The Radon transform of Boehmians

Mikusiński, Piotr; Zayed, Ahmed

doi:10.1090/S0002-9939-1993-1145949-6

The Radon transform of Boehmians
HTML articles powered by AMS MathViewer

by Piotr Mikusiński and Ahmed Zayed

Proc. Amer. Math. Soc. 118 (1993), 561-570

DOI: https://doi.org/10.1090/S0002-9939-1993-1145949-6

PDF | Request permission

Abstract:

The Radon transform, which enables one to reconstruct a function of $N$ variables from the knowledge of its integrals over all hyperplanes of dimension $N - 1$, has been extended to Schwartz distributions by several people including Gelfand, Graev, and Vilenkin, who extended it to tempered distributions. In this paper we extend the Radon transform to a space of Boehmians. Boehmians are defined as sequences of convolution quotients and include Schwartz distributions and regular Mikusiński operators. Our extension of the Radon transform includes generalized functions of infinite order with compact support. The technique used in this paper is based on algebraic properties of the Radon transform and its convolution structure rather than on their analytic properties. Our results do not contain nor are contained in those obtained by Gelfand et al.

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
Stanley R. Deans, The Radon transform and some of its applications, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. MR 709591
I. M. Gel’fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Academic Press, New York-London, 1964. Translated by Eugene Saletan. MR 0166596
I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831
Sigurdur Helgason, The Radon transform, 2nd ed., Progress in Mathematics, vol. 5, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1723736, DOI 10.1007/978-1-4757-1463-0
Donald Ludwig, The Radon transform on euclidean space, Comm. Pure Appl. Math. 19 (1966), 49–81. MR 190652, DOI 10.1002/cpa.3160190207
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
Jan Mikusiński and Piotr Mikusiński, Quotients de suites et leurs applications dans l’analyse fonctionnelle, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 9, 463–464 (French, with English summary). MR 646866
Piotr Mikusiński, Convergence of Boehmians, Japan. J. Math. (N.S.) 9 (1983), no. 1, 159–179. MR 722539, DOI 10.4099/math1924.9.159
P. Mikusiński, Boehmians and generalized functions, Acta Math. Hungar. 51 (1988), no. 3-4, 271–281. MR 956979, DOI 10.1007/BF01903334

Über die Bestimmung von Funktionen durch ihre Integralwärte längs gewisser Manningfaltigkeiten

69

Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. MR 0344043

Théorie des distributions

A. H. Zemanian, Generalized integral transformations, 2nd ed., Dover Publications, Inc., New York, 1987. MR 896486