Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
multiplication of Schwartz distributions; Colombeau generalized functions
multiplication of Schwartz distributions; Colombeau generalized functions
Summary:
The differential $\Bbb C$-algebra $\Cal G(\Bbb R^m)$ of generalized functions of J.-F. Colombeau contains the space $\Cal D'(\Bbb R^m)$ of Schwartz distributions as a $\Bbb C$-vector subspace and has a notion of `association' that is a faithful generalization of the weak equality in $\Cal D'(\Bbb R^m)$. This is particularly useful for evaluation of certain products of distributions, as they are embedded in $\Cal G(\Bbb R^m)$, 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 $\Bbb R^m$, 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.
References:
[1] Colombeau J.-F.: New Generalized Functions and Multiplication of Distributions. North Holland Math. Studies 84, Amsterdam, 1984. MR 0738781 | Zbl 0761.46021
[2] Fisher B.: The product of distributions. Quart. J. Oxford 22 (1971), 291-298. MR 0287308 | Zbl 0213.13104
[3] Fisher B.: The divergent distribution product $x_+^\lambda x_-^\mu$. Sem. Mat. Barcelona 27 (1976), 3-10. MR 0425606
[4] Friedlander F.G.: Introduction to the Theory of Distributions. Cambridge Univ. Press, Cambridge, 1982. MR 0779092 | Zbl 0499.46020
[5] Jelínek J.: Characterization of the Colombeau product of distributions. Comment. Math. Univ. Carolinae 27 (1986), 377-394. MR 0857556
[6] Korn G.A., Korn T.M.: Mathematical Handbook. McGraw-Hill Book Company, New York, 1968. MR 0220560 | Zbl 0535.00032
[7] Oberguggenberger M.: Multiplication of Distributions and Applications to Partial Differential Equations. Longman, Essex, 1992. MR 1187755 | Zbl 0818.46036
Search
Partner of
