We construct a diieomorphism invariant (Colombeau-type) diierential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing diierential calculus in innnite dimensional (convenient) vector spaces, previous attempts in this direction are uniied and completed. Several classiication results are achieved and applications to nonlinear diierential equations inv...

We introduce an intrinsic notion of Hölder-Zygmund regularity for Colombeau generalized functions. In case of embedded distributions belonging to some Zygmund-Hölder space this is shown to be consistent. The definition is motivated by the well-known use of Littlewood-Paley decomposition in characterizing Hölder-Zygmund regularity for distributions. It is based on a simple interplay of different...

The main purpose of this project is to construct a particular differential algebra ρE(Ω) of generalized functions which, among other things, satisfies: (a) ρE(Ω) is an algebra of Colombeau type, i.e. ρE(Ω) contains a copy of the space D′(Ω) of Schwartz distributions (Schwartz generalized functions on an open set Ω ⊆ Rd). (b) The set of the scalars ρC of ρE(Ω) (ρC consists of the functions in ρE...

Results on products of Schwartz distributions are obtained when they have coinciding point singularities and only sums of the products exist in the distribution space. These results follow the pattern of a well-know distributional product published by Jan Mikusiński in 1966, and are named Mikusiński type products. The formulas are derived as the distributions are embedded in Colombeau algebra o...

Results on products of the distributions xa± and δ (x) with coinciding singularities are derived when the products are balanced so that they exist in the distribution space. These results follow the idea of a known result on distributional products published by Jan Mikusiński in 1966. They are obtained in Colombeau algebra G(R)of generalized functions that contains the distribution space and th...

We study Banach C̃-algebras, i.e., complete ultra-pseudo-normed algebras over the ring C̃ of Colombeau generalized complex numbers. We develop a spectral theory in such algebras. We show by explicit examples that important parts of classical Banach algebra theory do not hold for general Banach C̃algebras and indicate a particular class of Banach C̃-algebras that overcomes these limitations to a lar...

The geodesic as well as the geodesic deviation equation for impulsive gravi-tational waves involve highly singular products of distributions (, 2 , 2). A solution concept for these equations based on embedding the distributional metric into the Colombeau algebra of generalized functions is presented. Using a universal regularization procedure we prove existence and uniqueness results and calcul...

Results on singular products of the distributions x ± and x −p for natural p are derived when the products are balanced so that their sum exists in the distribution space. These results follow the pattern of a known distributional product published by Jan Mikusiński in 1966. The results are obtained in Colombeau algebra of generalized functions, which is most relevant algebraic construction for...