Duke Mathematical Journal

Radon transform on real, complex, and quaternionic Grassmannians

Genkai Zhang

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Let $G_{n,k}({\mathbb K})$ be the Grassmannian manifold of $k$-dimensional ${\mathbb K}$-subspaces in ${\mathbb K}^n$, where ${\mathbb K}={\mathbb R}, {\mathbb C}, {\mathbb H}$ is the field of real, complex, or quaternionic numbers. For $1\le k \lt k^\prime \le n-1$, we define the Radon transform $({\mathcal R}f)(\eta)$, $\eta \in G_{n,k^{\prime}}({\mathbb K})$, for functions $f(\xi)$ on $G_{n,k}({\mathbb K})$ as an integration over all $\xi \subset \eta$. When $k+k^\prime \le n$, we give an inversion formula in terms of the Gårding-Gindikin fractional integration and the Cayley-type differential operator on the symmetric cone of positive ($k\times k$)-matrices over ${\mathbb K}$. This generalizes the recent results of Grinberg and Rubin [4] for real Grassmannians

Article information

Duke Math. J. Volume 138, Number 1 (2007), 137-160.

First available in Project Euclid: 9 May 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A33: Fractional derivatives and integrals 44A12: Radon transform [See also 92C55] 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]
Secondary: 57S15: Compact Lie groups of differentiable transformations 43A85: Analysis on homogeneous spaces


Zhang, Genkai. Radon transform on real, complex, and quaternionic Grassmannians. Duke Math. J. 138 (2007), no. 1, 137--160. doi:10.1215/S0012-7094-07-13814-6. http://projecteuclid.org/euclid.dmj/1178738562.

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