Radon transforms on affine Grassmannians
HTML articles powered by AMS MathViewer
- by Boris Rubin PDF
- Trans. Amer. Math. Soc. 356 (2004), 5045-5070 Request permission
Abstract:
We develop an analytic approach to the Radon transform $\hat f (\zeta )=\int _{\tau \subset \zeta } f (\tau )$, where $f(\tau )$ is a function on the affine Grassmann manifold $G(n,k)$ of $k$-dimensional planes in $\mathbb {R}^n$, and $\zeta$ is a $k’$-dimensional plane in the similar manifold $G(n,k’), \; k’>k$. For $f \in L^p (G(n,k))$, we prove that this transform is finite almost everywhere on $G(n,k’)$ if and only if $\; 1 \le p < (n-k)/(k’ -k)$, and obtain explicit inversion formulas. We establish correspondence between Radon transforms on affine Grassmann manifolds and similar transforms on standard Grassmann manifolds of linear subspaces of $\mathbb {R}^{n+1}$. It is proved that the dual Radon transform can be explicitly inverted for $k+k’ \ge n-1$, and interpreted as a direct, “quasi-orthogonal" Radon transform for another pair of affine Grassmannians. As a consequence we obtain that the Radon transform and the dual Radon transform are injective simultaneously if and only if $k+k’ = n-1$. The investigation is carried out for locally integrable and continuous functions satisfying natural weak conditions at infinity.References
- Carlos A. Berenstein, Enrico Casadio Tarabusi, and Árpád Kurusa, Radon transform on spaces of constant curvature, Proc. Amer. Math. Soc. 125 (1997), no. 2, 455–461. MR 1350933, DOI 10.1090/S0002-9939-97-03570-3
- I. M. Gel′fand, Integral geometry and its relation to the theory of representations. , Russian Math. Surveys 15 (1960), no. 2, 143–151 (Russian). MR 0144358, DOI 10.1070/RM1960v015n02ABEH004218
- Fulton B. Gonzalez, Radon transforms on Grassmann manifolds, J. Funct. Anal. 71 (1987), no. 2, 339–362. MR 880984, DOI 10.1016/0022-1236(87)90008-5
- Gonzalez, F., and Kakehi, T., Pfaffian systems and Radon transforms on affine Grassmann manifolds, Math. Ann., 326 (2003), no. 2, 237-273.
- —, Dual Radon transforms on affine Grassmann manifolds, Transactions of the Amer. Math. Soc. (to appear).
- M. I. Graev, A problem of integral geometry associated with a triple of Grassmann manifolds, Funktsional. Anal. i Prilozhen. 34 (2000), no. 4, 78–81 (Russian); English transl., Funct. Anal. Appl. 34 (2000), no. 4, 299–301. MR 1818288, DOI 10.1023/A:1004165625544
- Grinberg, E. L., and Rubin, B., Radon inversion on Grassmannians via Gårding-Gindikin fractional integrals, Annals of Math. (to appear).
- Sigurdur Helgason, A duality in integral geometry; some generalizations of the Radon transform, Bull. Amer. Math. Soc. 70 (1964), 435–446. MR 166795, DOI 10.1090/S0002-9904-1964-11147-2
- 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
- Alexander Hertle, Continuity of the Radon transform and its inverse on Euclidean space, Math. Z. 184 (1983), no. 2, 165–192. MR 716270, DOI 10.1007/BF01252856
- Árpád Kurusa, Support theorems for totally geodesic Radon transforms on constant curvature spaces, Proc. Amer. Math. Soc. 122 (1994), no. 2, 429–435. MR 1198457, DOI 10.1090/S0002-9939-1994-1198457-1
- W. R. Madych and D. C. Solmon, A range theorem for the Radon transform, Proc. Amer. Math. Soc. 104 (1988), no. 1, 79–85. MR 958047, DOI 10.1090/S0002-9939-1988-0958047-7
- Eric Todd Quinto, Null spaces and ranges for the classical and spherical Radon transforms, J. Math. Anal. Appl. 90 (1982), no. 2, 408–420. MR 680167, DOI 10.1016/0022-247X(82)90069-5
- Radon, J., Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math. - Nat. Kl., 69 (1917), 262–277 (Russian translation in the Russian edition of S. Helgason, The Radon transform, Moscow, Mir, 1983, pp. 134–148).
- Boris Rubin, Fractional integrals and potentials, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 82, Longman, Harlow, 1996. MR 1428214
- —, Inversion formulas for the spherical Radon transform and the generalized cosine transform, Advances in Appl. Math., 29 (2002), 471-497.
- —, Reconstruction of functions from their integrals over $k$-planes, Israel J. Math., 141 (2004), 93-117.
- —, Notes on Radon transforms in integral geometry, Fract. Calc. Appl. Anal., 6 (2003), no. 1, 25-72.
- Donald C. Solmon, A note on $k$-plane integral transforms, J. Math. Anal. Appl. 71 (1979), no. 2, 351–358. MR 548770, DOI 10.1016/0022-247X(79)90196-3
- Donald C. Solmon, Asymptotic formulas for the dual Radon transform and applications, Math. Z. 195 (1987), no. 3, 321–343. MR 895305, DOI 10.1007/BF01161760
- Robert S. Strichartz, Harmonic analysis on Grassmannian bundles, Trans. Amer. Math. Soc. 296 (1986), no. 1, 387–409. MR 837819, DOI 10.1090/S0002-9947-1986-0837819-6
- Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94
Additional Information
- Boris Rubin
- Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
- MR Author ID: 209987
- Email: boris@math.huji.ac.il
- Received by editor(s): May 13, 2003
- Received by editor(s) in revised form: September 11, 2003
- Published electronically: June 29, 2004
- Additional Notes: This work was supported in part by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 5045-5070
- MSC (2000): Primary 44A12; Secondary 47G10
- DOI: https://doi.org/10.1090/S0002-9947-04-03508-1
- MathSciNet review: 2084410
Dedicated: Dedicated to Professor Lawrence Zalcman on the occasion of his 60th birthday