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 [Formula: see text] contains the space [Formula: see text] of Schwartz distributions as a subspace, and has a notion of 'association' that allows us to evalu...

In arXiv:0806.4682 the self-energy and self-angular momentum (i.e., electromagnetic mass and spin) of a classical point-electron were calculated in a Colombeau algebra. In the present paper these quantities are calculated in the better known framework of ‘regularized distributions,’ i.e., the customary setting used in field-theory to manipulate diverging integrals, distributions, and their prod...

the present study aims at indicating the existence and uniqueness result of system in extended colombeaualgebra. the caputo fractional derivative is used for solving the system of odes. in addition, rieszfractional derivative of colombeau generalized algebra is considered. the purpose of introducing rieszfractional derivative is regularizing it in colombeau sense. we also give a solution to a n...

We define a general notion of set of indices which, using concepts from pre-ordered sets theory, permits to unify the presentation of several Colombeau-type algebras of nonlinear generalized functions. In every set of indices it is possible to generalize Landau’s notion of big-O such that its usual properties continue to hold. Using this generalized notion of big-O, these algebras can be formal...

In arXiv:0806.4682 the self-energy and self-angular momentum (i.e., electromagnetic mass and spin) of a classical point-electron were calculated in a Colombeau algebra. In the present paper these quantities are calculated in the better known framework of ‘regularized distributions,’ i.e., the customary setting used in field-theory to manipulate diverging integrals, distributions, and their prod...

This contribution is the first in a series of three: it reports on the construction of (a fine sheaf of) diffeomorphism invariant Colombeau algebras G(Ω) = EM (Ω)/N (Ω) on open sets of Eucildean space ([Gro01]), which completes earlier approaches ([Col94, Jel99]). Part II and III will show, among others, the way to an intrinsic definition of Colombeau algebras on manifolds which, locally, repro...

In a recent paper, we gave a topological description of Colombeau type algebras introducing algebras of sequences with exponential weights. Embeddings of Schwartz spaces into the Colombeau algebra G are well known, but for ultradistribution and periodic hyperfunction type spaces we give new constructions. We show that the multiplication of regular enough functions (smooth, ultradifferentiable o...

This paper gives a comprehensive analysis of algebras of Colombeau-type generalized functions in the range between the diffeomorphism-invariant quotient algebra G = EM/N introduced in part I and Colombeau’s original algebra G . Three main results are established: First, a simple criterion describing membership in N (applicable to all types of Colombeau algebras) is given. Second, two counterexa...

Positivity and positive definiteness in algebra of generalized functions are studied. Basic definitions and notions of Colombeau algebra of generalized functions are given and some special classes of positive definite generalized functions on those algebras are introduced. Their relation to distributions is also investigated. AMS Mathematics Subject Classification (2000):