Real analytic Radon transforms on rank one symmetric spaces

Quinto, Eric Todd

doi:10.1090/S0002-9939-1993-1135080-8

Real analytic Radon transforms on rank one symmetric spaces
HTML articles powered by AMS MathViewer

by Eric Todd Quinto PDF

Proc. Amer. Math. Soc. 117 (1993), 179-186 Request permission

Abstract:

Using microlocal techniques, we prove support theorems for Radon transforms with real analytic measures on horocycles in rank one symmetric spaces. We generalize Helgason’s support theorem to this case and prove a new local support theorem.

References

Jan Boman, Uniqueness and non-uniqueness for microanalytic continuation of ultradistributions, Analysis, geometry, number theory: the mathematics of Leon Ehrenpreis (Philadelphia, PA, 1998) Contemp. Math., vol. 251, Amer. Math. Soc., Providence, RI, 2000, pp. 61–82. MR 1771259, DOI 10.1090/conm/251/03860
Jan Boman and Eric Todd Quinto, Support theorems for real-analytic Radon transforms, Duke Math. J. 55 (1987), no. 4, 943–948. MR 916130, DOI 10.1215/S0012-7094-87-05547-5

Support theorems for real analytic Radon transforms on line complexes in

A. M. Cormack, The Radon transform on a family of curves in the plane, Proc. Amer. Math. Soc. 83 (1981), no. 2, 325–330. MR 624923, DOI 10.1090/S0002-9939-1981-0624923-1
David V. Finch, Uniqueness for the attenuated x-ray transform in the physical range, Inverse Problems 2 (1986), no. 2, 197–203. MR 847534
Josip Globevnik, A support theorem for the X-ray transform, J. Math. Anal. Appl. 165 (1992), no. 1, 284–287. MR 1151072, DOI 10.1016/0022-247X(92)90079-S

Microlocal analysis and Radon transforms on higher rank symmetric spaces

Allan Greenleaf and Gunther Uhlmann, Nonlocal inversion formulas for the X-ray transform, Duke Math. J. 58 (1989), no. 1, 205–240. MR 1016420, DOI 10.1215/S0012-7094-89-05811-0

Some remarks on integral geometry

Victor Guillemin, On some results of Gel′fand in integral geometry, Pseudodifferential operators and applications (Notre Dame, Ind., 1984) Proc. Sympos. Pure Math., vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp. 149–155. MR 812288, DOI 10.1090/pspum/043/812288
Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, Mathematical Surveys, No. 14, American Mathematical Society, Providence, R.I., 1977. MR 0516965
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
S. Helgason, Duality and Radon transform for symmetric spaces, Amer. J. Math. 85 (1963), 667–692. MR 158409, DOI 10.2307/2373114
Sigurđur Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1–154 (1970). MR 263988, DOI 10.1016/0001-8708(70)90037-X
Sigurdur Helgason, The surjectivity of invariant differential operators on symmetric spaces. I, Ann. of Math. (2) 98 (1973), 451–479. MR 367562, DOI 10.2307/1970914
Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
A. Kaneko, Introduction to hyperfunctions, Mathematics and its Applications (Japanese Series), vol. 3, Kluwer Academic Publishers Group, Dordrecht; SCIPRESS, Tokyo, 1988. Translated from the Japanese by Y. Yamamoto. MR 1026013
Jeremy Orloff, Invariant Radon transforms on a symmetric space, Integral geometry and tomography (Arcata, CA, 1989) Contemp. Math., vol. 113, Amer. Math. Soc., Providence, RI, 1990, pp. 233–242. MR 1108657, DOI 10.1090/conm/113/1108657
Eric Todd Quinto, The dependence of the generalized Radon transform on defining measures, Trans. Amer. Math. Soc. 257 (1980), no. 2, 331–346. MR 552261, DOI 10.1090/S0002-9947-1980-0552261-8
Eric Todd Quinto, The invertibility of rotation invariant Radon transforms, J. Math. Anal. Appl. 91 (1983), no. 2, 510–522. MR 690884, DOI 10.1016/0022-247X(83)90165-8
Eric Todd Quinto, Tomographic reconstructions from incomplete data—numerical inversion of the exterior Radon transform, Inverse Problems 4 (1988), no. 3, 867–876. MR 965651
Mikio Sato, Takahiro Kawai, and Masaki Kashiwara, Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), Lecture Notes in Math., Vol. 287, Springer, Berlin, 1973, pp. 265–529. MR 0420735
L. A. Shepp and J. B. Kruskal, Computerized Tomography: The New Medical X-Ray Technology, Amer. Math. Monthly 85 (1978), no. 6, 420–439. MR 1538734, DOI 10.2307/2320062
Donald C. Solmon, The $X$-ray transform, J. Math. Anal. Appl. 56 (1976), no. 1, 61–83. MR 481961, DOI 10.1016/0022-247X(76)90008-1
François Trèves, Introduction to pseudodifferential and Fourier integral operators. Vol. 2, University Series in Mathematics, Plenum Press, New York-London, 1980. Fourier integral operators. MR 597145
Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Pure and Applied Mathematics, No. 19, Marcel Dekker, Inc., New York, 1973. MR 0498996