Generalized function algebras as sequence space algebras

Delcroix, Antoine; Hasler, Maximilian; Pilipovic, Stevan; Valmorin, Vincent

doi:10.1090/S0002-9939-04-07306-X

Generalized function algebras as sequence space algebras
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by Antoine Delcroix, Maximilian F. Hasler, Stevan Pilipovic and Vincent Valmorin PDF

Proc. Amer. Math. Soc. 132 (2004), 2031-2038 Request permission

Abstract:

A topological description of various generalized function algebras over corresponding basic locally convex algebras is given. The framework consists of algebras of sequences with appropriate ultra(pseudo)metrics defined by sequences of exponential weights. Such an algebra with embedded Dirac’s delta distribution induces the discrete topology on the basic space.

References

Hebe A. Biagioni, A nonlinear theory of generalized functions, 2nd ed., Lecture Notes in Mathematics, vol. 1421, Springer-Verlag, Berlin, 1990. MR 1049623, DOI 10.1007/BFb0089552
H. A. Biagioni and J.-F. Colombeau, New generalized functions and $C^\infty$ functions with values in generalized complex numbers, J. London Math. Soc. (2) 33 (1986), no. 1, 169–179. MR 829397, DOI 10.1112/jlms/s2-33.1.169
Jean-François Colombeau, New generalized functions and multiplication of distributions, North-Holland Mathematics Studies, vol. 84, North-Holland Publishing Co., Amsterdam, 1984. Notas de Matemática [Mathematical Notes], 90. MR 738781
A. Delcroix, M. F. Hasler, S. Pilipović, and V. Valmorin, Embeddings of ultradistributions and periodic hyperfunctions in Colombeau type algebras through sequence spaces, Université Antilles–Guyane (2001) arXiv.org/abs/math.FA/0210240.
A. Delcroix, M. F. Hasler, S. Pilipović, and V. Valmorin, Algebras of generalized functions through sequence spaces: Functoriality and associations, Université Antilles–Guyane (2002) arXiv.org/abs/math.FA/0210249.
Antoine Delcroix and Dimitris Scarpalezos, Topology on asymptotic algebras of generalized functions and applications, Monatsh. Math. 129 (2000), no. 1, 1–14. MR 1741037, DOI 10.1007/s006050050001
Yu. V. Egorov, A contribution to the theory of new generalized functions, Russian Math. Surveys 45:5 (1990) 1–49; translated from Uspehi Mat. Nauk 45:5 (1990) 3–40.
Michael Grosser, Eva Farkas, Michael Kunzinger, and Roland Steinbauer, On the foundations of nonlinear generalized functions I and II, Mem. Amer. Math. Soc. 153 (2001), no. 729, xii+93. MR 1848157, DOI 10.1090/memo/0729
Michael Grosser, Michael Kunzinger, Michael Oberguggenberger, and Roland Steinbauer, Geometric theory of generalized functions with applications to general relativity, Mathematics and its Applications, vol. 537, Kluwer Academic Publishers, Dordrecht, 2001. MR 1883263, DOI 10.1007/978-94-015-9845-3
A. H. Lightstone and Abraham Robinson, Nonarchimedean fields and asymptotic expansions, North-Holland Mathematical Library, Vol. 13, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0414354
Jean-Andre Marti, $({\scr C},{\scr E},{\scr P})$-sheaf structures and applications, Nonlinear theory of generalized functions (Vienna, 1997) Chapman & Hall/CRC Res. Notes Math., vol. 401, Chapman & Hall/CRC, Boca Raton, FL, 1999, pp. 175–186. MR 1699819
M. Oberguggenberger, Multiplication of distributions and applications to partial differential equations, Pitman Research Notes in Mathematics Series, vol. 259, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR 1187755
M. Oberguggenberger, The Carleman system with positive measures as initial data—generalized solutions, Transport Theory Statist. Phys. 20 (1991), no. 2-3, 177–197. MR 1114095, DOI 10.1080/00411459108203901
Michael Oberguggenberger and Todor Todorov, An embedding of Schwartz distributions in the algebra of asymptotic functions, Internat. J. Math. Math. Sci. 21 (1998), no. 3, 417–428. MR 1620298, DOI 10.1155/S0161171298000593
Abraham Robinson, Function theory on some nonarchimedean fields, Amer. Math. Monthly 80 (1973), no. 6, 87–109. Papers in the foundations of mathematics. MR 330126, DOI 10.2307/3038223
Elemer E. Rosinger, Nonlinear partial differential equations, North-Holland Mathematics Studies, vol. 164, North-Holland Publishing Co., Amsterdam, 1990. An algebraic view of generalized solutions. MR 1091547
D. Scarpalézos, Some remarks on functoriality of Colombeau’s construction; topological and microlocal aspects and applications, Integral Transform. Spec. Funct. 6 (1998), no. 1-4, 295–307. Generalized functions—linear and nonlinear problems (Novi Sad, 1996). MR 1640520, DOI 10.1080/10652469808819174
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Todor Todorov, An existence result for linear partial differential equations with $C^\infty$-coefficients in an algebra of generalized functions, Trans. Amer. Math. Soc. 348 (1996), no. 2, 673–689. MR 1316863, DOI 10.1090/S0002-9947-96-01450-X