Analytic multipliers of the Radon transform

Peters, James V.

doi:10.1090/S0002-9939-1978-0509239-6

Analytic multipliers of the Radon transform

Author: James V. Peters
Journal: Proc. Amer. Math. Soc. 72 (1978), 485-491
MSC: Primary 44A15; Secondary 43A85
DOI: https://doi.org/10.1090/S0002-9939-1978-0509239-6
MathSciNet review: 509239
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We define the Radon transform over the real and complex domain, and list some of its simplest properties. For the complex domain ${{\mathbf{C}}^n}$ , a theorem of the Paley-Wiener type is obtained to determine the support of a continuous, integrable function from its Radon transform. The construction requires defining an associated function of the Radon transform for each $z \in {{\mathbf{C}}^n}$ . The support of the function is then obtained via analytic multipliers of the associated functions.

[1] I. M. Gel'fand, M. I. Graev and N. Ya Vilenkin, Integral geometry and representation theory, Academic Press, New York, 1966.
[2] 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
[3] -, Functions and generalized spaces, Academic Press, New York, 1966.
[4] Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. MR 0075429
[5] J. V. Peters, Radon transforms over the real and complex domain, Doctoral Thesis, Stevens Institute of Technology, Hoboken, N. J., 1976.
[6] Georgi E. Shilov, Generalized functions and partial differential equations, Authorized English edition revised by the author. Translated and edited by Bernard D. Seckler, Gordon and Breach, Science Publishers, Inc., New York-London-Paris, 1968. MR 0230129
[7] Kennan T. Smith, Donald C. Solmon, and Sheldon L. Wagner, Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc. 83 (1977), no. 6, 1227–1270. MR 490032, https://doi.org/10.1090/S0002-9904-1977-14406-6
[8] Donald C. Solmon, A note on 𝑘-plane integral transforms, J. Math. Anal. Appl. 71 (1979), no. 2, 351–358. MR 548770, https://doi.org/10.1016/0022-247X(79)90196-3