Sigurdur Helgason

The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds

1. Totally Geodesic Submanifolds of Xand of S. Reduction to the Hermitian Hyperbolic Case

The Surjectivity of Invariant Differential Operators on Symmetric Spaces I

2. Conical Distributions. Elementary Properties

Eigenspaces of the Laplacian; integral representations and irreducibility

3. Conical Functions and Finite-Dimensional Representations ......

Radon-Fourier transforms on symmetric spaces and related group representations

4. The Determination of the Conical Distributions

A duality in integral geometry; some generalizations of the Radon transform

5. The Spherical Representations and the Fourier Transform on X .... 96. Construction of the Intertwining Operators

Invariant differential equations on homogeneous manifolds

As proved by Radon [16] and John [13], a differentiable function / of compact support on a Euclidean space R n can be determined explicitly by means of its integrals over the hyperplanes in the space. Let J(~o, p) denote the integral of / over the hyperplane <x, co> = p where r is a unit vector and <,> the inner product in R n. I f A denotes the Laplacian on R n, do) the area element on the unit sphere S n-1 then (John [14], p. 13)

The Abel, Fourier and Radon transforms on symmetric spaces

An hydraulic jack for an elevator installation comprises a pair of cylinders and a common plunger. One cylinder is fixedly mounted in a vertical position in a hoistway; while one end of the plunger is mounted within the fixed cylinder for vertical reciprocating movement therein. The other cylinder, which carries an elevator cab up and down in the hoistway, is slidably mounted about the other end of the plunger for vertical reciprocating movement thereon.

Invariant differential operators on Grassmann manifolds

Abstract An integral representation is given for eigenfunctions of the Laplacian on a noncompact two-point homogeneous space. For the flat case an irreducibility criterion for the eigenspace representations is proved, complementing an earlier result [6a, p. 143] for the nonflat cases.

Support of Radon Transforms

First published in the Bulletin of the American Mathematical Society in Vol.71, 1965, published by the American Mathematical Society