Results on Colombeau product of distributions

Blagovest Damyanov

Commentationes Mathematicae Universitatis Carolinae (1997)

  • Volume: 38, Issue: 4, page 627-634
  • ISSN: 0010-2628

Abstract

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The differential -algebra 𝒢 ( m ) of generalized functions of J.-F. Colombeau contains the space 𝒟 ' ( m ) of Schwartz distributions as a -vector subspace and has a notion of ‘association’ that is a faithful generalization of the weak equality in 𝒟 ' ( m ) . This is particularly useful for evaluation of certain products of distributions, as they are embedded in 𝒢 ( m ) , in terms of distributions again. In this paper we propose some results of that kind for the products of the widely used distributions x ± a and δ ( p ) ( x ) , with x in m , that have coinciding singular supports. These results, when restricted to dimension one, are also easily transformed into the setting of regularized model products in the classical distribution theory.

How to cite

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Damyanov, Blagovest. "Results on Colombeau product of distributions." Commentationes Mathematicae Universitatis Carolinae 38.4 (1997): 627-634. <http://eudml.org/doc/248052>.

@article{Damyanov1997,
abstract = {The differential $\mathbb \{C\}$-algebra $\mathcal \{G\}(\mathbb \{R\}^m)$ of generalized functions of J.-F. Colombeau contains the space $\mathcal \{D\}^\{\prime \}(\mathbb \{R\}^m)$ of Schwartz distributions as a $\mathbb \{C\}$-vector subspace and has a notion of ‘association’ that is a faithful generalization of the weak equality in $\mathcal \{D\}^\{\prime \}(\mathbb \{R\}^m)$. This is particularly useful for evaluation of certain products of distributions, as they are embedded in $\mathcal \{G\}(\mathbb \{R\}^m)$, in terms of distributions again. In this paper we propose some results of that kind for the products of the widely used distributions $x_\{\pm \}^a$ and $\delta ^\{(p)\}(x)$, with $x$ in $\mathbb \{R\}^m$, that have coinciding singular supports. These results, when restricted to dimension one, are also easily transformed into the setting of regularized model products in the classical distribution theory.},
author = {Damyanov, Blagovest},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {multiplication of Schwartz distributions; Colombeau generalized functions; multiplication of Schwartz distributions; Colombeau generalized functions},
language = {eng},
number = {4},
pages = {627-634},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Results on Colombeau product of distributions},
url = {http://eudml.org/doc/248052},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Damyanov, Blagovest
TI - Results on Colombeau product of distributions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 4
SP - 627
EP - 634
AB - The differential $\mathbb {C}$-algebra $\mathcal {G}(\mathbb {R}^m)$ of generalized functions of J.-F. Colombeau contains the space $\mathcal {D}^{\prime }(\mathbb {R}^m)$ of Schwartz distributions as a $\mathbb {C}$-vector subspace and has a notion of ‘association’ that is a faithful generalization of the weak equality in $\mathcal {D}^{\prime }(\mathbb {R}^m)$. This is particularly useful for evaluation of certain products of distributions, as they are embedded in $\mathcal {G}(\mathbb {R}^m)$, in terms of distributions again. In this paper we propose some results of that kind for the products of the widely used distributions $x_{\pm }^a$ and $\delta ^{(p)}(x)$, with $x$ in $\mathbb {R}^m$, that have coinciding singular supports. These results, when restricted to dimension one, are also easily transformed into the setting of regularized model products in the classical distribution theory.
LA - eng
KW - multiplication of Schwartz distributions; Colombeau generalized functions; multiplication of Schwartz distributions; Colombeau generalized functions
UR - http://eudml.org/doc/248052
ER -

References

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  1. Colombeau J.-F., New Generalized Functions and Multiplication of Distributions, North Holland Math. Studies 84, Amsterdam, 1984. Zbl0761.46021MR0738781
  2. Fisher B., The product of distributions, Quart. J. Oxford 22 (1971), 291-298. (1971) Zbl0213.13104MR0287308
  3. Fisher B., The divergent distribution product x + λ x - μ , Sem. Mat. Barcelona 27 (1976), 3-10. (1976) MR0425606
  4. Friedlander F.G., Introduction to the Theory of Distributions, Cambridge Univ. Press, Cambridge, 1982. Zbl0499.46020MR0779092
  5. Jelínek J., Characterization of the Colombeau product of distributions, Comment. Math. Univ. Carolinae 27 (1986), 377-394. (1986) MR0857556
  6. Korn G.A., Korn T.M., Mathematical Handbook, McGraw-Hill Book Company, New York, 1968. Zbl0535.00032MR0220560
  7. Oberguggenberger M., Multiplication of Distributions and Applications to Partial Differential Equations, Longman, Essex, 1992. Zbl0818.46036MR1187755

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