The Radon Transform
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Progress in Mathematics
Edited by
J. Goates and s. Helgason
Sigurdur Helgason
a
0
sform
Springer Basel AG
5
Author Sigurdur Helgason Department of Mathematics Massachusetts Institute ofTechnology Cambridge, MA 02139 U.s.A.
Ubrary of Congress Cataloging in Publication Data Helgason, Sigurdur,1927The Radon transform. (Progress in mathematics; 5) Bibliography: p. Indudes index. 1. Radon transforms. I. Trtle. 11. Series: Progress in mathematics (Cambridge); 5. QA649.H44 516.3'6 80-15951
ISBN 978-1-4899-6767-1 ISBN 978-1-4899-6765-7 (eBook) DOI 10.1007/978-1-4899-6765-7
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Helgason, Sigurdur: The radon transform / Sigurdur Helgason.-Boston, Basel, Stuttgart : Birkhäuser, 1980. (Progress in mathematics : 5)
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© Springer Science+Business Media New York 1980 Originally published by Birkhäuser Boston in 1980.
'ID ARTIE
PREFACE The tit1e of this booklet refers to a topie in geometrie
analysis which has its origins in results of Funk [1916] and Radon [1917] determining, respeetive1y, a synmatrie funetion on the twosphere 52 fran its great cire1e integrals and a funetion on the plane R 2 fran its 1ine integrals (See Referenees) .
Reeent deve1op-
ments, in particu1ar applieations to partial differential equations, X-ray techno1ogy, and radioastronany, have widened interest in the subjeet. These notes eonsist of a revision of 1eetures given at MIT in the Fall of 1966, based roostly on my papers during 1959 - 1965 on the Radon transfonn and some of its generalizations.
transfonn" is adopted fran John [1955]).
(The tenn "Radon
The viewpoint for these gene-
ra1izations is as [ollows. The set of points on S2
and the set of great circ1es on S2
are both hotmgeneous spaees of the orthogonal group ()(3). the set of points in
~
Simi1ar1y,
and the set of lines in R2 are both hotmof rigid rootions of R 2 .
geneous spaees of the group M(2)
This
rootivates our general Radon transfonn definition fran [1965A,Blwhich fonns the franework of Chapter II: G/K and G/H of the maps funetions spaee. f
For
E;, E
f
Given two haoogeneous spaees A
SaIOO
group G the Radon transfonn f ~ f
on the first spaee to funetions G/H,
f(n
over the set of points x
sense of Chern [1942].
A
fonthe seeond
is defined as the (natural) integral of E
G/K which are ineident to
The problem of inverting
out in a few eases. (v)
E;, A
f ~ f
in the is worked
vi It happens when G/K is a Euclidean space. and IIDre generally G/K is a Riemannian synmetric space. that the natural differen-
~
A
tial operators D on G/K are transferred by f--? f A
I1Dre manageable differential operators D on G/H; A
= Df.
(Df)
Then the theory of the trans form
into tmJCh
the camection is A
f ~ f
has signifi-
cant applications to the study of the pro
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