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ON THE DETERMINATION OF STAR BODIES FROM THEIR HALF-SECTIONS

Part of: Integral transforms, operational calculus

Published online by Cambridge University Press:  13 March 2017

B. Rubin
B. Rubin*
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. email [email protected]
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Abstract

We obtain explicit inversion formulas for the Radon-like transform that assigns to a function on the unit sphere the integrals of that function over hemispheres lying in lower-dimensional central cross-sections. The results are applied to the determination of star bodies from the volumes of their central half-sections.

MSC classification

Primary: 44A12: Radon transform
Secondary: 44A15: Special transforms (Legendre, Hilbert, etc.)
Type
Research Article
Information
Mathematika , Volume 63 , Issue 2 , 2017 , pp. 462 - 468
Copyright
Copyright © University College London 2017 

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References

Backus, G., Geographical interpretation of measurements of average phase velocities of surface waves over great circular and semi circular paths. Bull. Seism. Soc. Amer. 54 1964, 571–610.CrossRefGoogle Scholar
Gardner, R. J., Geometric Tomography, 2nd edn., Cambridge University Press (New York, 2006).Google Scholar
Gel’fand, I. M., Integral geometry and its relation to the theory of representations. Russian Math. Surveys 15(2) 1960, 143–151.CrossRefGoogle Scholar
Gelfand, I. M., Gindikin, S. G. and Graev, M. I., Selected Topics in Integral Geometry (Translations of Mathematical Monographs 220 ), American Mathematical Society (Providence, RI, 2003).CrossRefGoogle Scholar
Goodey, P. and Weil, W., Average section functions for star-shaped sets. Adv. Appl. Math. 36(1) 2006, 70–84.Google Scholar
Groemer, H., On a spherical integral transformation and sections of star bodies. Monatsh. Math. 126(2) 1998, 117–124.CrossRefGoogle Scholar
Helgason, S., Integral Geometry and Radon Transform, Springer (New York, Dordrecht, Heidelberg, London, 2011).Google Scholar
Rubin, B., Inversion formulas for the spherical Radon transform and the generalized cosine transform. Adv. Appl. Math. 29 2002, 471–497.CrossRefGoogle Scholar
Rubin, B., On the Funk–Radon–Helgason inversion method in integral geometry. Contemp. Math. 599 2013, 175–198.Google Scholar
Rubin, B., Overdetermined transforms in integral geometry. In Complex Analysis and Dynamical Systems VI. Part 1 (Contemporary Mathematics 653 ), American Mathematical Society (Providence, RI, 2015), 291–313.Google Scholar
Rubin, B., Introduction to Radon Transforms: with Elements of Fractional Calculus and Harmonic Analysis, Cambridge University Press (New York, 2015).Google Scholar