On the extension of Schwartz distributions to the space of discontinuous test functions of several variables. (English) Zbl 1183.46040
The space of distributions is considered if discontinuous test functions of several variables are used. These test functions allow the definition of the operation of integration of a distribution and the multiplication of a distribution with discontinuous functions. Examples from game theory are given.
Reviewer: Thomas Sonar (Braunschweig)
MSC:
| 46F10 | Operations with distributions and generalized functions |
| 34A36 | Discontinuous ordinary differential equations |
References:
| [1] | J.F. Colombeau, Elementary introduction to new generalized functions , North-Holland Publishing Co., Amsterdam, 1985. · Zbl 0584.46024 |
| [2] | ——–, Multiplication of distributions. A tool in mathematics, numerical engineering and physics , Lecture Notes Math. 1532 , Springer-Verlag, New York, 1992. · Zbl 0815.35002 |
| [3] | T.M.K. Davison, A generalization of regulated functions , Amer. Math. Monthly 86 (1979), 202-204. JSTOR: · Zbl 0426.54010 · doi:10.2307/2321523 |
| [4] | V.K. Domansky, On one application of the theory of distributions to the theory of zero-sum games , Dokl. Akad. Nauk SSSR 199 (1971), 515-518 (in Russian). |
| [5] | A.F. Filippov, Differential equations with discontinuous right-hand sides , Kluwer Acad. Publishers, Boston, 1988. · Zbl 0664.34001 |
| [6] | D. Jones, The theory of generalised functions , Cambridge University Press, Cambridge, 1982. · Zbl 0477.46035 |
| [7] | L.V. Kantorovich and G.P. Akilov, Functional analysis , Pergamon Press, Oxford, 1982. · Zbl 0484.46003 |
| [8] | P. Kurasov, Distributions theory with discontinuous test functions and differential operators with generalized coefficients , J. Math. Anal. Appl. 201 (1996), 297-323. · Zbl 0878.46030 · doi:10.1006/jmaa.1996.0256 |
| [9] | P. Kurasov and J. Boman, Finite rank singular perturbations and distributions with discontinuous test functions , Proc. Amer. Math. Soc. 126 (1998), 1673-1683. JSTOR: · Zbl 0894.34079 · doi:10.1090/S0002-9939-98-04291-9 |
| [10] | B.M. Miller, The method of discontinuous time substitution in problems of the optimal control of impulse and discrete-continuous systems , Automat. Remote Control 54 (1993), 1727-1750. |
| [11] | K.P. Rath, Existence and upper hemicontinuity of equilibrium distributions of anonymous games with discontinuous payoffs , J. Math. Econ. 26 (1996), 305-324. · Zbl 0871.90131 · doi:10.1016/0304-4068(95)00748-2 |
| [12] | C.O.R. Sarrico, Some distributional products with relativistic invariance , Portugal. Math. 51 (1994), 283-290. · Zbl 0806.46042 |
| [13] | ——–, The linear Cauchy problem for a class of differential equations with distributional coefficients , Portugal. Math. 52 (1995), 379-390. · Zbl 0841.34012 |
| [14] | ——–, Distributional products and global solutions for nonconservative inviscid Burgers equation , J. Math. Anal. Appl. 281 (2003), 641-656. · Zbl 1026.35078 · doi:10.1016/S0022-247X(03)00187-2 |
| [15] | A.N. Sesekin and S.T. Zavalishchin, Dynamic impulse systems : Theory and applications , Kluwer Academic Publishers, Boston, 1997. · Zbl 0880.46031 |
| [16] | G.E. Shilov, Mathematical analysis , Second special course , Moscow State University, Moscow, 1984. · Zbl 0097.03603 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
