-
A DUALITY IN INTEGRAL GEOMETRY; SOME
GENERALIZATIONS OF THE RADON
TRANSFORM 1
'IONNAIRE
SIGURDUR HELGASON
1. Introduction. It was proved by J. Radon in 1917 (see [26]) that
SECTION
a differentiable function f of compact support on a Euclidean space
can be determined from the integrals of f over each hyperplane in the
space. Whereas Radon was primarily concerned with the dimensions
2 and 3, the following formulation for an arbitrary Euclidean space
Rn
was given by John [23], [24]. If w is a unit vector let J(co, p) de-
note the integral of f over the hyperplane {xERnI (x, w)=p} where
( , ) denotes the inner product. Then, if d denotes the surface element on the unit sphere 2 = Sn-l, and A the Laplacian,
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f(x) =
M
zI
(2)
f(x) = (2 ri)
-n (n-2)/2
Ax
dw
f.
-oo
d(w, p)
p - (Co, x)
(n even),
where in the last integral, the Cauchy principal value is taken.
Applications. The applications of these formulas are primarily
based on the following property: Consider the integrand in the integral over 2, say the function J(co, (co,x)) in (1). For a fixed w this
function x--J(w, (, x)) is a plane wave, that is a function which is
constant on each member of a family of parallel hyperplanes. Aside
from the Laplacian, formulas (1) and (2) give a continuous decomposition of f into plane waves. Since a plane wave only amounts to a
function of one real variable (along the normal to the hyperplanes)
the formulas (1) and (2) can sometimes reduce a problem for n real
variables to a similar problem for one real variable. This principle
has been used effectively on partial differential equations with constant coefficients (see Courant-Lax [2], Gelfand-Shapiro [10], John
[24], Borovikov [1], Garding [4]) and even for general elliptic equations (John [24]).
Generalizations. The above representation of a function by means of
An address delivered under the title The Radon transform on symmetricspaces
before the Summer Meeting of the Society on August 30, 1963, in Boulder, Colorado
by invitation of the Committee to Select Hour Speakers for Annual and Summer
Meetings; received by the editors March 26, 1964.
1 Work supported by the National Science Foundation, NSF GP-149.
435
436
SIGURDUR HELGASON
[July
its plane integrals suggests the general question of determining a
function f on a space from the knowledge of the integrals of f over
certain subsets of the space. Radon himself discussed in [26] the
problem of determining a function on the non-Euclidean plane from
the integrals of the function over all geodesics, Funk [3] proved that
a function f on the 2-sphere, symmetric with respect to the center,can be determined by means of the integrals off over the great circles.
The theory of representations of noncompact semisimple Lie groups,
in particular the Plancherel formula, raises the problem of determining a function on a semisimple Lie group by means of its integrals
over all conjugacy classes and their translates (see Gelfand-Naimark
[9], Harish-Chandra [12], [14], Gelfand-Graev [6]). Other examples
can be found in Gelfand [5], Gelfand-Graev [7], Gelfand-GraevVilenkin [8], Hachaturov [11], Harish-Chandra [13], Helgason
[16], [19], [21], John [24], Kirillov [25] and Semyanistyi
i964]
A
action of G on X a
measures
v.
(iii) If f and g ar
tively, we can defin
We shall now dis
problems which wil
A. Relate function
formsf-o] and g--i
B. Does there ex
E- i of D(z) into
[30].
Notation. If M is any manifold, 8(M) denotes the space of C- functions on M, )D(M)the space of C functions on M with compact
support. If M is a complete Riemannian manifold, S(M) denotes the
space of rapidly decreasing functions on M as defined in [21]. We
recall that fES(M) if and only if the following condition is satisfied.
Let A1 , · · , A, be the Laplace-Beltrami operators of the various
factors in the local de Rham decomposition of M into irreducible
parts [27]. Each Ai can be viewed as a differential operator on M.
Then fES(M) if and only if P(A1 , · · · , An)f goes to zero at o faster
than any power of the distance from a fixed point.
for generalf and g?
2. Dual integral transforms. As mentioned above, formulas (1)
and (2) give a continuous decomposition of f into plane waves. We
shall now focus attention on a kind of a projective duality which
appears in (1). Formula (1) contains two integrations, dual to each
other: first one integrates over the set of points in a given hyperplane,
then one integrates over the set of hyperplanes passing through a
given point. Guided by this duality we adopt the following general
setup.
(i) Let X be a manifold and G a transitive Lie transformation
group of X. Let Z be a family of subsets of X, permuted transitively
by G; in particular, there is induced a G-invariant differentiable
structure on S. The manifold Z will be called the dual of X. Let
D(X) and D(z) denote the algebras of G-invariant differential operators on X and Z, respectively.
give the same t, th
(ii) Given xEX, let x= {tE IxE}. It is assumed that each f
and each x have measures, say
and v, respectively, such that the
C. In case the trt
inversionformulas.
(1) and betweeng
3. Points and hy
and let G be the gr
hyperplanes in X.
Euclidean measure
the compact set x, i
by (x) = 1. A hype:
coand a real number
Co functions on Z
satisfying F(w, p)=
of rapidly
decreaE
S"- XR, satisfying
are defined. Let SR
integer k > 0, the ir
geneous kth deg
= O(z)ns(z). Fi
satisfying ff(x)p(x,
functions FES(2)
integers k > 0.
THEOREM 3.1.
and the algebra D(5
d2/dp2 (g(o, p)).
r
[July
. of determining a
integrals of f over
:ussed in [26] the
clidean plane from
nk [3] proved that
pect to the center,
'er the great circles.
isimple Lie groups,
problem of detereans of its integrals
X
Gelfand-Naimark
i]). Other examples
7], Gelfand-Graevra [13 ], Helgason
nyanistyi [30].
te space of C* func1 M with compact
1,S(M) denotes the
efined in [21]. We
indition is satisfied.
tors of the various
M into irreducible
:ial operator on M.
to zero at oo faster
t.
bove, formulas (1)
:o plane waves. We
:tive duality which
:tions, dual to each
a given hyperplane,
passing through a
.e following general
Lie transformation
rmuted transitively
riant differentiable
the dual of X. Let
at differential oper;sumed that each e
vely, such that the
action of G on X and Z permutes the measures/
measures
437
A DUALITY IN INTEGRAL GEOMETRY
i9641
and permutes the
v.
(iii) If f and g are suitably restricted functions on X and Z, respectively, we can define functions f on X, on X, by
() = f f(x)d(x),
g(x) = f
g()dv().
We shall now discuss several examples within this framework. The
problems which will be considered are:
A. Relate function spaces on X and Z by means of the integral transforms f-- and g--g.
B. Does there exist a map D--45 of D(X) into D(Z) and a map
E---> of D() into D(X) such that
(Df)A = Dj,
(Eg)v =
for generalf and g?
C. In case the transforms f--j and g--g are one-to-one,find explicit
inversionformulas. In particular,find the relationshipsbetweenf and
(J) and betweeng and ()".
3. Points and hyperplanes in a Euclidean space. Let X = R n (n > 1)
and let G be the group of all rigid motions of X and Z the set of all
hyperplanes in X. If xGX,
E4 the measure s/ on is the ordinary
Euclidean measure and the measure v on i is the unique measure on
the compact set x, invariant under all rotations around x, normalized
by v(x) = 1. A hyperplane fEZ is determined by a unit normal vector
X and a real number p such that pwhE. The pairs (co,p) and (-o, -p)
give the same ~, the manifold Sn- X R is a double covering of and
C- functions on Z will be identified with C* functions F on S-1 XR
satisfying F(wc,p) = F( -co, -p). Accordingly, let 8(Z) denote the set
of rapidly decreasing functions F on the Riemannian manifold
S"- 1 XR, satisfying F(o, r)= F(-co, -r). Similarly D(Z) and 8(z)
are defined. Let SH(Z) denote the set of FES(Z) such that for each
integer k>_0, the integral f2 F(w, p)pkdp can be written as a homoDH(Z)
CO,,n.Let
geneous kth degree polynomial in c1,
= sD(Z)fSR(Z). Finally, let 8*(X)=S*(Rn) denote the set of fG8(X)
satisfying ff(x)p(x)dx=O for each polynomial p and 8*(z) the set of
p)pkdp=O0for all w and all
functions FES(Z) satisfying fF(w,
integers /t> O.
THEOREM 3.1. The algebra D(X) is generated by the Laplacian A,
and the algebra D(Z) is generatedby the differential operator E: g(o, p)
2 /dp2 (g(Wc,p)). Moreover
-->d
438
SIGURDUR HELGASON
(Af)^ =
ll,
[July
1964]
A
(L]g)' = Ai
for fES(X), gEG(Z).
Cl
3.2. The Radon transformf--3 is a linear one-to-onemapping of D(X) onto DH(,), of S(X) onto SH(:), and of S*(X) onto S*(Z).
THEOREM
Theorem 3.1 is proved in [21]. The first statement of Theorem 3.2
holds in the following sharper form [21]: If fES(X) and if J(F)
=0
for all at distance > R from 0 then f(x) vanishes identically for
Ix I >R. The second part of Theorem 3.2 is, in a different form,
stated in Gelfand-Graev-Vilenkin [8]. The proof given there (pp.
34-39) seems incomplete in some respects; for example, it uses (1)
§1 and therefore leaves out the even-dimensional case. A different
proof is given in [21] using the relation
F(rw)
=
(1)
r(_2
The odd-dimensi(
of (1) §1 and Theor
from (2) §1 by mear
Pf.rl -2 n (see [21]).
Fourier transform.
4. Points and ar
geneous spaces. Le-
that is, a compact:
](, p)e-iprdp
group G of all isome
XX d(x, y) = r} fo
connecting the Radon transform ] and the Fourier transform J of f.
The last part of Theorem 3.2 is stated in Semyanistyi [29].
f = cA(-1)2((J)),
g= C(n-1)/2(Q)^)
f S(X);
gE s*(),
where the constant c is given by
point homogeneous
of antipodal manifo
the transitive actior
Lie group (ignoring
mannian structure
duced by the negat
r
(2ri) 1-n"n/2.
On
we choose th{
phism
If n is even,
(3)
is a to
Riemannian structu
If n is odd,
c=
antipodalmanifold,
x. This A
THEOREM 3.3. The following inversion formulas hold.
(2)
denote the diameter
: x-->A
is
f= cJ1(()),
fE s(X);
g = C2J2((g) ) ,
Beltrami operators
on the manifolds
structures of X an,
g E s*(Z),
quently
where J1 and J2 are given by analytic continuations
2(x) =
Jl:
f(x) -- anal. cont. ff(y)
a =1-2n
J2: F(w, p) -- anal. cont.
a=-n
I x -y
Rn
F(,
R
g()dv()
dy,
so
q) I p-
q
dq,
and
2 In the notation of Schwartz [28, p. 45], Jlf=(Pf.r
l-"
n) * f, J2 F=(Pf.r)
*F.
Because of this corr
and g--g it suffices
Concerning Probl
July
la"r(n- C1 -
ear one-to-one mapf S*(X) onto S*(%).
mntof Theorem 3.2
(X) and if I(t)=0
hes identically for
a different form,
f given there (pp.
sample, it uses (1)
I case. A different
or transform f of f.
istyi [29].
hold.
f s(x);
g
S*(),
f s(x);
g E S*(),
439
A DUALITY IN INTEGRAL GEOMETRY
I964]
,
(22
2)
C2 =
r(1-n)12
r (1- )
The odd-dimensional case in Theorem 3.3 is a direct consequence
of (1) §1 and Theorem 3.1. The inversion formula (3) can be derived
from (2) §1 by means of some computation involving the distribution
Pf.rl-2 " (see [21]). The formula for g then follows by use of the
Fourier transform.
4. Points and antipodal
manifolds in compact two-point homo-
geneous spaces. Let X be a compact two-point homogeneous space,
that is, a compact Riemannian manifold with the property that the
group G of all isometries of X acts transitively on the set {(x, y) EX
XX I d(x, y)=r} for each fixed r>0. Here d denotes distance. Let L
denote the diameter of X. If xGX, let A. denote the corresponding
antipodal manifold, that is, the set of points yEX at distance L from
x. This A, is a totally geodesic submanifold of X and with the
Riemannian structure on A, induced by that of X, Ax is another twopoint homogeneous space. Also x$y=~A 2 vAv. Let : denote the set
of antipodal manifolds, with the differentiable structure induced by
the transitive action of G. The Lie group G is a compact semisimple
Lie group (ignoring the trivial case dim X = 1). Changing the Riemannian structure on X by a constant factor we may assume it induced by the negative of the Killing form of the Lie algebra of G.
On we choose the Riemannian structure such that the diffeomorphism : x-->A is an isometry. Let A and a denote the LaplaceBeltrami operators on X and 5, respectively. The measures /Aand v
on the manifolds and will be those induced by the Riemannian
structures of X and S. If xEX, then = {+(y) IyEq(x)}. Conse-
quently
g(x) =
y l-dy,
f; g()dv(l)
=
I
)g(c(y))dv(c(y)) = f
ve0 X)
o(go
)(y)dM(y)
(X)
so
- q dq,
*f, J2F=(Pfr- f) *F.
g= (go ) o.
Because of this correspondence between the integral transforms f--J
and g--* it suffices to consider the first.
Concerning Problems A, B and C the following theorem holds.
440
[July
SIGURDUR HELGASON
THEOREM
i964]
4.1. The mapping f-->Jis a linear one-to-one mapping of
g(X) onto8(z) and
points and hyperpl
phism of X onto th
the polarity with r(
mapping of X -{
planes which do no
(Af)^=
Except for the case when X is an even-dimensionalreal projective space,
f= p(A)(),
A
f G(X),
underlying the app
where P is a polynomial, independent of f, explicitly given below. The
algebras D(X) and D(z) are generatedby A and , respectively.
in place of the spai
This theorem is proved in [21] by an elaboration of the method in
[16, p. 284], where the case of odd-dimensional projective spaces
is settled. The general case requires Wang's classification [31] of compact two-point homogeneous spaces as the following spaces: The
spheres S (n = 1, 2, .-), the real projective spaces P"(R)
(n=2, 3, · · · ), the complex projective spaces P"(C) (n=4, 6, · · · ),
Let n be an integer
5. Points and tot
Q(Z)= z
X = P(R)
on the Euclidean s
with the Riemann
O(n, 1) of linear t
Riemannian manifi
tive Lie group of i,
of X of dimension
planes in R 'n+ thr,
such a hyperplane
so the set of totall
be identified with
identified. This ide
quadratic form Q
(a - K(n- 2)2)(A- K(n- 4)4) ... (A - K2(n- 2)) ' X = P(C)
ture (n-1, 1) and
[( - K(n - 2)4)( - (n - 4)6) · · · ( - K8(n- 6))]
[(A - tK4(n- 4))(A - K4(n- 2))] X = P(H)
In group-theoretic
the quaternion projective spaces pn(H) (n=8, 12,
), and the
Cayley projective plane P16(Cay). The superscript denotes the real
dimension. The antipodal manifolds in the respective cases are a
point, P"-'(R), P-2(C), Pn-4(H), S 8. The polynomial P(A) above has
degree equal to one half the dimension of the antipodal manifold,
and is a constant multiple of
1 (the identity)
(A - K(n - 2)1)(A - K(n - 4)3) ·. (A - K(n - 2))
(A -
112K) 2 (A -
120K)
X = Sn
X = O(n, 1)
X = P16 (Cay).
2
where O(n) is the
two elements. Sinc
taken as the measu
of X and ; concert
In each case K= (r/(2L)) 2 , where L is the diameter of X.
In the exceptional case when X is an even-dimensional real projective space there is still an inversion formula f =K((J) ) but now K
is an integral operator. Considering functions on X as symmetric
functions on Sn, K is given by a suitably regularized integral operator
(K
)(x)= c
-,
F(
2
2
- ; sin2 (d(x,y))
(y)dw(y),
2
where d& is the volume element of Sn, c is a constant and F is the
hypergeometric function (Semyanistyi [30]).
REMARK. There is an important difference in the duality above between points and antipodal manifolds and in the duality between
Riemannian struct
of S.
Concerning Prot
I
THEOREM 5.1. 2.
o;
Laplace-Beltrami
If n is odd,
f = c(A + 1(n
[July
eal projectivespace,
fE (X),
ly given below. The
respectively.
5. Points and totally geodesic hypersurfaces in a hyperbolic space.
Let n be an integer > 1 and consider the quadratic form
,n of the method in
I projective spaces
)wing spaces: The
:ive spaces Pn(R)
)t denotes the real
,ective cases are a
lial P(A) above has
ntipodal manifold,
X = Sn
)) X = P(R)
)) x = P-(C)
6))]
X = O(n, )/O(n),
)] X = P"(H)
X = P'l(Cay).
))
.I
II
but now K
)w(y))
9
Z'l
Z=(, · ,zn+l,
t
1
z =
(n, 1)/((n
- 1, 1) X Z2),
where O(n) is the orthogonal group in RI and Z2 is the group with
two elements. Since the group G acts isometrically, /u and can be
taken as the measures induced by the pseudo-Riemannian structures
of X and :; concerning v one has to remark that for xEX the pseudoRiemannian structure of Z is nondegenerate on the submanifold
of .
Concerning Problems A, B and C we have the following results.
THEOREM 5.1. The algebras D(X) and D(z) are generated by the
Laplace-BeltramioperatorsA and A on X and , respectively,and
Lstant and F is the
e duality above bele duality between
+ Z'
on the Euclidean space Rn+'. Let X denote the quadric Q(Z) +1 =0
with the Riemannian structure induced by Q. Let G be the group
O(n, 1) of linear transformations of Rn+l leaving Q invariant. The
Riemannian manifold X has constant curvature -1 and G is a transitive Lie group of isometries of X. The totally geodesic submanifolds
of X of dimension n -1 are obtained as intersections of X with hyperplanes in R "+ through the origin. A normal (with respect to Q) to
such a hyperplane intersects the quadric Q(X)- 1=0 in two points
so the set of totally geodesic hypersurfaces-the dual space -- can
be identified with the quadric Q(X)-1 =0 with symmetric points
identified. This identification is consistent with the action of G. The
quadratic form Q induces a pseudo-Riemannian structure of signature (n-1, 1) and constant curvature +1 on Z (see [17, p. 146]).
In group-theoretic terms we have
C) (n=4, 6,- -),
12, - ), and the
n X as symmetric
.d integral operator
0
9
Q(Z) z + -
cation [31] of com-
(()v)
441
points and hyperplanes in §3. In the first case we have a diffeomorphism of X onto the manifold : of antipodal manifolds. In the second
the polarity with respect to the unit sphere IxI = 1 gives a one-to-one
mapping of X- 0 } onto the subset of Z consisting of those hyperplanes which do not pass through 0. This special role of 0 is the reason
underlying the appearance of the spaces D)H(2),SH(:) in Theorem 3.2
in place of the spaces D() and S(,).
'-to-one mapping of
r of X.
nsional real projec-
A DUALITY IN INTEGRAL GEOMETRY
i964]
(Af) = i for fE D(M).
If n is odd,
f = c(A +
l(n-
2))(A + 3(n - 4)) ... (A + ( - 2)1)(?)v
442
SIGURDUR HELGASON
[July
i964]
where c is a constant, independent of f.
A
and Z(G/K), can be
R" is invariant und
The last formula is proved in [16]. An inversion formula for the
even-dimensional case, more complicated to state, can be found in
in the hyperplane,
cycle in X.
[30].
Let W be the We
let S(A) be the syl
ments in S(A), inv;
The mapping a
I(A) [15, Theorem
to an isomorphism
described explicitly
and C we have the
For the space X the volume of a ball of radius r increases with r
like e(n-l)r. This explains the growth condition in the next result. Let
o be a fixed point in X; if xEX, EZ then d(o, x) and d(o, ) denote
the corresponding distances from o.
satisfies
(i) For each integer m > 0, f(x)emd(ox)is bounded;
THEOREM5.2. Suppose f G(X)
(ii) (~)=0 if d(o, ) > R.
Then
THEOREM 6.1. T,
f (x) =
for d(o, x) > R.
I
I
DO(X)into (z). T
such that
6. Points and horocycles in a symmetric space [19]. Let X be a
symmetric space of the noncompact type [18] and G the largest connected group of isometries of X in the compact open topology. The
group G has center reduced to the identity element and thus can be
identified with its adjoint group. This matrix group contains maximal
unipotent subgroups, all known to be mutually conjugate, and their
orbits in X are called horocycles.The group G acts transitively on the
set Z of all horocycles. The transform f-f is defined by integration
over each horocycle , the volume element being that induced by the
Riemannian structure of X. The transform g--g is defined by averaging over each x, xEX, the compact isotropy subgroup of G at x acting
transitively on x.
Let G=KAN be an Iwasawa decomposition of G where the subgroups K, A, N are compact, abelian and unipotent, respectively. Let
M denote the centralizer of A in K. Then we have the following
identifications from the natural action of G on X and ,
X = G/K,
A = G/MN.
The manifolds X and Z have the same dimension. The polar coordinate decomposition of Euclidean space, i.e. the mapping (w, p)--pC
of S n-1 XR onto R" has an analog for X, namely the differentiable
mapping 4: (kM, a)--÷kaK of (K/M) XA onto G/K. Both maps are
singular on certain lower dimensional sets. In contrast, the dual
Z(Rn) of Rn (in the sense of §3) is doubly covered by Sn-1 XR and
the dual Z(G/K) of G/K is diffeomorphic to (K/M) XA under the
map : (kM, a)--kaMN. This difference between the duals, Z(Rn)
The isomorphisrr
position above con
Euclidean space A
metric space X ar
positive restricted i
the automorphism
fD=t-1((D)') for
THEOREM 6.2. St
there exist operators
(1)
(2)
where dx and d, a?
normalized.
We shall indicate
uct of the positive
identified with its
algebra of G then
constant factors, A
the image of 7rund
Theorem 6.2 ho
class of Cartan sub
[July
1964]
443
A DUALITY IN INTEGRAL GEOMETRY
and Z(G/K), can be traced to the fact that whereas a hyperplane in
Rn is invariant under the geodesic symmetry with respect to a point
in the hyperplane, the analogous statement does not hold for a horocycle in X.
Let W be the Weyl group of G/K acting on the Euclidean space A,
let S(A) be the symmetric algebra over A and I(A) the set of elements in S(A), invariant under W.
The mapping ¢ above gives rise to an isomorphism r of D(X) onto
also gives rise
I(A) [15, Theorem 1], [18, p. 432]; the mapping
to an isomorphism of D(A) onto S(A). These isomorphisms can be
described explicitly in Lie algebra terms. Concerning Problems A, B
on formula for the
:e, can be found in
3 r increases with r
the next result. Let
} and d(o, ) denote
and C we have the following results [19], [22].
6.1. The mapping f--f is a linear one-to-onemapping of
20(X) into D(E). There is an isomorphism D--Di of D(X) into D(z)
such that
THEOREM
. [19]. Let X be a
)I
(Df)^ = Dffor f C
1 G the largest con-
>pen topology. The
nt and thus can be
p contains maximal
:njugate, and their
transitively on the
ned by integration
hat induced by the
defined by averagoup of G at x acting
f G where the subLt,respectively. Let
have the following
and Z,
(X).
The isomorphism Di--+ is defined as follows. The Iwasawa decomposition above corresponds to an ordering of the dual space of the
Euclidean space A. In particular, the restricted roots of the symmetric space X are ordered; as usual let 2p denote the sum of the
positive restricted roots, counted with multiplicity. Let p->p' denote
the automorphism of S(A) given by H' =H+p(H) for HEA. Then
D= -1(r(D)') for DED(X).
6.2. Suppose the Lie group G is a complex Lie group. Then
A ED(Z) with the following property:
G
there exist operators ED(X),
THEOREM
(2)
f
f = o ((]) 7),
(1)
X fX)
dx =
l
()
2d,
where dx and d are the G-invariant measures on X and
normalized.
. The polar coordilapping (co,p)-po
y the differentiable
/K. Both maps are
contrast, the dual
ed by Sn-i XR and
/M) XA under the
n the duals, Z(Rn)
(X),
fE D(X),
, suitably
We shall indicate the definition of 1Iand A. Let 7rdenote the product of the positive restricted roots (without multiplicity). If A is
identified with its dual by means of the Killing form of the Lie
algebra of G then rES(A) and r2 CI(A). Then, except for certain
constant factors, A equals P-l(7r') and [] equals r-(r2). (Here 7r'is
the image of 7r under the automorphism p-*p' above.)
Theorem 6.2 holds more generally if G has only one conjugacy
and A are
class of Cartan subgroups, but the differential operators
444
SIGURDUR HELGASON
[July
i9641
A
then defined somewhat differently. For a general real G, the operators
[] and A are integro-differential operators, and the analog of (2) is
still valid. (See [22].)
Applications. It was mentioned earlier how the inversion formula
(1) §1, to a certain extent, can reduce a partial differential equation
with constant coefficients to an ordinary differential equation. Similarly, since O naturally commutes elementwise with D(X), the inversion formula in Theorem 6.2 can reduce an invariant differential
equation on X to a differential equation on the Euclidean space A
(see [19]); this differential equation on A is obtained via the mapping
r, in particular it has constant coefficients, hence solvable.
In the case when G is not complex better results are obtained by
restricting the transform f-f to radial functionsf on X, that is func-
trary fibre F and k
formation [3] of (
tions invariant under the action of K on X. In fact it turns out that
similarly (interchai
Let g*(Rn) be as in
this transform converts each operator DED(X) into a differential
operator on A with constant coefficients, resulting in the existence of
a fundamental solution for D. The proofs of these results are based on
Harish-Chandra's work on harmonic analysis on the group G (see
[20] and the references given there).
7. p-planes and q-planes in R+q+l. Let p and q be two integers
>0 and put n=p+q+1.
A p-plane E in Rn is by definition a translate of a p-dimensional vector subspace of Rn. Let G(p, n) denote the
manifold of p-planes in Rn and G*(p, n) the set of p-planes which do
not pass through 0. The projective duality between points and hyperplanes in Rn, realized by the polarity with respect to Sn-l generalizes
to a duality between G*(p, n) and G*(q, n). In fact, if a#0 in R", let
E,_l(a) denote the polar hyperplane. If a runs through a p-plane
EvEG*(p, n) then the hyperplanes En_(a) intersect in a unique qplane EEG*(q, n) and the mapping E,--E,, is the stated duality.
We have now an example of the framework in §2, although convergence difficulties make the results (Theorem 7.1) very restrictive.
Let X= G(p, n) and G the group of rigid motions of Rn, acting on X.
Given a q-plane Eq consider the family ~ = ~(Eq) of p-planes intersecting Eq. If E' #E,' then (Eq') 0(E "); thus the set of all families
--the dual space -- can be identified with G(q, n). In accordance
with this identification, if E, =xEX then = (Ep) is the set of qplanes Eq intersecting x.
The manifold G(p, n) is a fibre bundle with base space G,,,, the
manifold of p-planes in Rn through 0, the projection r of G(p, n) on
G,,n being the mapping which to any p-plane associates the parallel
p-plane through 0. Thus the fibre of this bundle (G(p, n), G,., 7r) is
Rn-P. If fES(G(p, n)) let f] F denote the restriction of f to an arbi-
for each fibre F, i
convergence difficu
but define ? by
J(EQ
whenever these int
the Grassmann ma
dq is the Euclidea:
operator Lp:f-S>f
holds [21].
THEOREM 7.1. T
Rq, respectively.T
S*(X) onto S*(:) s
If n is odd then
where c is a constai
1. W. A. Boroviko-
with constant coefficient.
2. R. Courant and
ferential equations witA
Pure Appl. Math. 8 (1(
3. P. Funk,
ber,
Math. Ann. 77 (1916),
4. L. Girding, Tra
Math. France 89 (1961
5. I. M. Gelfand, I
lions, Uspehi Mat. Nal
6. I. M. Gelfand
classicalgroups, Trudy
7.
, The geo
geneous spaces and ques
Ob. 8 (1959), 321-39
8. I. M. Gelfand,
[July
d G, the operators
te analog of (2) is
inversion formula
Terential equation
al equation. Simia D(X), the inverariant differential
1964]
trary fibre F and let AF denote the Laplacian on F. The linear transformation Cp of 8(G(p, n)) given by
(aplf)I F =
F),
AF(f
f E (G(p,n)),
for each fibre F, is a differential operator on G(p, n). Because of
convergence difficulties we do not define the measures u and v directly
but definef by
(L,
a(E) =fj
euclidean space A
d via the mapping
olvable.
s are obtained by
)n X, that is func-
445
A DUALITY IN INTEGRAL GEOMETRY
Eq
aEE2
f(Ep) dorpEp) d. (a
into a differential
whenever these integrals exist. Here dup is the invariant measure on
the Grassmann manifold of p-planes through a with total measure 1,
dq is the Euclidean measure on Eq. The transform g--g' is defined
similarly (interchanging p and q). For p = 0 we get the situation in §3.
Let S*(Rn) be as in §3 and let S*(X) be the image of S*(Rn)under the
in the existence of
operator L:f--fE
.sults are based on
holds [21].
t it turns out that
the group G (see
f(a)dp(a) (fES*(Rn)). Then the following result
THEOREM 7.1. The algebras D(X) and D(a) are generatedby [Upand
I,r respectively. The mapping fq be two integers
definition a trans7(p, n) denote the
p-planes which do
points and hyper:o Sn- 1 generalizes
, if a-O in Rn, let
hrough a p-plane
.ct in a unique q-
stated duality.
§2, although
con-
) very restrictive.
' Rn, acting on X.
-planes intersectset of all families
a). In accordance
is a linear one-to-onemapping of
8*(X) onto S*(Z) such that
(opf)^ =
-qf.
If n is odd then
f = C(p)(n-l)2((j)
),
f *(X),
where c is a constant, independent of f.
BIBLIOGRAPHY
1. W. A. Borovikov, Fundamental solutionsof linear partial differential equations
with constant coefficients, Trudy Moscov. Mat. Obs. 8 (1959), 199-257.
2. R. Courant and A. Lax, Remarks on Cauchy's problemfor hyperbolicpartial differential equations with constant coefficientsin severalindependent variables, Comm.
p) is the set of q-
Pure Appl. Math. 8 (1955), 497-502.
3. P. Funk,
ber eine geometrische Anwendung der Abelschen Integralgleichung,
Math. Ann. 77 (1916), 129-135.
4. L. Ggrding, Transformation de Fourier des distributions homoglnes, Bull. Soc.
Math. France 89 (1961), 381-428.
5. I. M. Gelfand, Integral geometry and its relation to the theory of group representa-
,e space Gp,n, the
tions, Uspehi Mat. Nauk. 15 (1960), 155-164= Russian Math. Surveys, 1960.
6. I. M. Gelfand and M. I. Graev, Analogue of the Plancherel formula for the
n r of G(p, n) on
Siates the parallel
r(p, n), Gp.n, 7r) is
n of f to an arbi-
classical groups, Trudy Moscov. Mat. Obs. 4 (1955), 375-404.
7.
, The geometry of homogeneous spaces, group representations in homogeneous spaces and questions in integral geometry related to them. I, Trudy Moscov. Mat.
Obs. 8 (1959), 321-390.
8. I. M. Gelfand, M. I. Graev and N. Vilenkin, Integral geometry and its relation to
446
SIGURDUR HELGASON
problems in the theory of group representations, Vol. 5, Generalized Functions, Fizmatgiz,
HARMC
Moscow, 1962.
9. I. M. Gelfand and M. A. Naimark, Unitary representations of the classical
groups, Trudy Mat. Inst. Steklov. 36 (1950), 288.
10. I. M. Gelfand and S. J. Shapiro, Homogeneous functions and their applications,
Uspehi Mat. Nauk 10 (1955), 3-70.
1. Let E2 repre
coordinate system
11. A. A. Hachaturov, Determination of measuresin domains of an n-dimensional
space by their values on all half spaces, Uspehi Mat. Nauk 9 (1954), 205-212.
12. Harish-Chandra, The Plancherelformula for complex semisimple Lie groups,
that X is a 1-coch
Trans. Amer. Math. Soc. 76 (1954), 485-528.
number for every
in R, (b) X(-a)
, Fourier transformson a semisimpleLie algebra.I, II, Amer. J. Math.
13. --
79 (1957), 193-257; 653-686.
(c)X()= X(+)+
, A formula for semisimple Lie groups, Amer. J. Math. 79 (1957), 733-
14.
a, collinear, simil.
tended by linearil
simplex (i.e., orier
We shall call the
ing two condition,
(1) there exist
locally in L1 on R
(a) if a is a 1-si
760.
, Sphericalfunctions on a semisimpleLie group. I, II, Amer. J. Math. 80
15.
(1958), 241-310; 553-613.
16. S. Helgason, Differential operators on homogeneous spaces, Acta Math. 102
(1959), 239-299.
, Some remarks on the exponential mapping for an aine connection,
17.
Math. Scand. 9 (1961), 129-146.
18.
~, Differential geometry and symmetric spaces, Academic Press, New
York, 1962.
19.
~,
Duality and Radon transform for symmetric spaces, Amer. J. Math.
| X(a)
85 (1963), 667-692.
20.
~,
I _ fagl( x)dd
(15) if a is a 1-si
Fundamental solutions of invariant differential operatorson symmetric
spaces, Bull. Amer. Math. Soc. 69 (1963), 778-781.
The Radon transform on Euclidean spaces, compact two-point homo21.
~,
IX() I <fg 2(y)d
geneousspacesand Grassmannmanifolds (to appear).
22.
~ , The Plancherel formula for the Radon transform on symmetric
spaces(to appear).
23. F. John, Bestimmung einer Funktion aus ihren Integralen iibergewisseMannig-
such that if r is a
(2) there exists
to the x and y-ax(
faltigkeiten, Math. Ann. 100 (1934), 488-520.
24.
, Plane waves and sphericalmeans, applied to partial differential equa-
tions, Interscience, New York, 1955.
25. A. A. Kirillov, On a problem of I. M. Gelfand, Dokl. Akad. Nauk SSSR 137
(1961) =Soviet Math. Dokl. 2 (1961), 268-269.
26. J. Radon,
Let Q be a mee
IR-Q12 =
ber die Bestimmung on Funktionen durch ihre Integralwertelangs
gewisser Mannigfaltigkeiten, Ber. Verh. Schs. Akad. Wiss. Leipzig Math.-Nat.
69 (1917), 262-277.
(whe
ure). Using the n
K1.
in R is Q-good if
tained in R is Q-e
We shall call tl
local L' differenti.
27. G. de Rham, Sur la reductibilitgd'un espacede Riemann, Comment. Math.
Helv. 26 (1952), 328-344.
28. L. Schwartz, Thgorie des distributions. I, Hermann, Paris, 1950.
29. V. I. Semyanistyi, On someintegraltransformsin Euclidean space, Dokl. Akad.
met:
536-539=Soviet Math. Dokl. 1 (1960), 1114-1117.
Nauk SSSR 134(1960),
30. ~, Homogeneous functions and some problems of integral geometry in
spaces of constant curvature, Dokl. Akad. Nauk SSSR 136 (1961), 288-291 = Soviet
Math. Dokl. 2 (1961), 59-62.
31. H. C.Wang, Two-point homogeneous spaces, Ann. of Math. (2) 55 (1952), 177-
17, 1962, by invitatio
191.
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