Range characterization of the cosine transform on higher Grassmannians

On an analytic description of the $α$-cosine transform on real Grassmannians

The goal of this paper is to describe the $α$-cosine transform on functions on a Grassmannian of $i$-planes in an $n$-dimensional real vector space. in analytic terms as explicitly as possible. We show that for all but finitely many complex $α$ the $α$-cosine transform is a composition of the $(α+2)$-cosine transform with an explicitly written (though complicated) O(n)-invariant differential operator. For all exceptional values of $α$ except one we interpret the $α$-cosine transform explicitly as either the Radon transform or composition of two Radon transforms. Explicit interpretation of the …

On the Range of Cosine Transform of Distributions for Torus-Invariant Complex Minkowski Spaces

In this paper, we study the range of (absolute value) cosine transforms for which we give a proof for an extended surjectivity theorem by making applications of the Fredholm's theorem in integral equations, and show a Hermitian characterization theorem for complex Minkowski metrics on \mathbb{C}^n. Moreover, we parametrize the Grassmannian in an elementary linear algebra approach, and give a characterization on the image of the (absolute value) cosine transform on the space of distributions on the Grassmannian Gr_{2}(\mathbb{C}^{2}), by computing the coefficients in the Legendre series expansi…

高维格拉斯曼流形上的拟解析性、不确定性与积分变换

Quasianalyticity, uncertainty, and integral transforms on higher grassmannians

本文利用 O(n)-谱的不确定性与拟解析性原理，证明高维格拉斯曼流形上余弦、α-余弦及 Radon 等积分变换不能支集于单个开 Schubert 胞腔，并强化凸几何与几何断层学中的若干经典定理。

The Capelli identity and Radon transform for Grassmannians

We study a family $C_{s,l}$ of Capelli-type invariant differential operators on the space of rectangular matrices over a real division algebra. The $C_{s,l}$ descend to invariant differential operators on the corresponding Grassmannian, which is a compact symmetric space, and we determine the image of the $C_{s,l}$ under the Harish-Chandra homomorphism. We also obtain analogous results for corresponding operators on the non-compact duals of the Grassmannians, and for line bundles. As an application we obtain a Radon inversion formula, which generalizes a recent result of B. Rubin for real Gras…

Derivatives of the L^p cosine transform

The $L^p$-cosine transform of an even, continuous function $f\in C_e(\Sn)$ is defined by: $$H(x)=\int_{\Sn}|\ip{x}ξ|^pf(ξ) dξ,\quad x\in {\R}^n.$$ It is shown that if $p$ is not an even integer then all partial derivatives of even order of $H(x)$ up to order $p+1$ (including $p+1$ if $p$ is an odd integer) exist and are continuous everywhere in ${\R}^n\backslash\{0\}$. As a result of the corresponding differentiation formula, we show that if $f$ is a positive bounded function and $p>1$ then $H^{1/p}$ is a support function of a convex body whose boundary has everywhere positive Gauss-Kronekcer …