Equality of Ultradifferentiable Classes by Means of Indices of Mixed O-regular Variation

We characterize the equality between ultradifferentiable function classes defined in terms of abstractly given weight matrices and in terms of the corresponding matrix of associated weight functions by using new growth indices. These indices, defined by means of weight sequences and (associated) weight functions, are extending the notion of O-regular variation to a mixed setting. Hence we are extending the known comparison results concerning classes defined in terms of a single weight sequence and of a single weight function and give also these statements an interpretation expressed in O-regular variation.

Unified Treatment of the Krätzel Transformation for Generalized Functions

We discuss a generalization of the Krätzel transforms on certain spaces of ultradistributions. We have proved that the Krätzel transform of an ultradifferentiable function is an ultradifferentiable function and satisfies its Parseval's inequality. We also provide a complete reading of the transform constructing two desired spaces of Boehmians. Some other properties of convergence and continuity conditions and its inverse are also discussed in some detail.

Colombeau generalized ultradistributions

An algebra of Colombeau generalized ultradistributions in which the corresponding space of ultradistributions is embedded via regularizations and in which the multiplication of ultradifferentiable functions of an appropriate class is the ordinary multiplication is constructed. This enables the development of the local and micro-local analysis in the framework of a new scale of Colombeau algebras. Sufficient conditions for a generalized ultradistribution being an ultradifferentiable function are given.