Neocontinuous Mikusiński operators
HTML articles powered by AMS MathViewer
- by Carl C. Hughes and Raimond A. Struble PDF
- Trans. Amer. Math. Soc. 185 (1973), 383-400 Request permission
Abstract:
A class of Mikusiński-type operators in several variables, called neocontinuous operators, is studied. These particular operators are closely affiliated with Schwartz distributions on ${R^k}$ and share certain continuity properties with them. This affiliation is first of all revealed through a common algebraic view of operators and distributions as homomorphic mappings and a new representation theory, and is then characterized in terms of continuity properties of the mappings. The traditional procedures of the operational calculus apply to the class of neocontinuous operators. Moreover, the somewhat vague association of operational and distributional solutions of partial differential equations is replaced by the decisive representation concept, thus illustrating the appropriateness of the study of neocontinuous operators.References
- Thomas K. Boehme, The support of Mikusiński operators, Trans. Amer. Math. Soc. 176 (1973), 319–334. MR 313727, DOI 10.1090/S0002-9947-1973-0313727-5
- M. Gutterman, An operational method in partial differential equations, SIAM J. Appl. Math. 17 (1969), 468–493. MR 249969, DOI 10.1137/0117046 J. Horváth, Topological vector spaces and distributions. Vol. 1, Addison-Wesley, Reading, Mass., 1966. MR 34 #4863.
- J. L. Lions, Supports dans la transformation de Laplace, J. Analyse Math. 2 (1953), 369–380 (French). MR 58013, DOI 10.1007/BF02825641
- Jan Mikusiński, Operational calculus, International Series of Monographs on Pure and Applied Mathematics, Vol. 8, Pergamon Press, New York-London-Paris-Los Angeles; Państwowe Wydawnictwo Naukowe, Warsaw, 1959. MR 0105594
- J. Mikusiński, Convolution of functions of several variables, Studia Math. 20 (1961), 301–312. MR 140895, DOI 10.4064/sm-20-3-301-312
- Laurent Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, IX-X, Hermann, Paris, 1966 (French). Nouvelle édition, entiérement corrigée, refondue et augmentée. MR 0209834
- Raimond A. Struble, On operators and distributions, Canad. Math. Bull. 11 (1968), 61–64. MR 233199, DOI 10.4153/CMB-1968-008-1
- Raimond A. Struble, An algebraic view of distributions and operators, Studia Math. 37 (1970/71), 103–109. MR 303288, DOI 10.4064/sm-37-2-103-109
- Raimond A. Struble, Operator homomorphisms, Math. Z. 130 (1973), 275–285. MR 326381, DOI 10.1007/BF01246624 F. Tréves, Linear partial differential equations with constant coefficients, Math. and its Applications, vol. 6, Gordon and Breach, New York, 1966. MR 37 #557.
- Josef Wloka, Distributionen und Operatoren, Math. Ann. 140 (1960), 227–244 (German). MR 114119, DOI 10.1007/BF01361146
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 185 (1973), 383-400
- MSC: Primary 46FXX; Secondary 44A40
- DOI: https://doi.org/10.1090/S0002-9947-1973-0333719-X
- MathSciNet review: 0333719