, 2005), to integrals defined on membranes. We use this to prove a generalized version of the Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. A number of results from classical differential and integral calculus, like the inverse and implicit function theorems and Green's theorem, are transferred to the generalized setting. Further, we indicate that solution formulas for transport and wave equations with generalized initial data can be obtained as well.

Levy stable distributions via associated integral transform

We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Levy stable probability distributions g(alpha)(x), 0 <= x < infinity, 0 < alpha < 1. We demonstrate that the knowledge of one such a distribution g a ( x) suffices to obtain exactly g(alpha)p ( x), p = 2, 3, .... Similarly, from known g(alpha)(x) and g(beta)(x), 0 < alpha, beta < 1, we obtain g(alpha beta)( x). The method is based on the construction of the integral operator, called Levy transform, which implements the above operations. For a rational, alpha = l/k with l < k, we reproduce in this manner many of the recently obtained exact results for g(l/k)(x). This approach can be also recast as an application of the Efros theorem for generalized Laplace convolutions. It relies solely on efficient definite integration. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4709443]