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Schwartz kernel theorem in algebras of generalized functions

Vincent Valmorin — 2010

Banach Center Publications

A new approach to the generalization of Schwartz’s kernel theorem to Colombeau algebras of generalized functions is given. It is based on linear maps from algebras of classical functions to algebras of generalized ones. In particular, this approach enables one to give a meaning to certain hypotheses in preceding similar work on this theorem. Results based on the properties of G ∞ -generalized functions class are given. A straightforward relationship between the classical and the generalized versions...

Sequence spaces with exponent weights. Realizations of Colombeau type algebras

Antoine Delcroix;  Maximilian F. Hasler;  Stevan Pilipović;  Vincent Valmorin — 2007

We give a description of various algebras of generalized functions based on the introduction of pseudo-ultranorms on spaces of sequences in given locally convex function algebras. We study sheaf properties of these algebras, needed for microlocal analysis, and also consider regularity theory, functoriality and different concepts of association and weak equality in a unified setting. Using this approach, we also give new results on embeddings of ultradistribution and hyperfunction spaces into corresponding...

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