15. ÖMG-Kongress
Jahrestagung der Deutschen Mathematikervereinigung

16. bis 22. September 2001 in Wien

Sektion 7 - Funktionalanalysis, Harmonische Analysis
Dienstag, 18. September 2001, 17.00, Hörsaal 28


Sequence spaces with exponent weights, Realisations of Colombeau type algebras

Stevan Pilipovic, University of Novi Sad (Koautoren: A. Delcroix, M. Hasler, V. Valmorin)


Colombeau had constructed his well-known algebras by algebraic methods. No topology had appeared in his construction. Our aim is to give a purely topological description of Colombeau type algebras. We show that such algebras fit very well in the general theory of the well known sequence spaces forming appropriate algebras. All these classes of algebras are simply determined by the (locally convex) space $ E$ and a sequence of weights $ r:{\bf N}\rightarrow
{\bf R}_+$ which serves to construct an ultrametric on the sequence space $ E^{\bf N}$. The sequence $ r={(r_n)}_n$ is assumed to be decreasing to zero. This implies that sequence spaces under consideration ( $ \subset E^{\bf N}$) contain as a subspace $ E\sim {\rm diag} E^{\bf N}$ and that they induce the discrete topology on $ E$. Our analysis implies that if one has a Colombeau type algebra containing the Dirac delta distribution as an embedded Colombeau generalized function, then the topology induced on the basic space must be discrete. This is an analogous result to the Schwartz's ``imposibility result'' concerning the product of distributions. A major part of the talk is devoted to embeddings of ultradistribution and hyperfunction spaces into corresponding classes of sequence spaces.

[1] Colombeau, J. F.: Multiplication of Distributions Lect. Not. Math. 1532, Springer, Berlin, 1992.
[2] Oberguggenberger, M.: Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Res. Not. Math. 259, Longman Sci. Techn., Essex, 1992.
[3] Pilipovic, S.: Colombeau's generalized functions and pseudodifferential operators, University of Tokio, Lecture Notes Series, 1994.
[4] Pilipovic. S., Scarpalezos, D.: Colombeau generalized Ultradistributions, Math. Proc. Camb. Phil Soc., 130(2001), 541-553.

E-Mail: pilipovic@unsim.ns.ac.yu
Homepage: www.im.ns.ac.yu

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