Colombeau products of distributions
In this paper, some products of distributions are derived. The results are obtained in Colombeau algebra of generalized functions, which is the most relevant algebraic construction for dealing with Schwartz distributions. Colombeau algebra $${\mathcal{G}}({\mathbf{R}})$$G(R) contains the space $${\mathcal{D}'}({\mathbf{R}})$$D′(R) of Schwartz distributions as a subspace, and has a notion of 'association’ that allows us to evaluate the results in terms of distributions.
Results on the Colombeau Products of the Distribution x+−r−1/2 with the Distributions x−−k−1/2 and x−k−1/2
Results on the products of the distribution x+−r−1/2 with the distributions x−−k−1/2 and x−k−1/2 are obtained in the differential algebra G(ℝ) of Colombeau generalized functions, which contains the space D′(ℝ) of Schwartz distributions as a subspace; in this algebra the notion of association is defined, which is a faithful generalization of weak equality in G(ℝ). This enables treating the results in terms of distributions again.
Modelling and Products of Singularities in Colombeau's Algebra
Abstract Modelling of singularities given by discontinuous functions or Schwartz distributions by means of generalized functions of Colombeau has proved useful in many problems posed by physical phenomena. We introduce in a systematic way generalized functions that model singularities given by distributions with singular point support. Moreover, we evaluate various products of singularity-modelling generalized functions whenever the result admits an associated distribution.