Tokyo Journal of Mathematics

Generalized Hyperfunctions and Algebra of Megafunctions


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Sheaves of spaces of generalized hyperfunctions $\mathcal{B}\mathcal{G}$ and algebras of megafunctions $\mathcal{M}\mathcal{G}$ are introduced. The first one is a flabby sheaf. Moreover, there exist injective sheaf homomorphisms $\mathcal{G} \rightarrow \mathcal{B}\mathcal{G}$ and $\mathcal{B}\mathcal{G} \rightarrow \mathcal{M}\mathcal{G},$ where $\mathcal{G}$ is the algebra of Colombeau generalized functions.

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Tokyo J. Math., Volume 28, Number 1 (2005), 1-12.

First available in Project Euclid: 5 June 2009

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PILIPOVI\'C, Stevan. Generalized Hyperfunctions and Algebra of Megafunctions. Tokyo J. Math. 28 (2005), no. 1, 1--12. doi:10.3836/tjm/1244208275.

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  • J. Aragona, On existence theorems for the $\bar\partial$ operator on generalized differential forms, Proc. London Math. Soc. 53 (1986), 474–488.
  • J. Aragona, Théorèmes d'existence pour l'opérateur $\bar\partial$ sur les formes différentielles généralisées, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), 239–242.
  • H. A. Biagioni, A Nonlinear Theory of Generalized Functions, Springer (1990).
  • J. F. Colombeau, New Generalized Functions and Multiplications of Distributions, North Holland (1982).
  • J. F. Colombeau, Elementary Introduction in New Generalized Functions, North Holland (1985).
  • J. F. Colombeau and J. E. Galé, Holomorphic generalized functions, J. Math. Anal. Appl. 103 (1984), 117–133.
  • J. F. Colombeau and J. E. Galé, Analytic continuation of generalized functions, Acta Math. Hung. 52 (1988), 57–60.
  • N. Djapić, M. Kunzinger and S. Pilipović, Symmetry group analysis of weak solutions, Proc. London Math. Soc. 84 (2002), 686–710.
  • M. Grosser, M. Kunzinger, M. Oberguggenberger and R. Steinbauer, Geometric Generalized Functions with Applications to General Relativity, Kluwer (2001).
  • M. Grosser, M. Kunzinger, R. Steinbauer and J. A. Vickers, A global theory of algebras of generalized functions, Adv. Math., 166 (2002), 50–72.
  • R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall (1965).
  • A. Kaneko, Introduction to hyperfunctions, Kluwer (1982).
  • A. Khelif and D. Scarpalezos, Zeros of generalized holomorphic functions, preprint.
  • M. Morimoto, An Introduction to Sato's Hyperfunctions, Transl. Math. Monographs, V129, Providence, (1993).
  • M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Pitman. Res. Not. Math. 259 (1992), Longman Sci. Techn.
  • M. Oberguggenberger, S. Pilipović and D. Scarpalezos, Local and microlocal properties of Colombeau generalized functions, Math. Nachr, 256 (2003), 88–99.
  • S. Pilipović and D. Scarpalezos, Colombeau Generalized Ultradistributions, Math. Proc. Camb. Phil. Soc. 53 (2000), 305–324.
  • M. Oberguggenberger, S. Pilipović and V. Valmorin, Global reprensetatives of Colombeau holomorphic generalized functions, Preprint.
  • M. Sato, T. Kawai and M. Kashiwara, Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), Lecture Notes in Math., 287 (1973), Springer, 265–529.
  • V. Valmorin, Generalized hyperfunctions on the circle, J. Math. Anal. Appl. 261 (2001), 1–16.