Radon transforms on affine Grassmannians

Author:
Boris Rubin

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

Published electronically:
June 29, 2004

MathSciNet review:
2084410

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Abstract | References | Similar Articles | Additional Information

Abstract: We develop an analytic approach to the Radon transform , where is a function on the affine Grassmann manifold of -dimensional planes in , and is a -dimensional plane in the similar manifold . For , we prove that this transform is finite almost everywhere on if and only if , 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 . It is proved that the dual Radon transform can be explicitly inverted for , 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 . The investigation is carried out for locally integrable and continuous functions satisfying natural weak conditions at infinity.

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Additional Information

**Boris Rubin**

Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel

Email:
boris@math.huji.ac.il

DOI:
https://doi.org/10.1090/S0002-9947-04-03508-1

Keywords:
Radon transforms,
Grassmann manifolds,
inversion formulas

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).

Dedicated:
Dedicated to Professor Lawrence Zalcman on the occasion of his 60th birthday

Article copyright:
© Copyright 2004
American Mathematical Society