This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
'(.]. >'(.2: >.p)
(1.13)
= 1>'1- 1 "RI(elo~;p).
=
Thus, this (wlction d(!pclld'4 not only on the line {I Xl + ~ %2 - P 0 but also on the choice of the 1>lIl'8mcters ~l. ~2. p. III t.be case of (~I.e2) = (cosY'. sinY'). the fWlction 'RI coincides with the ftmetion 'R./(t;.p) introduced in the EucUdean case. Oue frequently writES the expression for 'R. I in the fonn "R I({I.
(.2;
p)
= Jfa
where the definition of 6({. Xl
+ 6 X2 - I»
+ ~ X2 - p) dx l dx2•
is similar to that given in 1.1.
Tile inversion fomnda Jor the Radon transJo"fn (1.12) ;s
TUEOREM 1.3.
(1.14)
l(xI. X2) 6(el Xl 2
I(Xl •.r2) 12 = --4 7r
111-""('R.f)~(el'{2.p+el.l'.+{2X2)P-ldl})({1t1!.2-{2d(d. r
(
_<Xl
wllere r is Ofl arintmry contour in JR2 \ 0 intersecting once almost every my iSS'Uing from the point O. The validity of this formula follows from t\\'U simple facts. 1) The di.fft"rcntial fono in the integrand can be pushed down to the manifold S of rays issuing from tht' point O. i.e.. to the base space of the bundle R2 \ 0 - S. This follow~ from the fact that this form is homogeneous with mJpect to { of degree zero. and the form wnishcs on the vectors tangent. to thfo libels of tht' bundle ((~l' ~2) = ({t· ~ll) d>.). 2) If for r we take the twit. (ircle oontered at the point 0, then (1.14) coincides with fonnula (1.11) proven above.
1.S. Relation to the Fourier transform and another proof of the inversion formula. Let F(el.{2) be the Fonrier transfoml oftbe fuoction I(Xl,X2), Lt'.. (1.15)
F({I.{2)
= (2",)-1 f
JR3
I(ZI,Z2)C·«(lzl+E;a"'2)dxldx2.
To establish the relation betwet.'Jl the Fonrit'I' trnnsform F(el. {2) and the naclon trall.'4fonn R./({1.{2;P) of the fUllction/. wc reprcm.>nt till! intt-gral (1.15) for ({1.{2) :f:. 0 08 an iterated integral. where the integration it; t.aken lUst over the lines {I Xl + {2 Z2 = P and then over p.
F({I'{2)=(27r)-·j+~(l. -OQ
tl
%1
-rEl Z2=P
J(Zloz2)d~)eiJldp.
1. RADON TRANSFORM
6
Thus. we have THEOREM 1.4. The Radon transform 'R. [ and the Fourier transform F o[ a function [ are related as follows:
F(~I'~2) =
(1.16)
00
(271")-11: 'R.[({I.{2;P) e ip dp
«{1'{2)::/: 0).
Since tbe function 'R. [ is homogeneous. one can represent (1.16) in the fonu +00 (1.17) F(A{I. A~2) = (271")-1 -00 'R.[({I'{2;p)ei~Pdp.
1
Thus, the two-dimensional Fourier transform is the composition of the Radon tr~ form on th(' plane and the one-dimensional Fourier transfonn. Let us now present another proof of the inversion formula for the Radon tran.'iform. This approach is based on the inversion formula
= (271")-1 f F f(~.. ~2)e-i(~IZI+~2Z2) ~I ~2 JR = (471")-11+ 2 F [(A COS". A sin")e-'~(zlCU9,,,+z2sinop!) IAI d", dA
[(x .. X2)
2
00
1
11'
-00
0
for the Fourier transform. Substitute the expression in (1.17) (in which th(' function = 'R.[(",:p» into this inversion formula. Wc obtain
F [(A cos",. A sin",) is represented in terms of 'R.[(cos..p. sin,,: p)
+001+01) -, 1 .~ =--4' 1 Fp (Xl,x2,p)e' PsgnAdpdA 71" I -00 -00 1 +00 = --2 ~(XI. X2. p)p -I dp. 71" -oc
1
Here the function F is given by (1.6). At the last step we used the following fact: the Fourier transform of the distribution sgnA e S'(R) is the distribution 2ip-1 (see 110», i.e., if'" e S(R) and .p(A) = r~: "'(p)ei~p dp. then
1::x. ~(A)
sgnAdA = 2i
1:
00
tJ'(p) p-I dp.
In general. the integral on the right-hand side is understood in th(' principal value sense. However. if '" is a smooth odd function, then this integral converges in the ordinary sense. We finally obtain the inversion formula (1.5). REMARK. Note that the Radon transform sends the Laplace operator
~
=
-£ + fir on the functions [(XI. X2) to the operator /i;s on the functions 'R.[(",:p).
More generally. ifthere is a distribution K (r 2 ). where r2 = xi +x~. then the function K(~) K( -r2) * [ becomes the convolution K( -PZ) * 'R. [ (with respect to the
t:l
2. RADON TRANSFORM IS THREE-DIMENSIONAL SPACE
7
variable p). These reasonings can Ix> used to invert the Radon transform of a nonsmooth function. If I is a compactly supported distribution (in particular, a nonsmooth function). then we can represent I in thf:' form I = {I - A)k 'P, where 'P is a smooth function and k is a positive integer. The Radon t.ransfonn of the function I can be defined 88 the convolution K (1 + y)" .. 'R..p. This definition agrees with that given aboVE' for compactly supported continuous functions. If I is a nonsmooth function, then \W take the convolution of 'R I with (I + y) -k and obtain a smooth function 'R rp. Applying the inversion formula. for the Radon transform to the function 'R 'P, we find the function 'P. Finally, applying the differential operator (1- A)1r to 'P, we obtain the desired function I.
2. Radon transform in three-dimensional space 2.1. Radon transform in Euclldean space. By the Radon transform in three-dimensional Euclidean space W(> mean the integral transform sending a function on the space to the integrals of this ftmction over all possible planes (with respect to the Euclidean area E'lement on each of these planes). All considerations bere are carried out along the same schCDle as for the Radon transform on the plane. Let us deBnt> the Radon transform by Bn explicit formula. We present every plane by an equation (w, x) - p = O. «w, x) =
WI Xl
where
Iwl = 1
+ W:2 X2 + ""'3 X3), or by parametric equations r = tl 0 1 + t20:2 + pw,
where 0: 1 , 0/2 is a pair of mutuaUy ortbogonal unit vectors, i.e., (0: 1 , 02) = (w, Q I ) = (w, Q2) = O. Euclidean measure on such a plane is equal to dtl dt2. Thus, every plane is defined by a pair (w,p), where vJ ill a point of the unit sphere and p is a number, and it is clear that t.he planes r.orresponding to different pairs (w,p) and (w'.P') coincide if and only if (w',P') = -(..."p). The Radon transform of a function lex) i~ giw.n by the relation (2.1)
'R. I(w,p) =
I I(tl a 1 + t2 0:2 + pw) tit1 dt2, 1",2
where 0 1 • 0 2 are arbitrary unit vectors such that (0/ 1 , 02 ) = (~', ( 1 ) = (101, a2) = 0 (and the integral does not depend on the choice of these vectors). Since 'R. I( -101, -p) ='R/(..."p). it foUoM! that the function 'RI can be pushed down to the manifold
or planes.
As in the two-dimensional case, one frequently rt"presents (2.1) in the form 'R/(w,p)
=
s la11(:z:)6«w,x)-p)dx
where c5(.) is a function on R. For simplicity, we ronsider the Radon transform on the functions I E'D(R3). In this case, the functions 'R I are rompactly supported and infinitely diflere.ntiable on the manifold of planes, i.e., these functions are infinitely differentiable when regarded 88 function.'i of w and p and are compactly supported with respect to p.
l. RADON TRANSFORM
8
Let US find the inversion formula for the Radon transform on three-dimensional Euclidean space. We first consider a function I(x) = F(r) depending on r = (X1 2 + X2 2 + X3 2)1/2 only. The Radon transform of I does not depend on w. i.e., 'R/(w,p) = F(P). According to (2.1) we have (2.2)
F(P)
=f
1R2
F(JtI2
+ t22 + p2)dtl dt2 = 271'1.x F(r) rdr. 1,,1
Thus, in contrast to the two-dimensional case where we obtained the integral of order 1/2, here we simply obtain the integration operator. Therefore. the inverse is the differentiation operator, and hence F(r)
(2.3)
= _..!.. F'(r) . 271'
r
In particular,
(2.4)
1(0)
= F(O) = - 2~ F"(O).
To obtain the inversion formula for an arbitrary function I. we use the fact that. as in the two-dimensional case, the Radon transform commutes with the motions of Euclidean space. Introduce the mean value of the function lover the spheres centered at a fixed point x, (2.5)
F(x; r)
=..!.. f I(x + rw) dw. 471' 11",1=1
where dw is the surface area element of the unit sphere. Further. let F(.r; p) be the mean value of the function 'RI over the planes equidistant from the point x, i.e., (2.6)
F(x;p)
= 41 f 'RI(w. p+ (w. r.»dw. 71' 11",1=1
Then it fonows from (2.4) and from the fact that the Radon transform commutes with the motions of Euclidean space that
1 :;" I(x) = F(x.O) = - 271' r" (w:O). Thus, we have proved the following assertion. THEOREM 2.1. The inversion lormula lor the Radon tronslonn on three-dimensional Euclidean space has the lollowing lorm:
(2.7)
I(X)=-SI 2 f ('Rf);(w;(w,r.»dw. 71' 11",1=1
We see that the three-dimensional case substantially differs from the twodimensional one. In contrast to the two-dimensional case, here the inversion formula is local. In other words, to reconstruct a function I at a point x. one must know only the integrals of I over the planes close to x (in fact, over the planes in an infinitesimal neighborhood of the set of all planes passing through x). In passing, we have obtained the following relations between the averages F
2. RADON TRANSFORM IN THREE-DlMKNSIONAL SPACE
P of the fuu('tions f
and
aud
'RI.
respectively «2.8) is the S()-called
9
Asgeirsson
relation for three-dimensional spare): F(x;p)
(2.8)
= 2r. I~Q F(x, r) rdr, ;pl
F(r) ;::
(2.9)
_..!.. F;(z: r) . 2r.
r
(Tbese equations are obtaiood frolD (2.2) and (2.3) by traMlation.~) The Radon transfonn of a function I is related to its f'(mricr trallsform:F f by the following formula:
:F I(Aw)
(2.10)
= (211')-3/2 /
+00 -C
(lwl = 1).
'R/(w, p)ei).Pdp
Starting from this equation and using the inversion formula for the Fourier trailSform. one cau obtain another proof of the inversion formula for the Radon transtonn, namely, I(x);:: (2r.)-3/2
= ~ (21r)-3 / 2
f :F/(~)e-i{(.~} d{
Ja
3
21.1... /+CIO :F I(Aw)e-'>'(""~) A2 dw dA /+00 f /+00 'R I(w. p) c..\ z» A2 tip dw dA 1=1
-oc;
;:: ! (211') -3/2 2
= 4\ 1r
(p- (....
-oc;
Jf..JI=1
-0:
/1'00 j+OCl F(x, p) et>.p A2 dp dA = -:!. F;'(x, 0). -0' dw for the space. REMARK. In the l.wo-dimerunOlUll case, the original functioD I is recoust.ructed from 'RI by a uonlocal formula. H~-er, the function ..rA/. where ~ stands for the Laplace operator, is reconstructed from 'R I by a local inversion formula:
../is. is an t>lliptic pseudodifl'erential operator, the singularities of the fmIC../is. I and I coincide. This fact is sometinle9 tlsed in tomography to n."CODstruct the singu1arities of the fWlction I from the kJ1oy.·n Radon transform 11. f. SiJl(~
tioll8
2.2. Radon transform iD the affine space. To define the Radon transform in three-dimensional 8p8(.'C, it suffiCCtl, as iD the two-dimen.'lional C88t", to have an affine rather than Euclidtlall structure. Let us define planes by equations of the form {IXI +~2X2 +~3Z3 -
and equip every plane ~1 Zl dXI d.r2
- P = 0 by a measure d#.~ sudl tbat p) dp~. For the coordinates on t.he plane one can
+ ~ X2 + e3 Z3
= d({l Xl + {2 X2 + {J Xa -
P= 0
I. RADON TRANSFORM
10
take any pair written as
Xi, X j,
i, j
= 1. 2, 3. and the measure dp.( in these coordinates can be (k"" i.j).
i.e., it depends on the equation of the plane. We define the Radon transform of a function relation
I
OD
the affine space by the
(2.11)
'RI(>'{; >.p)
= 1>'1- 1 'RI(e:p)·
e,2',
Thus. the function 'R I depends not only on the plane el XI + + e3 X3 - P = 0 but also on the choice of the parameters {I. e2, {3. p. For lel = 1, the function 'RI coincides with the function 'RI(w.p) introduced earlier in the Euclidean case. The expression for 'R I is frequently represented in the form
= JR3 ( l(xI. x"X3) 6(el.EI + e,x, + {3 X3 -
'RI({I, e"e3; p)
where the definition of 6({1 XI dimensional case. THEOREM 2.2.
+ {, X2 + e3 X3
-
p) dx l dx, dx3,
p) is similar to that in the
The inversion formula for the affine Radon transform
t~
(2.11) is
(2.12)
where
wee) = el d{, " ~ + e, d{3 "d{1 + e3 d{1 d{,. and r is an arbitmry surface in R3 \ 0 whose intersection with almost every my starting from the origin 0 consists 01 one point. As in the t'flo'lHlimensional case, the inversion formula (2.12) follows from two simple facts: first. the integral (2.12) does not depend on the choice of the surface r, and second. if r is the unit sphere, then (2.12) coincides with the inversion formula (2.7) for the Euclidean space. 2.3. Radon transform for a space of arbitrary dimension. The definition of Radon transform for R' and R3 can be extended to spaces of arbitrary dimension. Namely. the Radon transform of a function / e S(Rn) is defined by the formula (2.13)
'R I(e. p)
=
1 R"
I(x) 6( (e, x) - p) dx,
where (e. x) = EeiXi. dx = dx l one variable. The definition of 6( and n = 3.
··
ee(Rn),\O.
peR.
·dxn , and 6(P) stands for the delta function of p) is similar to that given above for Tt = 2
(e. x) -
2. RADON TRANSFORM IN THREE-DIMENSIONAL SPACE
11
We present without proof the inversion formulas for the Radon transform. H
!.p
= 'R. I, then
(2.14)
=
for odd ri, where (7«() ~""l(-I);-l{.Aj;ei~j and"Y C R" is an arbitrary surface intersecting once every ray in Rn issuing from the origin 0;
(2.15)
f(x)
=
(-1)" /2 1!~ a,,-l.p) =
(4.1)
1>'1- 1 'R/(F..
p).
f\u1;her, these functions are infinitely differentiable on (R2 \ 0) x R and rapidly decreasing with respect to p 88 P - 00 together with all derivatives. and this decrease is uniform with respect to F.. IF.I = 1. The following question arises: Are there other conditions on the functions 'RI in the image of S(R2)'! First we note that
I
(4.2)
+
CXl
'R. I(F., p) dp =
-00
( JR
I(x) dx.
2
i.e.• the left-hand side does not depend on F.. This is closely related to the classical Cavalieri principle. which enables one to calculate the area of a plane figure in terms of the lengths of the intersections of this figure with a pencil of parallel lines of any given direction. Namely, if / is a characteristic function of a plane figure. then 'RI(~. p) is equal to the length of the intersection of this figure with the line ~IXl + ~2X2 - P = 0 for any I~I = 1. and thus relation (4.2) expresses the area of a plane figure in terms of the lengths of the intersections of this figure with a pencil of parallel lines. In particular, the left-hand side of this relation does not depend on the direction of the pencil. For this reason, we refer to both the condition that the integral r~= 'R I(F.. p) dp does not depend on F. and some generalizations of this condition as Cavalieri's conditions. Consider the moments of the function 'R/(F.. p) with respect to p. (4.3)
l,,(F.) =
i:oc, 'R/(~.
p) plc dp.
k
= 0.1.2, ....
Substituting the expression for 'RI in terms of the original function I(x) into (4.3). we obtain (4.4)
1
+00 'R I(F.. p) pt dp =
-00
(
~2
l(xHF.. x)1c dx.
Relation (4.4) implies the following property.
Cavalierl's conditions. The integralllc(F.) is a homogeneous polynomial 01 degree k in F.1o F.2 (k = 0.1.2 •... ).1 lTbeIIe CODditions (for affiDe apaces or arbitrary dlmensioD) were first formulated iD [8).
4. CAVALlER1'S CONDITIONS AND PALEY. WIENER THEOREMS
IS
We gi\'e an interpretation of the integrals J,.({) in the language of the Fburier tran.'lfonn F I of thc ftmction I. It was shown in § 1 that the ftmctions :FI(~) and 'R. I(f.. p) are related as £0110\\'8:
F I().f.)
= (2r.)-1
f ..
oo
_'.Xl
'RI(E.. p)ei~p dp
(f. :/: 0).
This implies tbe fonnula
1,,(0 = 2r.i-"
(4.5)
8*~~~).f.) I~=o'
Assume now that the function F I is rapidly decreasing and infinitely diJl'ercntiable at the point 0 Dl any direction, i.c.. the ftmction F I().£.) is in6nitely cWW.rentiablc with respect to ). at the point ). = 0 for any ~ :f. O. We show that Cavalierfs conditions are equivalent to the condition that the function :F I(() Is infinitely differentiable at the point f. = O. Indeed, let :F f(f.) be infinitely differentiable at the point f. = O. Then the right-hand sidt> of (4.5) is equal to the homogeneous component of degree k in the ThyJor expansioll of F f in a neighborhood of the point f. = 0; thus. relations (4.5) imply Cavalieri's conditions. Conversely, assume that the function F I is infinitely diff(!ccntiablc at tbe point 0 in every direction and that Cavalieri's conditions are satisfied. Then it follows from the TayJor forJbllla for F I(). f.) regarded as a function of'\ that
F I().f.)
=
t
l)k
~~~).f.) I _+ 0(1).1") = (2r.)-1 ~=O
"'=0
t i:~k lk(~) k=D
+ 0(1).1")
•
for any positilo"C integer n and Bny { with 1(1 = 1. Since Ik<el is a homogeneous polynomial of degree k in ~I. ~2' tlu.'l equation can be repn>seoted in the fonn n.
F f«()
(4.6)
= (2r.}-1 L 1t=O
.1:
k' ,l,,«() + oOel"), •
We finally obtain a Thylor formula of order k for the function F ICe) at tbe origin O. Hence, F f(f.) is infinitely differentiable with respect t~ ~lo Q at the point O.
4.2. Paley-Wiener theorem for the space S(1R2 ). The assertions desai~ ing th£. images of various runction spaces under the Radon transfonn are often called Paley- Wiener t/u>orems for this trnnsfonn. TUEOREM 4.1. A functum CP({1o (2; p) on (R2 \ 0) x R is the &don tramlonn of some functiOlI I e S(lR2) if and only if
1)
ip sati~fies
the homogenetl.y condition (4.1);
2) '" is infinitely differentaGble;
:l) every denlJatire of the function t.p decrwucs for raptdly than any negative power of Ipl; 4) t.p satisfies Cavalieri's con4itions. PROOF.
I{I
= 1 as p -
00
more
Wc must prm-c the sufficiency of these conditions only. Let a function
Ip?({, p) satisfy the conditions of the theorem. Define the function F(~) on R2 \ 0 by t.he foml1lla
1 F(f.) = -2 11"
f+oc "'({, p)e -00
ip
dp.
I. RADON TRANSFORM
16
It follows from the homogeneity condition 1) that this relation is equivalent to the formula F(~~)
1 = -2
j+x
'P(~. p)ei>'p dp.
-oc
1r
Conditions 2) and 3) imply that the function F is infinitely differentiable at any point ~ ::F 0 and infinitely differentiable in any direction at ~ = 0: hence. it is infinitely differentiable at the point ~ = 0 by Cavalieri's conditions. Further, by conditions 2) and 3). the function F rapidly decreases together with all its derivatives. Thus. F E S(R2). and hence there is a function I E S(R2) such that F = F I. Then "'(~. p) coincides with the Radon transform 'R/(~. p) of the function / because the Fourier transforms of these functions with respect to p are equal to the same function F /(~ e).
4.3. Paley-Wiener theorem for the space "D(R2) of compactly supported in8nitely differentiable functions. THEOREM 4.2. A function 'P(~I. {2: p) on (R2 \ 0) x R is the Radon trons/onn 0/4 function / E "D(R2) i/4nd only i/ 1) 'P satisfies the homogeneity condition (4.1); 2) 'P is infinitely differentiable on (R2 \ 0) x R and compactly supported with respect to p: 3) ." satisfies Cavalieri's conditions. PROOF. If / E "D(R2), then the Radon transform 'RI of I clearly satisfies conditions 1) and 2). Cavalieri's conditions follow from the relation 1)(R2) C S(R2). Colwersely. let a function '" satisfy the conditions of the theorem. Then." satisfies the conditions of Theorem 4.1 as well. and hence 'P is the Radon transform of some function I E S(R2). It remains to prove that the function I is compactly supported. Let us present the proof which is due to Helgason 126). By assumption . .p is a compactly supported function. i.e ..
(4.7)
'R/(9. p) = R,/(cos9. sin 9;
p) = 0
if
Ipl > a
for some a > O. We prove that (4.8)
F(x. r) ==
L
2W
'R/(XI
+r
0088,
X2
+r
sin8: p)d9
=0
if r>
Ixl + 4.
Indeed. let F(x, p) be the mean value of the function." on the set of lines placed at the distance Ipl from the point x. It follows from (4.7) that F(x, p) = 0 for Ipl > Ixl + 4. Then F(x. r) = 0 for r > Ixl + 4 by the Asgeirsson relation (§ 1.2). Next. let us prove that if the function I satisfies condition (4.8). then the functions x, I. i = 1.2. also sati.olfy this condition. Indeed. it follows from (4.8) that
1.
I(x + y) dy
11I1>r
and hence
1.
11I1>r
/~. (x + 7/) dy =
=0
for r>
0 for r>
Ixl + a.
Ixl + a.
i
= 1.2.
.&. CAVALIERrs CONDITIONS AND PALEV-WlENER THEOREMS
17
By the Stokes theorem. this implies that
1,
fz.(x+y)dy=r 1
1111>"
2~
l(x+rw)",·;d8.
0
where w = (cos9, sinS). Thus,
12ft I(x + rw)(.l + TW), dB == Xj 12"ff I(x + rw) dJJ + r 88
12ft I(x + rw)
W;
dfJ = 0 for r >
Ixl + a,
required.
It follows from the abc»'E' that the nmction P(x) I(x) has zero mean value 0\'Cl'
the- circle (ccntered at the point 0) of every radius r > a for any polynoolial P(x). Hence, I = 0 011 this circle, and thus I vanishes in the entire domain 1.1:1 > a. 4.4. Inversion of the Radon transform of a function I e V{R2) using the moments. Let I be a compactly supported function. Then the Fourier tra.n. form :F I of I is all entire nmctiou and, by (4.6), the Ta-ylor series of the function :F f is given by the fonnula (4.9)
F
I(~) =
<Xl
(21i) -
I
L
.k
~! ft..«().
10=0
where I,,«() is the kth moment of the fulK'tion 'R. /(E., p) "Titb respect to P. which is a poll''Ilomial of degree k. TakiIJg the inversE> Fourier transform of F /, we obtain the way t.o reconstruct t.he fullction I from the moments 1". Thus, for a function I E V(JR2), we obtain an invennon formula usilJg the luomcnts I" only. TIJ.is formula substantially differs from that in § 1. Sinet' the moments I" are bU1nogencous polynomials, in order to rec:onst.ruct t.he function f, it suffices t.o know the values of I,. on an arbitrarily small arc of the circle 1f.1 = 1 or even on an infinite sequence of points (in general position). REMARK. It is worth mentioning the difI'erence bet\\'een various iDversion fo.... mulas. In an inversion fonnula v.-e have an operator defined on some space (which is generally larger than the image of the Radon transform) and determined by some kernel (generally, a distribution kernt>l). The operators in different iuversion formulas coincide on the image of too Radon troDSfonn. It follows from the Paley-Wiencr theorem for tht> space S(]R2) that every functional vanishing on the image of S(R2) is obtained from Cavalieri's CU11ditioDS, and the inversion formula for S(R2) is essentially wtiquc. Howe\'CI', the image of 1)(22) consists of compactly supported functions. and the family of annihilating functionals is richer in this case. Therefore, for V(1R2 ), it is possible to constrt1('t inversion formulas different from that in § 1. A version of such a formula, which uses the moments, was obtained in this subsc.>ction; for anotru-.r version, see 5.1.
4.5. Reconstruction of unknown directions from known 'YBlues of 'R. I. Here we present. another example in which Cavalieri's conditions arc used \"Tben inverting the Radon transform. IT the Radon transform 'R./(;p, p) of a compactly supported function f is known for some set of angles 'Pt. i = 1,2, ..• , n, then tIle function I ('an t~ reconstructed approximately.
1. RADON TRANSFORM
18
The following situation arises in electron microscopy of ribosomes: the values of 'R J( I{) , p) are known for some angles I{) 1 , 1P2, ••. ,I{)n and for all p, i.e., n functions of p are given, but the values of the angles I{)i themselves are unknown. Therefore, to reconstruct J, one must first reconstruct the unknown angles. or. to be more precise, to reconstruct these angles up to a common rotation. Goncharov (21) showed that using the fact that J is compactly supported and applying Cavalieri's conditions, one can solve this problem in the generic case provided that n is sufticiently large. Let J be a compactly supported function, let 'R J be the Radon transform of J, and let 1,,(1{) be the moments of the function 'RJ (k = 1,2, ... ). By Cavalieri's conditions, the moment I" is a homogeneous polynomial of degree k io COSI{). siol{); in particular, 11(1{) = ~1 COS I{) + ~2 sinl{),
(4.10)
12(1{)
= JJl cos2 I{) + 1'2 cos I{) sin I{) + 1'3 sio2 1{).
Note that the expressions for 11 and 12 can be represented in the fonn 11(1{) = a cos(1{) - a), (4.11) 12(1{) = b cos 2(1{) - /3) + c. All parameters in (4.10) and (4.11) depend on J. Suppose that 11(1{) ~ 0 and h(l{) ~ 0 and consider the following parametric curve r / in the plane (YIt 112):
Yl = /1 (I{) , LEMMA.
r/
1/2 = 12(1{).
is an algebraic cuMle oJ degree Jour.
Indeed. the degree of the curve r J is determined by the number of points in the intersection of this curve with a line in general position A Yl + B 1/2 + C = 0, i.e., the number of solutions of the system of equations
{
A (~1 tl + ~2 t2) + B (1'1 t~ + 1'2tlt2 + 1'3t~) t¥+t~=1.
+ C = 0,
We emphasize that different curves correspond to different functions J. A curve of degree four is determined by 15 points in general position on this curve (the number of coefficients of a polynomial P(Yl. 112) of degree four is 15). Therefore, if the functions 'RJ(I{), p) are known for 15 values of I{). then 15 points on the curve r / are known, and hence the curve is known. For a known curve r /' the coefficients a, b, c in (4.11) are defined by the formulas 1 1 a (Yl)max, b 2«Y2)max - (Y2)mln). c = 2«1I2)max + (lI2)mln).
=
=
Since a shift of the parameter I{) does not change the curve r /. we can assume that I{)I = O. Thus, we know cosa, cos2/3 and COS(l{)i - a), COS2(l{)i - (J) for i = 2,3, ... , 15. Hence, we can readily see that iJ a, /3, and I{)i are in general position, then the angles f/J2. f/J3 ••••• 1{)1Ii are detem&ined uniquely. REMARK. The number of angles n = 15 is excessive because the above construction can give only curves r / of special form rather than all curves r of degree four. In fact, it is enough to know only five angles I{)It 1{)2, .•. , 1{)5 (1{)1 = 0) for the
5. POISSON FORMULA AND THE DISCRETE RADON TRANSFORM
19
following reason. Since the angle I{)l is equal to O. it follows that the coefficients ~l and III in (4.11) are known. As a result. we obtain an overdetermined system of 2·4 = 8 equations in seven unknowns ~2. 112. 113 and 1{)2, 1{)3. 1{)4. 1{)5. One can show that this system has a unique solution in the generic case. REMARK. The results of this section can be immediately extended to the multidimensional Radon transform.
5. Poisson formula for the Radon transform, and the discrete Radon transform2 5.1. Poisson formula for the Radon transform on a plane. Along with Fourier integral, we have its discrete analog, namely, Fourier series. We shall see in this section that there is a discrete analog of the Radon transform, which is related to Fourier series in just the same way as the Radon transform is related to the Fourier transform. Recall that the Fourier series can be obtained from the Poisson formula for the Fourier integral. Namely, let I be a smooth compactly supported function on R2 and let F be the Fourier transform of I. Then, by the Poisson formula, +00
(5.1)
+00
L
L
I(Xl+k l .X2+ k2)=211' FI(211'k},211'k2)e-21ri(kl~I+k2%2) k l .k2 =-oc "1.k2=-OC
(both series are absolutely convergent). If a function I is supported on the square 0< X}, X2 < 1, then only the term with (klo k2) = (0, 0) on the left-hand side can be nonzero for points (Xl. X2) in this square, and we obtain a representation ofthe function I by a Fourier series. +00
(5.2)
I(x}, X2) = 211'
L
F/(211'k lo 211'k2) e- 21r i("1 %1+"2%2).
Thus, a function supported on the above square can be represented by both the Fourier integral and the Fourier series. Let us present an analog of the Poisson formula for the Radon transform. Introduce the subset A of points (k l • k2 ) E Z2 such that either kl > 0 and the numbers kl' k2 are coprime or (klo k2) = (0, 1). THEOREM 5.3. Let I be a smooth compactly supported function on the plane R2 and let 'R./(~I'~2;p) be the Radon translorm 011. Then
+00
(5.3)
L
I(XI
+ kl' X2 + k2) +00
=I('R.f)+
L (L
'R./(klok2;kIXI+k2x2+m)-I('R.f)),
("I."2)EA m=-oo
where (5.4) 2 See (11) and the computational applications in (20).
I. RADON TRANSFORM
20
PROOF. Recall that the Radon transform is obtained as a composition ofthe two-dimensional and onMimensional Fourier transfonns, namely.
'R1(~1'~2;P) = i~ FI().~)e-i>'P d)'.
(5.5)
where F 1 is the Fourier transform of I. Therefore. formula (5.3) can be obtained by combining the Poisson formulas for the two-dimensional and one-dimensional Fourier transfonns. Note first that each nonzero point of Z2 can be represented in the form m(klo k 2). where m E Z \ 0 and (klo k 2) E A. Thus. the Poisson formula (5.1) can be rewritten as +00
E
(5.6)
1(%1 +klo %2+ k2)
= 211'{FI(0)
kl.k2=-00 +00
+
E (E
FI(211'mk lo 211'mk2)e-2.. im(kl ZI+k2 Z 2 )
-
FI(O»)}.
(kl.k2)EA m=-OCl
It follows from (5.5) and the Poisson formula for the one-dimensional Fourier tra.nsform that +~
L
'RI(~; p+ m)
=
+~
L
FI(211'm~)e-2#i"'P.
m=-oc
Hence, +00
211' (5.7)
L
FI(211'mk lo 211'mk2)e-2 .. im(klzl+k2"2)
m=-oc
+00
L
=
'R/(k lo k2;kl Xl
+ k2%2 + m).
m=-oo
Since
FI(O) = (211')-1
i::x. 'R1(~.p)dp
for any {. relation (5.3) follows from (5.6) and (5.7). COROLLARY. 11 a function 1 then
u supported on the unit square 0 < XI.
%2
< I,
+00
(5.8) I(XI.%2) = I('R/)
+
L (L 'R/(klok2;kl%l+k2x2+m)-I('Rf))
«klo k2)EA m=-oo
lor any point (XIo X2) 0/ thu square, and the series on the right-hand side is absolutely convergent. Thus. we have obtained another inversion formula for a smooth function 1 supported on the unit square. Moreover. if I('RI) = 0, then the function 1 can be reconstmcted at a point % provided that only the integrals of lover the lines with rational angular coefficients passing through X and the parallel translations of these lines corresponding to the integral values of Il.p are known. Note that the distance between the parallel lines kl XI + k2 X2 = P and kl XI + k2 X2 = P + 1 is equal to (k12 + k22)-1/2 (in the ordinary Euclidean metric). The presence of integrals over shifted lines re8ects the fact that the inversion formula is nonlocal.
5. POISSON FORMULA AND THE
DL~
RADON TRANSFOR1..t
21
REMARK. Formula (5.8) can be used to munerically invert the Radon transform for not DeCeSIlMily smooth functions. &.2. Discrete Radon transform; relation to the Fowler series. Let I be a function supported OD tbe unit square 0 < Xl. X2 < 1 and let 'RI be the Radon transform of /. Starting from (5.8), it is natural (by analogy with the Fburier series) to refer to the nmctioD F on A x R given by the formula
F(klo ~; p) =
(5.9)
+oc
L
'R/(k 1• k,;p + m) -l('R1)
m=-oc
as the discrete Rtulon tnms/oma of f. This function is periodic in p with period 1. By (5.8), the function f can be expressed in terms of its discrete Radon tramrform as fOllows:
(5.10)
f(Xl, X2)
=[('RI) +
L
F(k" k,;kl Xl
+ 'v.lX2),
0 < Xl, X2 < 1.
(k:."z)EA
The ftmction F is related to tbt> FOurier st'Jies of the function f iD exactly the S8Jlltl way in which the Radon transform 'R! is related to the Fourier tnmsfonn :F!. Namely, let f(Xl. X2)
=
+00
L
a(ult n,)e2I1'j(ftl:r:I+"2~)
be tbe expansion of / in the Fourier series on 0 < Xl. F(kl, k,i p)
X2
< 1 and let
+00
L
=
b",(kt. k:z) e2ll'i mp
'n=-OQ
be the expansion of F in the Fourler series with respect to P OIl the iuterval 0 p < 1. Note that bo(ka. k2) O. Substituting these series into (5.10). we obtain
=
+00:: E
k, then the result is a differential (m - k)-form OD the variety of submanifolds rather than 8 numerical function. In this subsection we consider the integration of a 2-form w over the lines on R2; the result of this integration is a 1-form 'Rw on the manifold of lines. The informal idea to evaluate 'R.w is as follows. Th find the value 'R.w(l. r), where I is a line and r is a tangent vector to the manifold of lines, one must infinitesimally shift the line I in the direction of r and integrate w over the corresponding infinitesimal 2-dimensional domain. The same idea works in the general situation as well. We proceed with the corresponding computations. First we define the Radon transform of a differential 2-form For that. we introduce the three-dimensional ftag manifold A whose elements are pairs fonned by an oriented line on the plane and a point on this line. Since points on the plane are given by coordinates (Xlo X2), and oriented lines by coordinates (""p), it follows that one can equip A with two coordinate systems. The first system is given by the coordinates (XloX2' "'). where (XI.X2) determine a point on the plane and '" gives the direction of a line passing through the point, and the other system is given by the triple (",.p,t), where (""p) detennine an oriented line on the plane and t gives a point on this line. The coordinates (x I , X2. "') are expressed in terms of the coordinates (",.p,t) by relations (7.2). The inverse relations look as follows: t = X2COS", - XI
sin""
21
1. RADON TRANSFORM OF DIFFERENTIAL FORMS
Let US \iew tht' 2-form w = 1dx l A.dx2 85 a differential form on the manifold A with coordinat~ (X 10 X2.1P). Th define the Radon traosform of w, we 6rst repretiC1lt the formw in the coordinates (rp,p, t) by using (7.2) for Xh X2 and the corresponding expressions for d.r.1t dx2, dXl dX2
= - sinlPdt - t cos",d", - P 8inlP~ + coslPdp, = cos ip dt - t sin IP dIP + P cos IP dIp + sin IP dp.
We obtain
w= -f(-t sinlP+PCOSip, t coscp+psiu'P)dpA.dt
- t/(-t sin 1,0 + p oosrp, t coscp + p sinrp}~ A. tit + ... where the dots stand for tht' terms that do not contain dt.
,
We denote by Wo the differential form obtained from w by deleting all terms that do not contain dt. For fixed values of lp, p. dip. and dp, the resulting form Wo is 8 differential I-fonn on the oriented line corresponding to (rp,p). Integrating the form over the line, we obtain a differentiall-form on the manifold of oriented lines; this form is caUed the Radon trall8fOml of the original 2-form w and is denoted by 'Rw. Thus, WP ohtain the following definition. DEFINITION. The Radon transform
w
'Rw of a differential 2-form
= I(xl, X2) dXIA. dX2
on the plane is the differential 1-form 'R.w=a(rp,p)dp+b(IP.p)~
(7.5)
on tlw manifold H of oriented
a(.p.p)
(7.6)
1 =- 1
=
lin~
on the plane, where
+01:.
-00
1(-tsin O.
then, using (8.7), we obtain a new inversion formula for the affine Radon transform. which differs from the inversion formula in § 2. Namely, let 7rz C (R4)' be the subspace orthogonal to (Xl,X2,X3, 1) E R4, P(7rz ) the projectivization of this subspace, and 1z an arbitrary section of the bundle 7rz - P(7rz ). If lE S(R3) and if I{J = 'R. I is the affine Radon transform of the function I e S(R3). then I(x)
= -(27r)-2 c- 31
t
'Y~ i,j-1
(i(j
C::~~ 111,(,d11,d111· J
REMARK. A more general class of inversion formulas for the affine Radon transform can be obtained by taking a smooth vector field 11 e (R4)' \ 0 transversal to 7rz \ 0 (i.e., such that (x, «11» > 0 for any 11 E 7rz \ 0) at the points 11 E 7rz \ o. To every such field there corresponds an inversion formula for the affine Radon transform.
'{)::; (2nr" ~
I
X/(F., p)eiRe'\Pdpdj).
Hence, by the inversion formula (or the one-dimensional Fourier transform we obtain
X/CF..
p) = (2n"),,-2
~
I
F/(>.{)e- iRA!1." tU
tU.
I. RADON TRANSFORM
40
9.3. Inversion formula for the Radon transform. It is known that the operator :F- I inverse to F is
(9.6)
F-I 0 and t~ = 0 for t < 0; the ~,'s stand for arbitrary complex numbers. For Re ~ > 0 the function ~ is defined 88 a classical funttion; it extends to a distribution meromorpbic in ~ (see (10]). We establish a relation between this Jobn transform and the Gauss bypergeometric function F(a.b,c; z).
46
2. JOHN TRANSFORM
By definition, the function "'). is given by the integral "').(010 02, Plo.82) =
100 (01X3 + Pd!I-1 (02X3 + ~)!2-1 X~3-1 dx
3•
The integral converges if Re ~i > O. i = 1,2,3, and Re (~1 + ~2 + ~3) < 2. and in this domain the integral defines an analytic function of ~1' ~2' ~3' Therefore, the integral can be defined for any values of the ~i 's 88 the analytic continuation of this function with respect to the ~i 's. Note that the interval on which the integrand is nonzero depends on the signs of the numbers 0i and Pi, i = 1,2. To be definite, consider the domain on which 0i < 0 < Pi, i = 1,2, in the manifold of lines H. In this domain. changing the variable X3 = t, we can reduce the integral to the form
-!!
1/)).(01002,P.. .82)
= p~I-1 P:2+).:s-11 0 2r).:s
1
1(1- xt)!I-l (1 - t»).2- 1 t).:s-1 dt,
:!t.
where x = Compare this expression with the weU-known representation of the Gauss hypergeometric function F( a, b, e; x) for Ixl < 1 88 the EuJer integral. F(a,b,c; x)
= r(b)~~~ _ b)
11
t"-1 (1- W-"-1 (1- xt)-a dt;
this integral converges for Re c > Re b > 0, and for the other values of b and e it must be treated 88 the analytic continuation with respect to b and e. We conclude that, for 0i < 0 < Pi, i = 1,2, and < 1. the function 1/)). can be expressed in terms of the hypergeometric ftmction,
:!ta
"').(0., 02, PI,.82)
= f(~2) f(~3) r(~2 +~)
a).I-1 a).2+).,-11 1-).3 F (_~ + 1 ~ ~ + ~ . 0 1.82) "'2 02 I , 3· 2 3. 02Pl .
"'I
This implies that F(a.b,e;x)
f(e)
= f(b)r(c-b) 1/)).(-x,-I,I,I)
for O<x< I,
where ~ = (1 - a, c - b, b). Similar expressions for the function 1/)). in terms of the hypergeometric hypergeometric function can be obtained for other combinations of the signs of 0i and Pi, i = 1.2.
1.3. Theorem OD the image of the operator.1. This subsection, together with the next three, is devoted to the description of the image of the space S(R3) of Schwartz functions under the John transform .1. First, it is clear that the functions in the image of the space S(R3) satisfy conditions of smoothness and rapid decreasing. FUrther, since the John transform acts from a space of functions of three variables to a space of functions of four variables, one can naturally expect that the functions in the image satisfy additional conditions.
1. JOHN TRANSFORM IN THE REAL AFFINE SPACE
47
We prove that each function I;' in the image of the John transform satisfies the foUowing system of differential equations:
( 6) 1.
CP'(J 80" 8/Jj
CPrp
..
= 80j 1}{3. '
I,}
123
= , ..
Indeed, if I;' = .1/. then it follows from (1.1) that 1}2rp(a, P) =
aa. apj
j+«> t I" -00
(at + IJ) dt
z • .zj
•
a!.'&fJ,
Thus. the function is symmetric with respect to the transposition of t.he indices i and j. We claim that relations (1.5) are not only necessary but also suftlcient for a function '" to belong to the image of the space S(J!l3). Let us proceed with detailed definitions and statements. Denote by H the manifold of lines in a 3 and by G ~ p2 the manifold of one-dimensional subspaces in 3 , i.e., the submanifold of lines passing through the point O. The manifold H is equipped with tJre structure of a canonical vector bundle over G. Namely, the projection 11' : H - G takes every line h eH to the parallel subspace a e G. Thus, the fiOO of the bundle 11' over a is the quotient space Ha =R3/a . Since H is a vector bundle with compact base space, it follows that the notion of rapidly decreasing function OIl S is \\-eJl defined, namely, a function on H is cal1ed mpidly decreasing if this function rapidly decreases on each liOO of the bundle
a.
H-G. DEFINITION. Denote by S(H) the space of functions rp(a, (1) on E satis£ying conditioDS (1.2) and (1.3) and such that the function ",(a,p), 10'1 = I, regarded as a function 00 H, is infinitely differentiable and rapidly decrea5ing together with aD its derivatives.
It immediately follows from the definition of the operator .1 that the image of the space S(J!l3) under the John transform belongs to S(H),
.1: S(R3) - S(H). THEOREM 1.1. A /unction r.p e S(H) belongs to the image of the space S(23) under the John tmnsfcmn .1. I.e., is Tl!p1'esentable in the fcmn 'P = .1/, where / e S(IR3), if and only if this function 8atisfies system (1.6).
The proof is given in 1.4-1.6. REMARK. It is interesting to compare this desttiptioo of the image of the John translform with the description of the image of the Radon tnmsform given in Cbapter 1. In the latter case, the main component in the description of the image is the nonloc:al Cavalieri conditions. whereas in the former case, there are only local conditions (1.6). Of coun;e. it is p06Sible to write down analogs of Cavalieri's conditions for thE' John transifrom, but they wiU be consequences of the differential equations (1.6).
We now present another equivalellt formulation of the image theorem. Introduce the operator" which takes e\.'ery smooth function ",(a, {3) on E to the following
2. JOHN TRANSFORM
differential I-form on E: ~
8~
It~ = 8{J1 do.
(1.7)
8~
+ 8fJ2 do2 + 8133 do 3 •
LEMMA. If ~ e S(H), then It~ con be pushed down from E to the manifold ii of oriented lines in R3.
Indeed, it follows from (1.2) and (1.3) that the form It~ is invariant with respect to the transformations {J - {J + ato and a - .\a, .\ > o. Note that the form ~ changes the sign if the orientation of the line alternates (a- -a). For any point x e R3, denote by HZ the submanifold of lines p88Sing through x, and by Itz~ the restriction of the form Itz~ to HZ. PROPOSITION 1.1. Condition (1.5) is equivalent to the condition that the differential form Itz~ on the submanifold HZ is closed for any x e R3, i.e., to the condition dttz~ = O.
PROOF. We have
_
Itz~ -
~ 8~(a,{J) IJla.
L-
~.
i=1
Hence, the exterior differential of the form
dltz~
~ 1P~(a,{J) = L80.8/3. iJ I,
I_
l3-z
=~(IP~(a,{J) _
f..2 a2 + >"(hl +~» 001" dPl (see the proof of Proposition 2.2). Since n Ip= 0 by assumption and >.. is arbitrary, it follows that al = a2 = 0 and hi = -~. Thus, we have proved that the form n can be represented as
n = W:i(o, P) 001 "002 + "'I (0, P) (do:l "dl12 Since
0021\ dpt) + t/Jo(o, P) dPl 1\ dl12.
n is closed, the functions "'i are related as follows: i = 1,2.
(2.5) Therefore, the function t/Jo satisfies the equation
fPt/Jo 801112
=8
fPt/Jo 2 8 PI .
Q
Hence, by the theorem on the image of the John transform (see § 1), this function is the John transform of some function f e S(R3): 7/10(0, P)
=
i:
f(ol Z3
+ Plo 02 Z3 + /J:z, Z3) dz3.
We now prove that n = .1w, where w = f(z)dzl 1\ dz2 1\ dz 3 • Indeed, it immediately follows from formulas (2.3) defining the coefficients l{Ji of .1w that t/Jo = 1{Jo. Further, these formulas, together with (2.4) and (2.5), imply that ~
Ii,
i = 1,2. Hence, since "'I and 'PI are rapidly decreasing, it follows that "'1 We can similarly see that W:i = 1{J2, and hence n = .1w.
=
= I{Jl'
Note that the form w = f(z) dz1 1\ dz2 1\ dz3 can be reconstructed from only one coefficient I{Jo of the form .1w because the function 'Po is the John transform of the function f.
2. JOHN TRANSFORM
2.3. John tr8DSfonn of 2-forms OD Ra. Consider an arbitrary differential 2-form w
= hex) dx2" dx3 + hex) dxa "dx. + h(x) dx. "dx2
with coefficients in SeRa). In the coordinates 0.,02, fJh 132, X3 on the flag space this form can be represented as
w = h (X3 do2 + dfJ2)" dxa + 12 dxa " (X3 do. + dlh) + fa (01 dx3 + Xa dol + dl3d" (02 dxa + X3 do2 + dfJ2). According to the definition in 2.1, the John transform of the 2-form w is the I-form
:Iw = al dol + a2 do2 + bl d131 + b, dfJ2
(2.6)
with the coefficients al = 021/Js -
(2.7)
= -0. 1/Js + '!/Jl' b, = -01 'Pa + 'Ph
t/I-z.
a2
61 = 02 'Pa - 'P2,
where
i = 1,2,3,
(2.8) (2.9)
i
= 1,2,3.
Consider first the problem of reconstructing the form w from its John transform
:lw. This problem can be reduced to the corresponding problem for functions
by
using the following assertion. PROPOSITION
2.3. The functions 'Pi are e:rpretlsed in terms of the coefficients
01 the lorm :Iw as lollows: (2.10)
lJ~ lJo l 'Pa = - lJ0 2 lJfJ2 '
'PI
= 01'P3 + b"
'P2
= 02'P3 -
6•.
Relations (2.10) immediately follow from (2.7)-(2.9). Since the functions 'Pi are the John transforms of the coefficients I. of w. it suffices to apply the inversion formula for the John transform of functions to the 'Pi'S to reconstruct the coefficients li. We now proceed with the description of the image of the John transform :I. To this end, we first establish relations for the coefficients of the differential form
:lw. PROPOSITION 2.4. The coefficients ai, lollowing relations:
~
01 the differentiallorm :Iw satisfy the
(2.11) and
(2.12)
l _ a(lJa lJfJ2
Dbl ) _
lJ02 -
0
,
where
a = lJo)lJ{3, lJ2 -
lJ2
lJ02 lJfJ) .
Indeed, relations (2.11) immediately follow from (2.7), (2.8), and (2.9). Relation (2.12) follows from the above equation 'Ps = ~ - ~ because a'P3 = O.
,. JOHN TRANSFORM OF DIFFERENTIAL FORMS ON
57
R3
PROPOSITION 2.5. Relations (2.11) are equivalent to the condition that the restriction .1w Ip 0/ the differential/onn .1w to the sulnnani/old o/line8 that all belong to a plane P is a closed/onn on P, i.e., d.1w Ip= o. PROOF. We have
d.1w = (8b 1 _ 8a 1) do 1 1\ d/31 + (8~ _ 8a l ) dol 1\ df32 8al
8/31 8a l 8132 bt _ 8a,) do, 1\ d/Jl + (~ _ 8a,) do, 1\ df32 + .... 8a, 8/31 8a, 8132
+ (8
Therefore, relations (2.11) are equivalent to the condition that the form d.1w contains no terms with do i I\d/3i' i 1.2. and the coefficients of do l l\dfJ2 and do,l\dPl differ in the sign only. In turn, according to 2.2, this condition is equivalent to the one formulated in the proposition.
=
THEOREM 2.2. Suppose 0 is a differentiall-/onn with infinitely differentiable rapidly decreasing coefficients on the manifold o/line8 in R3,
o = a1 do 1 + a2 do, + b1d/31 + b, df32. For the fonn 0 to be the John tmnsfonn 0/ a differential 2-/onn w on R3 with coefficients in S(R3). it is necasary and sufficient that its restriction Olp to submanifolds Po/lines that belong to the same plane be closed /onns for any plane and that the coefficients % satisfy the additional condition (2.12). PROOF. We have already proved the necessity. Let us prove the sufficiency. Suppose a differential form 0 satisfies the hypothesis of the theorem. Then the coefficients of this form satisfy conditions (2.11) and (2.12). Define functions !Pi. i 1,2,3, by (2.10). It follows from (2.12) that 61{)3 O. We prove that also 6IPl = 0 and 6!p2 o. Indeed,
=
6!p2
=
=
= a, 6IP3 - :~ -
6bl
8 ( 8b1 80 1 ) Blb1 Blbl 8a2 - 8132 - 8a18f32 + 8a,8/31
= - 8PI
!!.... (8a 1 _ 8b1 ) = 8132 8/31 8al One can similarly prove that 6IPl = O. =
o.
This implies (see § 1) that the functions IPi are the John transforms of some functions /i E S(R3), i.e., the functions !Pi are given by (2.8). Introduce the differential form w = h(x)dx2 Adx3
We prove that 0
+ h(x)dx3 Adx1 + la(x)dx 1 I\dx,.
= .1w. Indeed, .1w = 41 dol + 4, do, + b1 dPl + ~ df32,
where
bl
= a2 IP3 -
and the functions !Pi and w by (2.8) and (2.9).
tPi
!p2,
b, = -al !P3 + IPI'
are expressed in terms of the coefficients of the form
2. JOHN TRANSFORM
58
By Proposition 2.3,
861 - -Bal = -80 2 B/J2'
= 02 'P3 - b- I . Comparing (2.8) and (2.13), we conclude that bi = bi, i = 1,2. Hence, compar-
(2.13)
'1'3
'1'1
= 01 '1'3 +~.
'P2
ing relation (2.11) for 0i and bi with the similar relations for lit and bi , we see that = i,j = 1,2. Since the functions a. and ~ are rapidly decreasing, we have Oi, i = 1,2, and therefore {} = .1w.
I;-
Bt,
a. =
2.4. John tr8D8form of I-forms on R3. Consider a differential I-form on
By definition, the John transform 'I' manifold of lines in R3:
;p(o,P)
= .1w of "" is the following function on the
= L:OQ [h(ol %3 + PI. 02%3 + /J2. %3)01 + 12(01 %3 + Plo 02 %3 + fh, %3) 02 + 1a(01 %3 + PI. 02 %3 + /J2. %3)J dx 3 •
We find the kernel and the image of the transform .1. PROPOSITION 2.6. The kernel Ker.1 of the John transform coincides with the subspace of all exact differentiGll-forms on R3 with coefficients in S(R3). PROOF. Obviously, each exact I-form W on R3 belongs to Ker.1. Conversely, let w e Ker.1, i.e., .1w = O. Then .1 dw = d.1w = O. Since the John transform .1 has zero kernel on the differential 2-fornlS. it follows from the relation .1 dw = 0 that dw = 0, i.e., the form w is closed. Since the form w is closed, it is exact. PROPOSITION 2.7. A function 'I' in the Schwartz space on the manifold of lines H belongs to the image Im.1 of the John transfoma of l-fofT1&8 if and only if the differential dip belongs to the image of the John transform of differential 2-fofT1&8 onH.
PROOF. Let w e Im.1, i.e., 'I' = .1wl, where WI is a I-form with coefficients in S(R3). Then dip = d.1Wl = .1dwlo i.e., dip e Im.1. Conversely. let dip e Im.1, i.e., dip = .1W2, where W2 is a 2-form on R3 with coefficients in S(R3). Then .1~ = d..11M2 = 0, and since the John transform has zero kernel on the a-forms in R3, it follows that ~ = 0, i.e., the form W2 is closed. Since each closed 2-form on R3 with coefficients in S(R3) is exact, it follows that W2 = dwlo where Wl is a I-form on R3 with coefficients in S(R3). Thus, dip = .1dw I = d.1WI, and hence d ('I' - .1WI) = o. Therefore, 'I' = .1WI • THEOREM 2.3. A function 'I' in the Schwarlz space on H is the John transform of a I-foma on R3 if and only if this function satisfies the equation
(2.14)
2
~ cp = 0,
where ~
IJ2
= IJolfh -
IJ2
1Jo2PI·
PROOF. It follows from Proposition 2.7 and the description of the image for the John transform of differential 2-fomlS (see 2.3) that 'I' e Im.1 if and only if the coefficients aj = and bi = i = 1,2, of the differential form dip
8£
Gf"
69
3. JOHN TRANSFORM IN apS
satisfy relations (2.11) and (2.12). Obviously. relations (2.11) hold identically for the differential form dJ.p. and then (2.12) coincides with (2.14).
3. John transform in the three-dimensional real projective space In this section. the main definitions and results of § 1 are extended to t.he case of projective space p3. The passage from the affine space to the projective space has two important advantages. First, the main definitions and results in the projective case have a simpler and more invariant form as compared with those in the affine (''&se. Second, since the projective space p3 itself and the manifold of lines in ps are compact, the function spaces related to these manifolds are more uatW'8l as compared with those in the affine case. We also give a group-theoretic interpretation of the John transform.
3.1. Manifold of lines in PS. Let us define points in pS by homogeneous coordinates X1>X2.X3,X4. i.e., by vectors:E E R" \ 0 defined up to a oonzero f~ tor. In other words, the points of the space p3 are interpreted as one-dimensional subspaces of R4. To the set of points on a line there corresponds the set of onedimensional subspaces in R4 belonging to some ~imensional 8Ubspace. Thus, the lines in p3 are interpreted as two-dimensional subspaces of 4 , and hence the manifold of lines in ps is identified with the Grassmannian G2,4 of two-dimensional subspaces in 1Il4. We introduce a coordinate system on the Grassmannian G2,4. Let h E G2,4 and let. % = (%I,%2,X3,X4). Y = (Yt.Y2,1I3,Y4) be an arbitrary basis in the subspace h.
a
DEFINITION. By the Plucker COOrdifUltes of a two-dimensional subspace h in R" one means the coordinates of the bivector :E A 'g, i.e., the follcn\ing number set:
Pi;
= Xi Y; -
X; 'U.,
i,j = 1,2,3,4,
i ::f:.j.
Under any change of a basis in h, in which case the vectors % and '11 are replaced by some linear combinations of them, all PlUcker coordinates are multiplied by the same number. Thus, the Pliicker coordinates of a subspace h are defined. uniquely up to a factor. Conversely. if the Pliicker coordinates of two subspaces are proportional, then these subspaces coincide. Thus, each two-dimensional subspace of R4 is uniquely determined by its PlUcker coordinates, and they can be taken as homogeneous coordinates on tbe Grassmanni8D G2,4' Since Pli = -Pji. it foUows that wc can restrict ounlelves to the six coordinates Pi;. i < j. The Pliicker coordinates Pij satisfy the quadratic relation
(3.1) PI2P3-1- P13P24 + P14P23 = O. This relation can be obtained, for instance, by expanding the following determJnaot of order four with respect to the first two rows: Xl
X2
X3
:E4
111
'Y2
113
11..
%1
:E2
%3
:E4
= O.
111 112 Y3 Y4 We prove that, conversely, if some numbers Pi;. i::f:. j, are not simnltaneously zero and satisfy (3.1), then these numbers are the PlUcker coordinates of a subspace h E G2,4. Indeed, to be definite, let P12 ::f:. 0; then one can assume that P12 = 1.
2. JOHN TRANSFORM
60
Consider the subspace spanned by the vectors x = (1.0,-P23.-P24) and y = (0,I,P13,P14)' Using relation (3.1), one can readily see that the Plucker coordinates of this subspace are equal to Pij' Thus, there is an embedding G2,4 ....
pS
that assigns, to each two-dimensional subspace with Plucker coordinates Pij' i < i, a point of the projective space pS with the homogeneous coordinates Pi]' The image of G2,4 under this embedding is the quadratic surface (3.1) in pS. 3.2. John transform in p3. The first problem that arises when passing from the affine space to the projective one is to choose a function space on which the John transform :I must be constructed. As in the case of Radon transform, spaces of functions on p3 are not convenient for this purpose. The natural object on which the John transform will be defined is the space F( -2) of infinitely differentiable functions on R4 \ 0, i.e., on a one-dimensionaJ bundle over p3, that satisfy the following homogeneity condition: (3.2)
J(>.x)
= >.-2 J(x)
for each
>. #: O.
Let us define the John transform of an arbitrary function J E F( -2). Take he G2 ,4. Choose an arbitrary basis u, v in h and denote by s and t the coordinates of points on h with respect to this basis, x=su+tv. To a function
J E F( -2) we assign the following differential J(su + tv) (sdt - tds).
I-form on h;
It follows from the homogeneity condition (3.2) that this form is homogeneous of degree zero, and the integral of the form over a contour S in h enclosing the point does not depend on the choice of this contour.
o
DEFINITION. By the John transform of a function J E F we mean the following function of a pair of linearly independent vectors u, v E R4;
(3.3)
(:If) (u, v) =
~
Is
J(su
+ tv)(sdt -
tds);
the integral is taken over an arbitrary contour S surrounding the point 0 in the subspace h spanned by u and tI. It has already been mentioned that the integral does not depend on the choice of the contour S. The function", = :IJ thus defined depends not only on the subspace h E G2 ,4 but also OD the choice of the basis in h. It follows from (3.3) that, when passing from one basis u, v to any other basis u' = 0 u + {J v, v' = "Y u + 6 v, the function '" is transformed by the following law: ",(ou + (Jv, "YU + 6v)
= 106 -
(J"Y1- 1 ",(u, v).
In particular, if the area elements in the bases u, v and u' , v' coincide, i.e., if
106 - {J"YI = I, then ",(0 u + (Jv, "Y u + 6v) = ",(u, v). Thus, the John transform of "'(u, v) depends only on the Plucker coordinates Pi; = utV; - u]v.. i.e., this transform is a function on the quadratic cone Kc R6
3 JOHN TRANSFORM L~ Rp3
given by (3.1). Obviously,
K\ 0
8a
is a one-dimeusional bundle over the Grassmanniao
G2,4. In what follows, \\'e often write :p(p) instead of tp(u, v) having in mind that P is a point of the cone K. i.e., a point of R6 with coordinates Pij, i < j, satisfying (3.1). DEFINITION. Denote by 'H( -1) the space of COO functions tp on K\ 0 satisfying the homogeneity condition
(3.4) It follows from the definition of the John transform .:I that .:I
1Jf.2 8'11
dF. 2
('=1
Differentiating the resulting relation "ith respect to
~
"
I
fil
and applying the formula
({l)-1~1"dF.2=O.
(s =1
We now return to the proof of the theorem. Using the above lemma, we define a function I(x) on R' by the formula
I(z) = I,,(z), 1] is all arbitrary vector in ()R4)' orthogonal to z. The function I is h0mogeneous, I(;\z) = ;\-2/(z). and infinitely differentiable on L" \ 0, where L" is a 3-dimensional subspace of 1Il4, and hence I is coo-dift'erentiable 011 Rot \ O. Therefore, I E F( -2). MoteO\"el'. it is clear that 3'1 because this relation holds for the restriction of '" to the set of subspaces belonging to an arbitmry three-dimensional space.
where
= '"
3.7. John transform as an Intertwining operator. The John transform has a simple grol1~theoretic interpretation. Consider the group G = GL(4,R) of nondegenerate matrices of order four. Any element 9 E G defines a linear transform z ...... X9 of Rnding a point x = (Zt,Z2,Z3,X,,) to the point with coordinates
a"
zj
= E:.1 Xi !}.J'
r
2. JOHN TRANSFORM
66
To any element 9 E G we assign linear operators TJ1) and TJ2) on the spaces F(-2) and 1t(-1), respectively, (TJ1) f) (x)
= I(xg),
(TJ2) rp) (u, v) = rp(ug, vg).
Obviously,
T91(') .....ti) 1.92
= T(i) 9192'
• •
= 1,2.
r,1)
for any 91,92 E G, i.e., the operators and T~2) form representations of the group G in the corresponding function spaces. The next assertion immediately follows from the definition of the operator :J. PROPOSITION
3.5. The operator:J T~2):J
(3.19)
= :JTJ1)
01 John translorm satisfies the relation lor each 9 E G.
In terms of representation theory, relation (3.19) means that the operator :J : F(-2) -1t(-1) is an intertwining operator for the representations T(l) and T(2) ofthe group G. One can prove that an intertwining operator F( - 2) - 'H( -I) is unique up to a factor. Thus, the condition that :J is an intertwining operator uniquely determines the John transform up to a factor. We now establish the group-theoretic meaning of the additional relations (3.7) for a function cp = :Jf. Let 9 be the Lie algebra of the group G and let {ei,} be the standard basis in 9 (i.e.• etj is the matrix with only one nonzero entry which is equal to 1 and stands at the (ij)th place). The representations 7' for rD we take the submaoifold of lines passing through the point
2. JOHN TRANSFORM
72
XO
and intersecting the curve A, i.e., the manifold of lines
x = (xo - ~) t By the definition of the operator (5.3)
"'I'
I..,.0=
+ ~,
~E
A.
le,
~ lP'{J(o, (J) ~ 8fJ,.8i).
iJ=l
I
J
In.O_A d~i"d~J""-A
I
Since the restriction '1'(0, (J) (j=A is equal to +(0, ~), the right-hand side of (5.3) is equal to (8A A8A)+(Q,~) lo=zO-A' Hence, by the inversion formula (4.7) for the John transform,
i
(8A
A8A)+(0,~) lo=zO-A = c(-yzo)/(xo).
Let us find the coefficient c(-yzo). H the order of the algebraic curve A is equal to k, then the cycle "'"(zo is homology equivalent to k -y, where "'"( is an Euler cycle. Since c(-y) 211" 2 i by §4, it follows that c(ro) 2k1l"2 i.
=
=
REMARK. If the curve A is given parametricaUy, ~
lII(o,s)
= u(s), then we set
= +(o,u(s».
In this case, the inversion formula takes the form
I(x)
= (2k1l"2i)-1 lef IIIZ,(x-u(s),s)d8Adi. .
REMARK. Real analogs of the inversion formula (5.2) play an important role in tomography. The real case is more complicated, and the reason is not only that real inversion formulas for line complexes are nonloca1. As a rule, it is impcl6Sible to reconstruct a functioin in R3 in terms of integrals along the lines intersecting a curve. The reconstruction is possible only for functions with strong restrictions on their supports. We do not consider such problems in this book.
5.2. Deflnition of admissible Une complexes in C 3 • Consider an arbitrary line complex K in C3. As above, the problem is to reconstruct a function I E S(C3 ) from the integrals of this function over the lines of the complex. We need to find conditions on K under which the reconstruction problem can be solved similarly to the case of a complex of lines intersecting a curve. using the inversion formula of § 4 for the John transform. Let HZ c H be a submanifold of lines passing through a chosen point x E C3 and let = K n HZ. The manifold is a cycle of real dimension two for almost every point x. Therefore, the following inversion formula holds for the John transform.1: if 'I' = .11, then
r
r
1..,.
(5.4)
"'I' = cf(x),
where " is the operator introduced in § 4. By the assumption, we know only the restriction • = 'I' IK of the function tp to the manifold of lines of the complex K. Thus, formula (5.4) gives a solution of the problem of integral geometry for the complex K if and only if the differential form can be expressed in terms of the function. and its derivatives along K only. An algebraic 3-dimensional subvariety of lines for which this condition holds
"'PI..,.
5. PROBLEMS OF INTEGRAL GEOMETRY FOR LINE COMPLEXES IN
CS
73
is called an admissible line complex. One can prove that this definition does not depend on the specific choice of the form K'P. An example of an admissible line complex is the complex of lines intersecting an algebraic curve in C 3 , which was treated in 5.1. If the complex K is admissible, then, substituting the explicit expression for KtPl'Y6 in terms of the function. into (5.4), we obtain an inversion formuJa for the integral transform related to this complex. REMARK. Our condition of admissibility is a purely local condition on a comis a cycle plex K. The requirement that K is an algebraic variety implies that for almost all x and the number of lines in K passing through a generic point x is constant. Combining these global geometric conditions with the local admissibility condition, we can write down the inversion formuJa for smooth complexes.
r
The next three subsections are devoted to the study of the analytic and geometric structure of admissible complexes. In particular, we claim that each admitmible complex is either the variety of lines intersecting an algebraic curve in C3 or a complex of lines tangent to some algebraic surface in C3.
5.3. Necessary and sufBcient conditions for a complex K to be admissible. Consider the John transform .1 in the local coordinates al. a2. Ph fJ'l on the manifold of lines H (see 4.1). If tP = .11 is the John transform of a function I e S(C3), then, according to § 4, the following inversion formuJa holds:
1
~
fPtP
-
L.J ~ 00; 1\ diij 'Y" i.j=I.2 a.a)
(5.5)
= C IX312 I(x),
where "{Z C HZ is an arbitrary cycle of real dimension two. Let K be an arbitrary complex. We set = HZ n K and, for each h e "(Z, denote by T"H, T"K, and T"r the tangent spaces at the point h to the manifold of lines H, the complex K C H, and the cycle respectively.
r r,
DEFINITION.
Introduce the endomorphism A" : T"H - T"H by setting
A,,(OOI,002,d{JI.dfJ'l) LEMMA.
= (0010002,0,0).
il and only il A"(T,,,,{Z) c T"K
A complex K is admissible
lor almost every point x e C3 and almost
every line her.
Indeed, the integrand in (5.5) is the derivative of tP taken in the complex direction A"(T,,r) for any h e Hence, the form in the integrand in (5.5) can be expressed in terms of. = '" IK if and only if this direction belongs to the 8Ubspace
,.z.
T"K. Let a complex K be given in the local coordinates (a, P) by the equation F(a, 13) = 0. PROPOSITION
5.1. The complex K is admissible
(5.6) at any point h
OF OF _ OF OF Oal OfJ'l 002 0131
= (a,l3)
01 K.
=
il and only if
°
2. JOHN TRANSFORM
74
= (001l002,d{3.,dlJ.z) be a vector in ThH. Then the condition
PROOF. Let ~
that
~ E
TIaK becomes
8F
8F
8 0 1 001
and the condition that
~ E
where al
=
~
-
x311;
8F
+ 81J.z djj.z = O.
d{J.
TIaHz is
+ d{Jl = 0,
X3oo.
Hence, the vector ~ E Tlar
8F
+ 8 0 2 002 + 8{31
X3 d0 2 + dlJ2
= TIa(HZ n K) is of the fonn ~ = (a.,a2, -X3aJ, -X3(2), and a2 = -(I!; - X3 :~).
= O.
It follows that AIa(~)
=
(al,a2,O,O). By the lemma, for the complex to be admissible, it is necessary and + = 0, which is equivalent to (5.6). sufficient that al
I!;
(12::'
We now present another proof of Proposition 5.1; this proof uses direct computations. First we assume that the complex K is given by an equation solved with respect to 1J.z:
IJ.z = U(01. 0 2,{J.). On the manifold of lines H we introduce new coordinates al. 02, {3J, S = IJ.z instead of the coordinates 011 02, {31,1J.z and denote by X(01.02, {JI, s) the function t/J in these new coordinates, i.e., U(01.02'~)
X(01,02, {311
IJ.z -
U(01l02,{3d)
= 1IJ(0.,02,{31,1J.z).
Note that in the new coordinates the complex K is given by the equation s = O. Let us represent the fonn ,,," in the new system of coordinates. We have i = 1,2.
Hence,
For the complex K to be admissible, it is necessary and sufficient that the restriction of this form to contain no term with 8x/8s. i.e., that
r
fJu- dOl 0
(5.7)
On the other hand, on hence (5.8)
fJu- 002 +0 V02
VO.
r
( -X3:;'
we have X3oo.
+ ::1)
001
I= ..,,,
+ d{3.
O.
= 0 and X3OO2 + dlJ.z = O. and
+ (X3 + ::2)
002
= O.
It follows from (5.7) and (5.8) that the function U(0},02,.8.) satisfies the equation
fJu (5.9)
80.
fJu fJu
+ 8 0 2 1J{3. = o.
If the complex K is given by an equation of the form F(o, {J) = O. then fJu/8~ = - Pa, / F's-;,. i = 1.2, and fJu/8{31 = - F~II F~. Substituting these expressions in (5.9). we obtain (5.6).
5. PROBLEMS OF INTEGRAL GEOMETRY FOR LINE COMPLEXES IN
CS
7&
5.4. Geometric structure of admissible complexes. The admissibility condition obtained in 5.3 for a line compll",x in (:3 has an ioteresting geometric interpretation. Let us first present some geometric notions and ract~ related to the manifold of lines H. To any line h E H Vo-e assign a three-dimeoslonal manifold B h or all lines intersecting h. The family of one-dimensional subspaces or Th H tangent to H h forms a tbrctHlimensional cone rh c TIIR. One can readily see that the manifold Hh, where h = (oP,po). is giveD in the local coordinates (0, fJ) by the equation (01 - Q~) (P-J -~) - (02 - Q~) (.81 - ~)
= O.
Hence, the equation of the cone rh is dol d~ - 002 dPt
= O.
The generatrices of the cone rh can be interpreted as isotropic directions with respect to the holomorphic metric
d82 = dot ~ - do2 dp'.
For this reason, it is natural to call rh c ThR t.he isotropic cone at the point h.
rh-
We state several simple facts concerning the isotropic cone Note first that the planes tangeot to rh are given by the equatiODs
al dol
+ ~ ~ + 01 0 of the plane of complex variable t = x + iy. where the motions corresponding to matrices 9 =
(~~)
E SL(2,1ll) are given as follows:
QZ+,
z 1-+ z 0 9
= fJz + 6 .
1Tbe main Facts lIhout hyperbolic geometry are presented without proofs. 79
3. INTEGRAL GEOMETRY IN 1:. 2 AND 1:.3
80
Obviously, the stabilizer of the point The Riemannian metric on £2 is
%=
i is the subgroup SO(2).2
(1.1)
The invariant distance d between points defined by the relation
%1
= Xl + iYI
and
Z2
= X2 + i1l2
is
(1.2)
The area element in the metric (1.1) is (1.3)
dv
= y- 2 dx dy.
Another way to define the hyperbolic plane is to consider the manifold of real symmetric matrices (1.4)
u
= (: :).
where a> 0 and
detu = 1.
with the motions g E SL(2, Ill), where g' stands for the transpa;ed matrix. Note that the manifold of matrices of the form (1.4) can be represented 88 a sheet of a hyperboloid of two sheets in R3: u ...... g'ug,
(1.5)
QC -
f12
= I.
a,e> O.
The correspondence between the points % of the upper halF-plane and the DUV trices u = u( %) is described 88 follows. The point % = i corresponds to the identity matrix e because the point i and the matrix e have the same isotropy subgroup. Hence, for each g = (~~) E SL(2, R), the matrix u = g'g corresponds to the point
. I . lor ~ g = (11112 = 8"+1 i+ . In partlCU ar,'log = X +'&y %11 2 11-0) 2 ,and we see that to -I( 2+ 2) -I ) a point % = X + iy corresponds the matrix ( 11 ~ 11 11 in other words. to a 1I:1t 11 matrix (~ :) corresponds the point z = ~ + ~. Geometrically, the passage from the
. log
1/
I
1/
_1%
;
sheet of the hyperboloid (of two sheets) to the half-plane is carried out as follows. We first project the hyperboloid (1.5) parallel to the axis a to the plane (b,c); this projection is the half-plane e > O. After this we make a projective transformation in the plane (b,e) by setting (b,e) ...... (~, ~), and this transformation sends the half-plane e > 0 into itself. It is convenient to pass in (1.5) to the new coordinates Xl = ~. %2 = G;C, X3 = b. Then we obtain a realization of the hyperbolic plane £2 as the upper sheet of the hyperboloid of two sheets in R3 given by (1.6) [x, x) == x~ - x~ - .r~ = 1. Xl > O. In this realization, the motions become linear transFormations in R3 that preserve the upper sheet of the hyperboloid, i.e., the elements of the group SOo(I,2) e! SL(2,R)/{±e}. The Riemannian metric in this model is (1.7)
ds 2
= -~ + dx~ + dx~.
2Note tbat tbe c:enter of the group 5£(2, R). which consists of two elements ±e, act.s ineffectively (preeerves any point). Therefore, it is convenient to view tbe quotient group 5L(2,R)/{±e} as tbe motion group.
1. ELEMENTS OF HYPERBOLIC PLANIMETRY
81
The invariant distance d between points x and , is (1.8)
coshd = [x, ,I.
The area element in the coordinates (1.9)
X2, X3
on £2 is
dv = dx2 dx3 • Xl
1.2. Horocycles. On the hyperbolic plane one can define circles of infinitely large radius (horocycles), which are the limits of non-Euclidean circles as the center and the radius of these circles consistently tend to infinity. In the Poincare model, the horocycles are represented either as (Euclidean) circles tangent to the real axis (the absolute) or as lines parallel to the real axis. The latter objects can naturally be interpreted as circles of infinitely large radius tangent to the real axis at the point X = 00. It follows from this description that every horocycle is given by an equation of the form
(1.10)
1(2z - (11 2 = y,
~
= (~1'~2) E R2 \ O.
Thus, the manifold of horocycles is parametrized by the points (~lt(2) E R2 \ 0, and the horocyc1es corresponding to the points (~1t~2) and (-~1'-~2) coincide. Denote by heel = h(elt ~2) the horocyc1e given by (1.10). REMARK. If a plane £2 is represented as the upper sheet of the hyperboloid given by (1.6), then the horocycles are the plane parabolic sections of this hyperboloid, namely, the plane sections of the form [~, x] E (1 Xl - (2 X2 - e3 x3 = 1, where ~ is a point of the upper nappe of the cone [~, ~I = 0, > 0, ~ E R3.
el
Obviously, the motion group 8L(2,R) acts on the manifold of horocycles H, and this action is transitivej the motion corresponding to a matrix 9
= (~ ~) sends
a horocycle h«(lt~2) to the horocycle h(ael + -r(2,P(1 + 6e2). In the language of motions, the horocycles are described as follows. They are the orbits3 of the unipotent subgroup Z of the matrices (~y) and all their translations. A group conjugate to Z acts on every horocycle, and the horocycle is a Euclidean line with respect to this action. Thus, the manifold H of horocycles is a homogeneous space of the group 8L(2, R), namely, the homogeneous space with the isotropy subgroup ±Z. Therefore, when passing from the hyperbolic plane £2 to the manifold of horocycles H, we obtain a new homogeneous space with the same motion group but with a different stabilizer. Constructions of this type play an important role in geometry. Every horocycle has a unique common point with the absolute, which is called the center of the horocyc1e. In the Poincare realization, this is the tangency point with the real line. Horocycles with common center are said to be pamlleL Note that a horocycle h(~lt~2) is tangent to the real axis at the point X = ~j hence, every pencil of parallel horocycles is of the form {h(~elt ~2) 1 0 < ~ < oo} for some chosen pair (el,e2). All points of a horocycle h(~el,~e2) are placed at the same distance 211og~1 from the parallel horocyc1e h(~1te2}' We refer to the expression p(Zje) -210g~
=
3In the Poincare model, these are the lines parallel to the real axis.
82
3. INTEGRAL GEOMETRY IN 1:. 2 AND 1:.3
as the oriented distance from a point z E h(~) to a horocycle heel. This distance is given by (1.11) (because ~ = (y-II{2Z - {112)-1/2). Every horocycle heel partitions the hyperbolic plane c,2 into two parts c,+({) and C,_({), where c,+({) abuts on the absolute and the boundary of c,-(e) has only one common point with the absolute, namely, the center of the horocycle heel. The domains c,+({) and c,-(e) consist of the points z E C,2 for which p(z;{) > 0 and p(z; e) < 0, respectively. REMARK. The homogeneous spaces c,2 (c,2 = SO(2) \ SL(2,R» and H (H = ±Z\SL(2, R» are dual to each other. By construction, to points { E H correspond curves heel in c,2 (horocycles). At the same time one can assign to a point z E c,2 the curve l( z) E H corresponding to the set of horocycles passing through z. Each of these curves is a trajectory of the subgroup SO(2) or a subgroup conjugate to SO(2). This trajectory partitions the manifold H into two parts, namely, the domain H+(z) in which p(z;e) > 0 and the domain H_(z) in which p(z;e) < O. In the realization H = R2 \ 0, the curves I(z) are all possible ellipses on R2 centered at the origin 0 and bounding a domain of unit area, and H+(z) and H_(z) are the exterior and interior domains of the ellipse 1(%), respectively. Note that the manifold of all orbits of the subgroup SO(2) and the subgroups conjugate to SO(2) is not homogeneous; in the realization at hand, this manifold consists of all ellipses on R2 centered at the origin O.
1.3. Geodesics. Consider now the "lines" (geodesics) on the hyperbolic plane c,2. In the Poincare model, these are the semicircles on the half-plane Im z > 0 with centers on the real line, and the half-lines orthogonal to the real line. Hence, every "line" is given by an equation of the form
(1.12)
(x - a)(x - b)
=
+ y2 = O.
=
a, bE R U (00),
a
-=F
b.
In particular, for a 00 (or b 00) we obtain the family of (Euclidean) half-lines orthogonal to the real line. It follows from (1.12) that the manifold of "lines" can be parametrized by the pairs (a, b), a -=F b, where a, b are the points of the projective line pI, and the "lines" corresponding to the pairs (a, b) and (b,a) coincide. In the language of motions, every "line" is an orbit of the subgroup D of diagonal matrices or of a subgroup conjugate to D, and all "lines" can be obtained from one another by motions. Thus, the manifold of "lines" is the homogeneous space D \ SL(2, R). We thus obtain another homogeneous space with the same motion group SL(2, R) and a different isotropy subgroup. Note that the manifold of all orbits of the subgroup D and subgroups conjugate to D is not homogeneous. In the Poincare model, this manifold coincides with the manifold of all half-lines on the half-plane Im z > 0 and of circular arcs having exactly two common points with the real line. IT the hyperbolic plane is represented as the upper sheet of the hyperboloid of two sheets given by (1.6), then, in this realization. the geodesics are the curves of intersection (one branch of a hyperbola) with planes in a3 passing through the origin 0, i.e., with the planes (1.13)
2. HOROCYCLE TRANSFORM
Denote by "Y«() the geodesic given by (1.13). Obviously, a plane (1.13) intersects the hyperboloid (x. xl = 1 if and only if < O. Moreover, since "Y(~) = "Y(e} for each
~.;:.
le,e]
0, we can set
le. e) = -1. Thus, the manifold of geodesics on tbe hyperbolic plane can be parametrized by the point-s of tbe byperboloid [~. () = -1 of w.'O sheets in R3, and the geodesics corresponding to the points and coincide.
e
e
-e
2. Horocycle trausform
2.1. Definition of the operator 'R,.h. In this section we study an analog of the Radon transform related to borocycles on the hyperbolic plane. DEFINITION. By the horocycle translorm we mean the integral tnmBfmm 'R,.h on £2 sending each smooth compactly supported function I OD £2 to the integrals of I over the horocycles.
(2.1)
'R,."
ICe) = j,,{O f I(Z)d8,
where h(el is tJle horocycle given by (1.10) and d8 the hyperbolic plane. The expression for 'R,." can also be written
88
= I¥ is the length element on
follows:
(2.2)
where p(z;{) is the distance from the point % to the borocycle h({) (see (1.11», and 6(·) is the delta function on R. It follows from the definition that the operator 'R,./a commutes with the arlion of tbe group SL(2,R). Thus, we have a pair of homogeneous spaces of the same group SL(2, R), namely. tbe hyperbolic plane and the manifold ofboroc:ycles, and an operator sending functions on one space to functions on the other and. commuting with the motions. Note that tbe integration is over the orbits of the isotropy subgroup of t.he second space and over the shifts of these orbits. 2.2. Inversion formula. Let us find t.he inversion formula for the integral transform 'R,.h following the same scheme as in tbe case of Euclidean plane (see § 1 of Chapter 1). First, take a radially symmetric function I depending only on the IlOIl-oEuclldean distance from a point z to the point i:
I(z}
= F(1I- 1/ 2 Iz - il}·
Since the operator "R" commutes with the motions of the plane £,2. it follows that "Rh I(e) depends only OD the distance from the point i to the borocycle h(~),
i.e., 'R,." I({)
= F(f')'
where r
= P(i;~) =
log({~ +~~).
3. INTEGRAL GEOMETRY IN t:. 2 AND 1:.3
84
Thus, the transform F( r) is equal to the integral of the function lover the horocycJe h(er / 2 ,O) for which 11 = er, i.e.,
F(r)
=
(2.3)
1 1
+00
F(e- r / 2.jx2 + (er - 1)2)e- r dx
-00
F(s)sds
+00
-2
21 s1nh il er / 2 Js2 - 4sinh2
-
~
(cf. (1.2) in § 1 of Chapter 1). Hence, the passage from F(s) to er / 2 F(r) reduces to the Abel transform. and it follows from (2.3) that
(2.4)
F(s)
11
(er / 2 F(r»' dr . 7r 2unb i>O J4sinh2 ~ _ s2
= --
In particular. I(i)
(2.5)
= F(O) = -1.1 (er / 2 !(r~)' dr. 00
27r
0
Slnh
2
Note that e r / 2 F(r) is a smooth even function by (2.3). Therefore. the integral (2.5) converges (because (er / 2 F(r»~=o = 0). and the expression for I(i) can also be represented in the form
1+
• = _1.
1(·)
(2.6)
00
47r
(e
r/
-00
2 F(r»' dr. sinh ~
Assume now that I is an arbitrary smooth compactly supported function on £2; we apply to I the averaging I(z) = F(y-l/2Iz - iD over the nOD-Euclidean circles centered at the point i,
F(
y
-1/2Iz _ il) = 1.1211" I 211"
0
( z~9 + sin 9 ) -zs1D9 + cos 9
dJJ:
the expression for F can also be represented in the form
F(s)
=~ f
7r It:.2
I(z) 6(,I- 1 Iz
- il2 -
s2) ~:y. 11
Note that I(i) = F(O). Let F(r) = 'Rh I«(), where r = log«(~ + ~~). Since the operator 'Rh commutes with the action of the group 8L(2, R), it follows that
F(r)
= 1.1211" 'Rh l(er/2 cos9,e-r/2 sin 9) d9 27r
= 21
(2.7)
11"
0
e- r
1
'Rh 1«()«(1 £1(2 - (2 £1(1)
p(i;()=r
= .!.e-r f 'Rh I«() 6(p(i; () - r) £1(1 £1(2, 7r la2 where p(i;() is the distance from the point z = i to the horocycle h(~). and (2.6), we obtain (2.8)
. _ - . _
1
I(t) - 1(1) - --2 11"
100 (e 0
r / 2 F(r»'
'oh r
SI
2
_ 1 dr - -47r
1+00 -00
Using (2.5)
(e r / 2 F(r»' 'oh r dr SI 2
2. HOROCYCLE TRANSFORM
85
(cf. (1.4) in § I of Chapter I). Substituting the expression in (2.7) for F(r} into (2.8), after elementary manipulations we obtain 1 [ e- p (l;()/2 ( { ) {) ) h (2.9) f(i)=-411"2JH±(i)Sinh~ ~1{){1 + {2{){2 +1 'R f({)d(1d(2,
where H±(i) are the domains of H2 in which p(i; {) > 0 and p(i;~) < 0, respectively. To obtain an expression for f at an arbitrary point z, one must apply the group translation taking the point i to the point z. We thus obtain the following assertion. THEOREM 2.1. A smooth compactly supported function I on t,2 can be expressed in terms of its horocyele transfonn 'Rh f by the following inversion fonnul4:
(2.10)
__ 2-1°C 2
f(z) -
11" 0
2-j+OC (e
(e r / 2 F(z,r»'r _ _ . hr dr 4 Stn 2 11"
-00
r / 2 F(z,r»'r 'oh r SI 2
dr,
where (2.11)
1
(2.12)
[
e-P(z;()/2 ( { )
{)
)
h
f(Z}=-411"2JH±(%)Sioh~ {1{)~1 + {2{){2 +1 'R f({)d(1d(2
(e/. (1.5) and (1.6) of Chapter I), where H±(z} are the domains of H in which p(x; {) > 0 and p(z; {) < 0, respectively.
it + {2i&
Note that {I parallel horocycles.
is the operator of differentiation along a pencil of
2.3. Asgeirsson relations. To invert the transform 'Rh, we used (2.4) for 8 f. 0, this formula contains an additional information on the relation between the averages of the function f and the horocycle transform 'Rh f of f. 8
= 0 only. For
THEOREM 2.2. The following Asgeirsson relations hold for the hyperbolic plane:
(2.13)
11
(e
F(Z;8} = - 11"
r / 2 F(z,
2sinbr/2>_ J4Sioh2
r»), r
i - 82
dr,
where F(r; s) is the mean 0/ the function / over the non-Euclidean circles centered at the point z,
(2.14) and
21
F(z,s}=-
11"
£2
2 dx'dy' f(z')6(" -1 y' -1 Iz'-zl 2 - s )--,
11'2
F( z, r)
from
is the mean of the function 'Rh f over the set of horocycles equidistant the point z; see (2.11). We also have
(2.15)
F-( z,r ) --
21
00
21ainb 51
er / 2
F(Z,S)8ds . 8 2 - 4 sinh2 i
J
3. INTEGRAL GEOMETRY IN 1:. 3 AND 1:. 3
86
Formulas (2.13) and (2.15) are obtained from (2.4) and (2.3) by a group translation; compare with the Asgeirsson relations (1.8) and (1.10) in § 1 of Chapter 1. 2.4. Symmetry relation. It follows from (2.15) that r ..... er / 2 F(z.r) is an even function of r, Le..
er / 2F(z, r) = e- r / 2F(z, -r) for any z and r. Substituting the expression (2.11) for F(z, r). we obtain the following symmetry relation for the function 'Rh I:
(2.16) e- r / 2
f
'Rh 1«()6(p(z;()-r)~1 ~2
= er / 2
f
'Rh 1«()6(p(z;{)+r)~1 ~2'
2.5. Inversion formula for the horocycle transform in another model of the hyperboUc plane. Let the hyperbolic plane {,2 be represented as the upper sheet of the hyperboloid in R3 given by
=
[x, x) x~ - x~ - x~ = 1. XI > 0. In this model, the horocycles are the sections of the hyperboloid by the planes
[(,xl
=eIXI-e2X2 -{3X3
= 1,
where { is a point of the upper nappe «(I > 0) of the cone K given by ((,{I The horocycle transform is then given by the following formula:
'Rhl«()
(2.17)
=f
= 0.
(eK,
l(x)6([{,xl-l)dx.
11:. 2
where dx
= x.l dx2 dx3 is an invariant measure on {,2.
THEOREM
2.3. A MTWOth COfIIpoctl,l8upportedjunction Ion {,2 can be expressed
in terms 01 its horocycle translorm 'Rh I by the lollowing inversion lormula: (2.18)
I(x)
= (2!)2 ['Rh I«() ([{,xl-l)-2~.
where
~ = {l' ~2 ~3.
The integral must be understood in the sense of regularized value [101. namely, as the value at I' = -2 of the analytic function of I' given for ReI' > by the convergent integral
°
(211')-2 ['Rh I«()
H~,xl- 111' cl(.
Note that formulas (2.17) and (2.18) are similar to the corresponding formulas for the Radon transform in Chapter 1. Similarly to the case of Radon transform, it suffices to prove the inversion formula (2.18) for functions on {,2 depending only on the distance from the point XO = (1,0,0). We can readily see that, for these functions I, the inversion formula (2.18) coincides with the one proven above.
3. Analog of the Fourler transform on the hyperbolic plane and the relation between this analog and the horocycle transform 3.1. Fourier transform on R2. At the naive level. the classical Fourier integral on R2 is the expansion of a function in L2(R2) in the eigenfunctions of the operators of parallel translation, Le., in the exponential functions ei(~IZ+~211). However, this integral is also closely related to a larger group, namely, the full motion group of the Euclidean plane. Let us indicate this relation.
3. ANALOG OF THE FOURIER TRANSFORM
Consider the Laplace operator fl. on the Euclideao plane, i.e., t.he second-order differential operator ~ = + commuting with the motions. It is known that this operat.or on L2(R2) has continuous Lebesgue spectrum coinciding "ith the half-line (-oo, 0). The problem is to construct sufficiently many distributional eigenf'unctions of tltis operator such that any function I E L2(22) can be expanded in these eigenfunctions. We sball argue at the naive level without specifying what object must be regarded as a distributional eigenfunction in the case of continuous spectrum and what is an expansion of a function / E L2(R2) in these distributions. Choose a one-parameter subgroup of translations in the motion group of the Euclidean plane; for instance. take the subgroup of translations parallel to the y axis. Each of the eigenspaces of the operator ~. say, the eigenspace corresponding to the eigenvalue _).2 (). E R \ 0), contains a pair of distributjons that are iDV8riant 'CIIritb respect to this subgroup, namely, e:t:·Az. Take one of these functions. for instance, eW:, and apply to it all possible motions. Obviously, the function e&A= is multiplied by a number under any translation and turns into e'~(z_9~aia') under the rotation through tJu.. angle IJ about the origin. Thus, for every eigenvalue _).2 of the operator ~ we obtain the family of eigenfunctions e i A.{zmatl+1/alnl) (in part.icular, the funct,ion e-j.\z). There are no linear relations among these functions, and every eigennmction 1>. of the operator A corresponding to the eigenvalue _).2 can be expanded in these functions,
£ lis
(3.1)
b.(x,y)
= 2...
f u~(IJ)e·~(zc:od+l/tdn9)
2w 0
d8
(). > 0).
Wc introduce the following norm in the space of eigenfunctions of the operator A that correspond to the eigenvalue _).2:
1I/l1l2 =
{3.2}
2~
1 lu,,(IJ)12 2 •
dB
and consider tht' related Hilbert spare H". In H" we obtain & unitary representation T" of the motion group of the Euclidean plane; namely, iD terms of the functions u~, to each translation (x. 7/) - (x + Xo,1I + 110) corresponds the operat.or of multiplication by the function eU(zo .-8+w aid), and to the rotat.ion through the aogIe a about the origin corresponds the operator of rotation u" (.) 1-1< u~ (. + a). It can readily be seen t.bat the representation T" is irreducible. The representation of the motion group in the space L2(R2) can be decomposed into the representations T". Let us describe this decomposition. Every function / E L2{R2) can be expanded in the integral
1= LOQ )./~ d>..,
(3.3)
where /l is given by (3.1) witb u~(8) = j(>.cos8,).sinIJ) (; stands for the Fourier transfonn of f). Moreover, the following Plancberel formula holds: (3.4) where
1l1A.1I is gi~-en by
(3.2).
3. INTEGRAL GEOMETRY IN
88
t:,2
AND
t:,3
3.2. Fourier transform on the hyperbolic plane. We now carry out similar considerations (at the same nonrigorous level) in the case of hyperbolic plane. Consider the Laplace-Beltrami operator A. on the hyperbolic plane 1:.2 , i.e., a second-order differential operator commuting with the motions of 1:.2 ; it is of the form
2(1fl Ifl) 8x2 +{Jy2'
A.=y
Let L2(.c2) be the space of measurable functions on 1:.2 with square-integrable absolute value (with respect to the invariant measure dv = On L2(I:.2) the operator A. has continuous Lebesgue spectrum coinciding with the interval (-00, -1/4). Let Z be the unipotent subgroup of matrices of the form ( : Y). By the Fourier integral on the hyperbolic plGne we mean the expansion of a function I e L2(.c2) in some special distributional eigenfunctions of the operator A., namely, in the distributional eigenfunctions that are invariant with respect to the subgroup Z, and the shifts of these functions. Just these functions are analogs of the exponential functions e i «(l Z+(211). Each eigenspace of the operator A. corresponding to some eigenvalue of the form (s e R) contains two functions invariant with respect to the subgroup Z, namely, the functions y~. Let us take one of them, say y~; under all possible translations, this function transforms to functions of the form
11").
_lit
(3.5)
+.(%;{)
= (yl{2% -
{lr 21¥ )
= exp (l+iS --2- P(%;{) ) , a point % = x + iy to the horocycle h({).
where p(%; {) is the distance from We call functions y~ %onal horocycle functions and translations of the form (3.5) are simply referred to as horocyde Junctions. We see that if { and {' are proportional, then the corresponding functions •• differ by a factor only. There are no other relations among the functions +. for S > 0, and if we take a family of functions +.(%;{) in which s ranges over the positive reals and { over a set having exactly one representative on each line in R2 passing through the origin 0, then we obtain a complete function set in which every function in L2(I:.2) can be expanded. Let us define the Fourier transform of a function on the hyperbolic plane 1:.2 by analogy with the classical Fourier transform. DEFINITION. By the Fourier translorm of a smooth compactly supported function I on .c2 we mean the inner product of I and the generalized horocycle function (3.5),
(3.6)
F I(£.,s)
=
1 t:,2
dxdy
1(%)·.(%; {)-2 .
Y
Since the functions •• are homogeneous with respect to {, it follows that the function F I is also homogeneous with respect to {,
(3.7)
F I(>.{;s)
= l,\ri . - 1 F I({;s).
3.3. Relation to the horocycle transform and the inversion formula.
The completeness of the family of horocycle functions +. means that an inversion formula for the Fourier transform F must exist. Let us construct it. First, we establish a relation between the Fourier transform F I and the horocycle transform 'Rh I. AB in the Euclidean case, the passage from a function I to
89
3. ANALOG OF THE FOURJER TRANSFORM
F / can be carried out in two steps. We first integrate the function / over the horocycles p(z;e- l / 2() == p(z;() - ~ = 0, and then integrate the resultiDg function multiplied by e-!!f!l with respect to ~. The first integration gives
Thus, the following assertion holds. THEOREM 3.1. The Fourier trans/onn on £,2 and the horocycle trans/onn are related as follows:
(3.8)
By the inversion formula for the one-dimensional Fourier transform, this implies that
1
+00
'Rh /«()
(3.9)
= (211r 1
-00
F /«(; 8) ds.
One can obtain the inversion formula for the Fourier transform :F from the lmown inversion formula (2.12) for the horocycle transform 'Rh. Note that in the Euclidean case we obtained the inversion formula for the Radon transform from the inversion formula for the Fourier transform; here we proceed in opposite direction. Substituting in (2.12) the expression for 'Rh / in terms of F / given in (3.9) we obtain (3.10)
f(z)
1 f 1+
= - 8r 1R2
00
-00
e-p(,::()/2
sinh ~
(8 + 8(28 + 1)F /«(; (18(1
(2
8) dsd(1 d&.
It remains to simplify the resulting expression. We first note that the homogeneity condition (3.7) for :F J implies
Further, we pass in (3.10) to the "polar" coordinates ( = erE-', where -00 < r < +00 and F.' belongs to a contour r c R2 \ 0 that intersects once every ray issuing from the point O. We obtain
Integrating over r (the integral over r must be understood in the principal value sense), we obtain the foUowing assertion.
3. INTEGRAL GEOMETRY IN t;2 AND t;3
90
THEOREM 3.2. A smooth compactly supported junction Ion (,'l can be expressed in terms 01 its Fourier translorm :F I by the lollowing inversion /ormvla: 4
I(z)
= S7r12
(3.11)
1+ (£ 00
x
=
S!2
7rS
stanh"2
-00
1:
00
:F I(~i s) exp ( - 1 ~ is P(z;~»)
stanh
~s
(£
(~1 d{2 -
{2 d{d) ds
:F 1(~;s)ct_.(Z:{)(~l d{'l - {'l d{d) ds.
In the proof of (3.11) we changed the order of integration; however, we did not prove rigorously that this change is indeed possible. 3.4. Symmetry relation. We show that the function I can be expressed in > 0 only. This is related to the following fact: the shifts of the functions,,~ belonging to the same eigenspace of the operator ~ can be expressed in terms of one another. Namely, terms of the values :F I(e; s) for s
,,¥ = c(s)
1:
00
(ylz -
ell- 2 )A¥ d{lo
where c(s) = 7r- 1/ 2 r ('~'.) Ir (¥). This readily implies the following symmetry relation for :F I: the integml
£:F f(e;s)ct-.(Z;eHel de'l- e2d{1) is an even /unction 01 s. Thus, by the symmetry relation, the inversion formula for the Fourier transform F can be represented as (3.12)
I(z)
=
4!21
OO
stanh ~s
(t:F f(e;s)ct-.(Z;~Hel d{2 -
e2d{1») ds.
REMARK. The symmetry relation for :F I can also be obtained from the symmetry relation (2.16) for the horocycle transform 'Rh I. Namely, substituting in (2.16) the expression for 'Rh I in terms of I and passing to the "polar" coordinates ~ = e'e', e' Er, we see that the integral
(3.13) e- r / 2
t 1: 1: 00
00
:F
e(l-i.)t:F I(e: s)6(2t+p(Zie)- r)dsdt(el d{2-{2d{d
is an even function of r. Integrating over t, we prove that the integral
{1+
lr
00
e-ier:F I({;s)e-¥p(z~)ds(el d{2 - {2d{d
-00
is an even function of r. Obviously, this condition is equivalent to the symmetry condition for :F I.
4nus (onaws from P.v.f+ao -OD 'o __
L". Te _l.p 2 •
-1f1WIIIU
.,-.rOdr olnh(a+l>
o dr = i e 6·P f+ ao alnr. dr = = e'·PP.V.f+ ao "-il ola r alDhr -DO
-00
4. RELATION TO THE REPRESENTATION THEORY
91
3.6. Plancherel formula. The definition of the Fourier transform F the inversion formula (3.12) imply the foUowing aasertion.
I
and
THEOREM 3.3. The lollowing Plancherel formula holds lor any smooth compactly supported function f on C2:
(3.14)
(lI{z) 12 d:c: y = 4 12 (00 ( Btanh 7r2BIF 1{(;s)1 2«(1 d(2 y 7r 10 lr
11:.2
~2d(1)ds.
Indeed, replacing the function fez) in the integrand of J l{z)/(z)~t'l by the expression for I(z) in terms of F I, we obtain
(1/{z)12d:c:Y=412 ( (00 (stanh7r2sl(z)F/{(;S) Y 7r 1/:.210 lr l+iS] d:cdy xexp [--2- P(z;() (~1d(2-(2d(1)ds7
11:.2
= 4!2 1000
£
stanh ~sl.r 1{(;s)12«(1d(2 -(2d(1)ds.
By the Plancherel formula, the Fourier transform can be extended from the smooth compactly supported functions to all functions I E L2(C2) according to the classical scheme.
4. Relation to the representation theory of the group SL(2, R) Let us show the relation between integral geometry and the representation theory of Lie groups by the example of the hyperbolic plane C2. We have the Hilbert space L2(C2) and a unitary representation T of the group SL(2, JR) in L2(C2),
Tgl(z)
= I(z 0 g),
9 E SL(2,R),
where % ..... % 0 g is the motion (corresponding to g) of the hyperbolic plane. The problem of representation theory (harmonic analysis for the hyperbolic plane) is to decompCl8e L2(C2) into irreducible invariant subspaces. This problem is similar to that of decomposing a function in L2(1R2) in the Fourier integral, and irreducible invariant subspaces can be treated as analogs of exponential functions. We show that the solution of this problem is given by (3.12) and (3.14) (see §3). We first describe the irreducible invariant subspaces themselves. Consider the eigenspace corresponding to an eigenvalue - 1~.2 of the Laplace-Beltrami operator~. As was already said before, this space is spanned by the distributions •• (z;() = (yl(2% - (21-2)~, where ~ ranges over a set containing exactly one representative of every line in R2 passing through the origin O. Thus, the elements of this eigenspace can be presented in the form
(4.1) where the function
f.(z) = U.
£
u.«() •• (z; ()«(1 d(2 - (2 d(l),
is homogeneous, u.(~) = 1~1'·-lU.«(),
and the integral is taken over any contour r c R2 \ 0 intersecting at one point every ray issuing from the origin O. (The integral does not depend on the choice of r because the differential form in the integrand is invariant with respect to the
3. INTEGRAL GEOMETRY IN r:~ AND
£,3
homothetic transformations ( ...... ~i cf. (1.14) in Chapter I.) In the space of functions I" we introduce the norm
i
11/,112 = lu,«()1 2«(1 ~2 -
(4.2)
(2 ~J)
and define the Hilbert space H, as the L2 space with respect to this norm. It follows from what was said in 3.4 that the spaces H, and H _. coincide. The motions on (.2 determine a unitary representation T. of SL(2,R) in H •. In terms of the functions u.«(), this representation looks 88 follows:
(T,,(g)U.)«(l,e2)=U.(Q(1+'Y~,,8(1+6(2)'
g=
(~ ~).
It is known (see, e.g., 18]) that this representation is irreducible. Introduce the projection operator p. : L2(£2) - H. by the rule
p.I = I., where I, is given by (4.1) in which u.«() = F I«(,s). Obviously, these operators p. commute with the action of the group SL(2, R) on the spaces L2(£2) and H •. Thus, the problem of decomposing the space L2(£2) into irreducible subspaces consists in expanding the functions I E L2(£2) in their projections I •. The solution of this problem is given by (3.12) and (3.14), since these formulas can be written 88 follows: (4.3)
(4.4) where
1
1= 4'1r2
1 £,a
[00 stanb"2/• 'lrS
0
1I(z ) 12dzd1l Jj2"
ds ,
I.=p./i 'lrsll 12
1 foo = 4'1r2 10 s tanb"2 1.1
ds,
11/.11 is given by (4.2).
Recall that the proof of the main formula, (3.12), used the inversion formula for the operator 'R" of horocycle transform. We clarify the role of this operator from the groul>'theoretic point of view. The operator 'R" sends a function I(z) on the homogeneous space £2 = SO(2)\ SL(2,R) to the function 'R" I(e) on the homogeneous space H ±Z \ SL(2,R), and this operator commutes with the action of the group SL(2, R) on these spaces. Moreover, the irreducible components transform into irreducible components; however, a priori this mapping may have a nonzero kernel. In fact, the kernel of the mapping is trivial because there is an inversion formula for and the problem of decomposing the representation of SL(2, R) in L2«(.2) is reduced to the problem of decomposing the representation in a function space on H = ±Z \ SL(2, R). The solution of the latter problem is very simple for the following reason. The homogeneous space H is endowed with an action of the commutative group of homothetic ~ commuting with the action of the group SL(2, R). Consider transformations the corresponding commutative group of the operators AA:
=
'R",
e. . .
(AA/)(e)
= I(~).
The common eigenspaces of AA consist of homogeneous functions. Since the operators AA commute with the operators of the representation of SL(2, R), it follows that their eigenspace8 are invariant with respect to this group. It is known that the representations of SL(2, R) acting on these subspaces are irreducible. Thus, the decomposition of the function space on H into irreducible components is reduced
5. INTEGRAL TRANSFORM
93
to the decomposition into the eigenspaces of the operators A~, and since the group of operators A~ is commutative and isomorphic to the additive group R, this decomposition is reduced to the one-dimensional Fourier transform. The composition of the horocycle transform 'Rh and the above one-dimensional Fourier transform is the Fourier transform :F on the hyperbolic plane. Thus, the role of the horocycle transform is that this transform reduces the problem of expanding functions on the homogeneous space £2 = SO(2) \SL(2, R) to the expansion problem for functions on another homogeneous space ±Z\SL(2,R), where this problem admits an elementary solution. Similar transforms reducing decomposition problems for group representations related to homogeneous spaces to a problem of integral geometry can be defined not only for £2 but also for other homogeneous spaces. This tool, known 88 the method 01 horospheres, was first introduced in [5]. The construction used in this section to introduce irreducible representations of the group SL(2, R) can be included in the general scheme of realization of irreducible representations in function spaces on homogeneous spacesj this construction originates &om Frobenius. A finite-dimensional irreducible unitary representation T of a group G can be realized in functions on a homogeneous space U \ G if and only if the representation space contains a nonzero vector invariant with respect to the subgroup U i the dimension of the space of these vectors is equal to the multiplicity of occurrence of the representation T in the space L2(U\G). The Frobenius duality theorem explicitly defines an operator that embeds T in L2(U \ G). In the infinite-dimensional case one needs some analytic refinements. These are the functions 1/~ considered above; they are invariant with respect to the subgroup Z, and the construction of irreducible representations of the group G = SL(2, R) that correspond to these functions follows the Frobenius scheme. Since every irreducible space contains two (distribution) vectors invariant with respect to Z, it follows that this space is embedded in L2(±Z \ G) with multiplicity two. However, every irreducible representation enters the decomposition of L2(£2) with multiplicity one, due to the symmetry relation for :F I.
5. Integral transform related to lines (geodesics) on the hyperbolic plane £2
5.1. Deftnltlon and the inversion formula in the Poincare model. Consider another analog of the Radon transform for the hyperbolic plane, namely, the integral transform related to "lines" (geodesics) on £2. According to § 1, every "line" on £2 is given in the Poincare realization by an equation on the half-plane lID % > 0 of the form (5.1)
(z - a)(z - b) + 1/2
= 0,
where a :F b and a, b e R U (oo)j we denote this "line" by G(a, b). It readily follows from (5.1) that under the motion z .... a line G(a, b) transforms into the line
i:!l
G(~,~). DEFINITION. Denote by 'R' the integral transform taking each smooth compactly supported function 1 on the hyperbolic plane £2 to the integrals of this
3. INTEGRAL GEOMETRY IN t:,2 AND t:,'
function over the lines in £,2 (i.e., over the geodesics with respect to the metric in £,2),
(5.2)
'R,9I(a,b) = (
ds
I(z)ds,
= /dz/. 1/
10(11,11)
Obviously, this transform commutes with the action of SL(2, R). Let us find the inversion formula reconstructing I from 'R,' I. In the proof of this formula we follow the scheme used in the case of Radon transform on the Euc1idean plane and the horocycle transform on £,2. We first consider a function I E D(£,2) that is coDStant on the non-Euc1idean circles centered at the point i, i.e., on the orbits of the subgroup SO(2) of matrices 9 (coe' - .iD'). this condition means that I has the form
= sin' coe' '
Since the transform 'R,I commutes with the action of the group SL(2, R), it follows that the function 'R,I 1(0, b) is also invariant with respect to the tnmsforms in 80(2). One can readily see that the fraction ~*-T is a full invariant of a pair of points a, b E pI with respect to 80(2). Thus, 'R,I 1(0, b) = from (5.2) that
F(r)
='R,I I(oo,r) =
(00 F
(r2
10
=1,(r2+l)1/2 F( s 1(r2 +l)1/2
It follows
+ (1/ -1)2) dy 21/
1/
ds
00
= (00
F (~"!1 ).
1) """"'i'9;==ri====:=t= S2 - (r 2 + 1)
J
(s-lF(s
-1» .,f82 -sds(r2 + 1) .
Thus, F(r) is an even function of r, and it follows from the inversion formula for the Abel transform that S-1 F(s
- 1) = _! 1r
In particular, for s (5.3)
fooH=i ../r2F'(r) dr; +1- 2
s
~ 1.
8
= 1 we obtain I(i) = F(O)
= _! 1r
(00
10
Per) dr r
(the integral converges because F'(r) = oCr) as r - 0). We now assume that I is an arbitrary smooth compactly supported function on £,2 and 7 is the result of averaging I over the DOn-Euc1idean circles centered at the point i,
5. INTEGRAL TRANSFORM
95
Since the transform 'R9 commutes with the motions of (,2, it follows that
'R9j(a, b)
=F =
(1
1..
+ab) a-b
f'hr'R 9
21r 10
I
(aCOS9 + sin 9 bcos9 + sinB ) -asinB+cos8' -bsin8 + COS 8
d(J
'
and hence
F(r)
= 'R9j(oo,r) =
(5.4)
21 f2fr 'R' I (-cot8, rC~89+sin(9) d8 1r 10 -rsm +008
=.!. f+oo 'R9 1 (t, rt -1) ~. 1rl_oo
r+t
l+t
In view of (5.3), we obtain
I(i)
(5.5)
= -I(i) = --1r2 LOO -F'(r) dr, 0 r
where F(r) is given by (5.4). To find an expression for I at an arbitrary point z, it suffices to apply a group translation 9 sending the point i to the point z, for instance, 111/2 9 = ( rg-1/2
0)
y-1/2
•
Under this translation, I(i) goes to I(z) and the transform 'R9 1(o.,b) goes to 'R9 1(x + o.y, x + by). We obtain the following result. THEOREM 5.1. A smooth compactl,lsupported /unction I on {,2 can be u;pre8sed in terms 01 its translorm 'R9 I related to the lines in (,2 by the lollowing inversion
lormula: I(z) where
~
F(z; r)
= _~ fOO F~(z;r) dr, 1r 10 r
1f+OO 'R I ( x + ty, x + rtr +- 1)y
= ;: Loo
9
t
dt 1 + t2·
5.2. Relation to the Radon transform on the projective plane. We now assume that the hyperbolic plane {,2 is represented as the upper sheet of the hyperboloid given by (x, x] 1, where (x, xl xY - xi - x~. In this realization, the geodesics on (,2 are the curves 'Y(~) which are the intersections of the hyperboloid with the planes
=
=
and the integral transform 'R9 is given by the formula
(5.6)
'R91(~) =
1
I(x)ds,
(~,~l = -1,
-r(E)
where ds is the length element on (,2. One can readily see that the expression for
'RI can also be represented in the form (5.7)
'R9I(~)= f l(x)6«(~,x})dx2dx3,
l£a
where 6(·) is the delta function on R.
Xl
3. INTEGRAL GEOMETRY IN £2 AND £3
96
In this model, one can obtain the inversion formula for the integral transform 'R' similarly to that for the Minkowski-Funk transform related to the unit two-dimensional sphere; in other words, the inversion formula can be found as a consequence of the inversion formula for the Radon transform 'R on the projective plane; see § 8 of Chapter 1. Namely, to any function I E (,2Coc we assign a function Fin R3 \ 0 satisfying the homogeneity condition F(.u) = ..\-2 F(x) and such that F 11~.~1=1= I and F = 0 for [x,x] < O. It follows from the definition of the operator 'R that 'R F(e) = 'R'I(e) for [e,e] = -1 and 'RF(e) = 0 for [e,e] > O. Therefore, the following assertion is an immediate corollary of the inversion formula given in Chapter 1 for the integral transform'R (see (8.10) on page 36). THEOREM 5.2. A function I on (,2 can be expressed in terms 01 its translorm 'R' I related to the geodesics by the lollowing inversion lormulo.:
(5.8)
I(x) = 41 1r
f 'R' I(e) (e, x) -2 w(e). J1E•EJ =-1
For the meaning ofthis integral, see § 8 of Chapter 1.
6. Horospherical transform in the tJuoe.dimensional hyperboUc space {,3 The definitions and results of §§ 1-5 for the hyperbolic plane can be extended to hyperbolic spaces of higher dimensions: however, the inversion formula has different forms for spaces of odd and even dimensions. For this reason, we give an independent exposition for the three-dimensional hyperbolic space {,3. Here we put an emphasis on explicit formulas.
6.1. Models of the hyperbolic space. Let us briefly present the elements of geometry in {,3. The hyperbolic space (,3 is the homogeneous space of the group SL(2, C) with the isotropy subgroup SU(2) (the subgroup of unitary matrices). The following quatemion model of the space {,3 is convenient for our purposes: a point of {,3 is a quatemion with three components, w
= x + iy + zj,
Z
> 0,
where x. y. Z E R, ;2 = j2 = -1, and ;j = - ji; the motion of (,3 corresponding to a matrix (~~) E SL(2,C) is given as follows:
= (wiJ + 6)-I(WQ + 1). Obviously, the isotropy subgroup of the point w = j is SU(2). w ...... w'
The Riemaonian metric on (,3 is (6.1) ds 2 = z-2(dz2 + dy The distance d between points by the relation (6.2)
where (6.3)
Wl
+ dz 2 ).
= Xl + Yl; + zd and w, = X, + 1I2i + z,j is given
97
6. HOROSPHERlCAL TRANSFORM
and I . I stands for the norm of a quaternion t'lernent in the metric (6.1) is ofthe fonD (6.4)
dv
(lwl' = z2 + If + Z2).
The vo)wne
= z- 3 dxdydz.
In a di1ferent way, the hyperbolic space can be given by the manifold of Hermitian-symmetric matrices (6.5)
u=(: :),
O.C
> 0,
detu = 1,
with motions u ...... g"ug, 9 e SL(2,q. where gO stands for the Hermitian--conjugate matrix. Note that the manifold of matrices (6.5) can be interpreted as a sheet of the following byperboloid of two sheets in R":
(6.6)
QC -
~
-
~
= 1,
a,e> 0 (bJ + i~ = b).
The correspondence ~-een the quaternions w and the matrices u follows:
.
.
x+yt+zJ......
(z-I(X 2 +y2+.:2) -1 (') If: X -111
= u( w) is as
») ;
Z-l(X+iY z-1
!
in other words. to a matrix (5!) corresponds the ql1atemion 1V = + ~j (cf. the case of the hyperbolic plane). In particular, the identity matrix corresponds to the point w:::: j. It Is convenient to pass in the equation of the hyperboloid given by (6.6) to the new coordinates Xl = ~, X2 = ~. X3 = bt, X4 = b:z. Then we obtain a realization of the hyperbolic space as the upper sheet of the following hyperboloid:
(6.7)
Ix,x] == x~ - x~ - ~ - x~ = 1,
Xl
> O.
In tbis realization. the motions arc given by linear transforms in R4 presening the upper sheet of the hyperboloid. i.c.. by elements of the group 800(1,3) 9!! 8L(2, C)/{±e}. The Riemannian metric in this model is ds 2 = -~ + dz~ + dzi + dz~i the in,'lUiant distance d between points X and 11 is
ooshd = (X, 11); the volume element in the coordinates dv
x2. X3
on
1:,2
is
= dz, tlz3 U4 . XI
6.2. Horospheres. In the hyperboUc SpaL'e one can define spheres of in6nit.ely largt' radius (horospheres), which arc limits of non-EucHdean spheres as the eent« and the radius of these spheres simultaneously tend to infinity. A horosphere can be represented either as a (Euclidean) sphere (in the balf-spac:e .z > 0 of R3) that is tangent to the plane z = 0 (the absolute). or 88 a plane (in this haIf-space) parallel to the plane % = O. The latter can be naturally interpreted 88 a sphere of infinitely large radius tangent to the plane If: = 0 at a point at infinity. In the quatemion model, every horosphere is given by an equation of the form (6.8)
IW{2 - {112
= Z.
W
= x + ui + zj.
3. INTEGRAL GEOMETRY IN C 3 AND
98
,3
where ~ = ({1'~2) E C 2 \ O. Thus, the manifold of horospheres is parametrized by the points ~ E C2 \ 0, and the horospheres corresponding to the points { and ~, IAI = 1, coincide (cf. the equation of horocyc1es on the hyperbolic plane). Denote by h({) the horosphere given by (6.8). REMARK. H the space £3 is represented as the upper sheet of the hyperboloid (6.7), then the horospheres are the plane parabolic sections of this hyperboloid, namely, the sections by the planes of the form [{,x] = {I Xl +{2 X2+{3X3+{4 X4 = 1, where ~ is a point of the upper nappe of the cone [{,~I = 0, and thus {I > 0 ({ E R4 \ O).
The motion group 5L(2, C) acts on the manifold of horospheres, and this action is transitive; the motion corresponding to a matrix 9 = (~~) takes a horosphere !(~.. {2) to the horosphere !(O{l
+ "'~2, .B~l + 6{2). In the language of motions, the horospheres are either orbits of the subgroup Z of the matrices of the form (: Y), tEe (these are the planes parallel to the plane z = 0) or shifts of these orbits. Thus, the manifold of horospheres is the homogeneous space of the group 5L(2, C) with the isotropy subgroup ±Z. Each horosphere has a unique point of tangency with the absolute; this point is called the cemer of the horosphere. In our interpretation, these are the points of tangency with the plane z = O. Horospheres with a common center are said to be paralleL Note that the horosphere h(~} is tangent to the plane z = 0 at the point ~; hence, every pencil of parallel horospheres is of the form {h( A{) I 0 < A < oo}, where { is fixed. The (oriented) distance p(w,{) from a point w E £3 to a horosphere h(~) is defined in the same way as the distance from a point to a horocycle on the hyperbolic plane, and is given by a similar formula w =x+yi+ zj.
(6.9)
6.3. Horospherical transform. DEFINITION. By the horospherical transform in £3 we mean the integral transform 'Rh taking a smooth compactly supported function f on £3 to the integrals of this function over the horospheres,
(6.10)
'Rh f(~)
=[
f( w) M,
lh(f.)
where h({) stands for the horosphere given by (6.8) and M for the (non-Euclidean) area element on h(~). The expression for 'Rh can also be written as (6.11)
'Rh f({) = [
lC3
f(w)6(p(w;~»dxd~dz, z
where p( w; {) is the distance from the point w = X + yi + zj to the horosphere h({); see (6.9). It follows from the definition that the operator 'Rh commutes with the action of the group 5L(2, C).
6. HOROSPHERlCAL TRANSFORM
99
6.4. Inversion formula. We now proceed with the coDStruction of an inversion formula for the integral transform Rh by repeating the arguments used in the case of hyperbolic plane. We first 88SUlDe that f is a radial function dependiDg only on the distance from the point w to the point j,
few)
= F(Z-1/2Iw -
jl).
In this case, Rh f({) depends only on the distance from the point j to the horosphere h(~), i.e., Thus, the value F( r) is equal to the integral of the function . h( e r/2 , 0)· given by z = e r , I.e.,
. . = f.
F(r) (6.12)
R2
F
f
over the horosphere
(JX2 + y2 +/2(er - 1)2) -:;dxdy r er
e-
= 271"100 F (Jt2 + (er
er / 2
o
-1)2)
tdt
= 27r
e2r
(00 121uab 11
F(s)sds. er
First, this implies that er F( r) is an even function of r,
er F(r)
= e- r F( -r),
and second, (6.13)
(erF(r»'
= -27r( coshrF(2sinh i) + sinhrcosh iF' (2 sinh i))·
It follows from (6.13) for r = 0 that
IU> =
(6.14)
F(O)
1
....
= - 27r (er F(r»'lr=o
(d (2.5) in this chapter and (2.4) in §2 of Chapter 1). Assume now that arbitrary smooth compactly supported function on £3 and J(w)
= F(Z-1/2Iw -
I
is an
jl)
is the function obtained by averaging I over the non-Euclidean spheres c:entered at the point j. Since these spheres are orbits of the subgroup of unitary matrices 9
= ( ~tt :) , lal2 + IPI2 = 1, it follows that f(w)
= F(z-1/2Iw -
jl)
=/
l«wP + O)-l(wa
-P» dp.(a,p),
where dp.(a,p} is the invariant measure on the group of unitary matrices, i.e., on the three-dimensional unit sphere lal 2 + IPI 2 = 1, normalized by the condition
/ d,,(a, P)
= 1.
We can readily see that the expression for F can be written in the following convenient form:
(6.15)
3. INTEGRAL GEOMETRY IN £.2 AND £.3
100
=
=
=
Let F(r) 'Rh f(e), where r p(j;e) log(leil 2 + le21 2). Since the transform 'Rh commutes with the action of the group 5L(2, C), it follows that
F(r) (6.16)
=
1.
'Rh l(er / 2Q, er/ 2fJ) d/J(Q, fJ)
101 2 +1111 2 =1
= 11"2 (~r
12
'Rh l(e r/ 2el,e r / 2e2) 6(p(j;e» del del de2de2,
where pU; e) is the distance from j to the horosphere heel. By (6.14),
IU)
~ = -IU) = - 211"1 (er F(r»"lr=o.
Substituting here the expression for F(r) given in (6.16), we obtain
I(j)
r
= - 81
11" JI~112+1~212=1
(6.17)
=-
8~
L2 'Rh I(e) d/J(e)
L
L2'Rh I(e) 6(p(j;e» dv,
where (6.18)
B L = el Bel
B
-B
-B
+ e2 ae2 + el ~1 + e2 ae2 + 2,
dv
= (i)2 2 del del de2 de2.
This gives an expression for IU) in terms of its horospherical transform 'Rh I. Hence, applying a translation that takes j to an arbitrarily chosen point w, we obtain the following assertion. THEOREM 6.1. A function I on £3 can be expressed in tenns 01 its horospherical translonn 'Rh I by the lollowing inversion lormula:
(6.19)
I(w)
where (6.20)
F(w,r)
= 11"-2
~ " = - 211"1 (er F(w, r)) Ir=o,
L
'Rh l(er/ 2e" er / 2e2) 6(p(w.e))dv;
hence,
(6.21)
I(w)
= - 8~ /(;2 L2 'Rh I(e) 6(p(w;e»
dv,
where the operator L is given by (6.18). Since 'Rh I(e i 'el. e i'e2) where L~ = el + e2~ + 1 and L~ REMARK.
ii
4L:'Rh 1= 4L:'Rh I
= 'Rh I(el, e2), we have L~ 'Rh I = l~ 'Rh I, = el ~ + e2at; + 1. Therefore, L2 'Rh 1=
= 4L~L~'Rh I.
Note that this inversion formula is local (in contrast to the case of hyperbolic plane); namely, to reconstruct a function I at a point w, one must know the integrals of I only over the horospheres infinitely close to w.
101
6. HOROSPHBRICAL TRANSPORM
6.5. Symmetry relation. We obtain the fobing result as a by-product, applying a group translation to (6.12). THEOREM 6.2. The following
Asgeirsson relation holdJJ for the hyperbolic space:
211'12111lnh il F(w; s)sds. 00
eTF(w:r) =
(6.22)
Here F( w, s) standa for the result of averaging the function f over the centered at the point w, i.e., (6.23)
F(w.s)
= 11'-1
La
=(211')-1
%1/2%,1/2110 -
LJsinh
sphere~
w'r 1f(w')6(z-l z ,-1 Iw - w'12 _ 82)dr'~dz'
d(w;w')
1-1 6(4sinh2 d(w~w') _ 82 )
f(w'}dz':!d.t,
uthere d(w.w') 18 the non·Euclide4n dUltance between w and 'Ill, a:nd P(W,T) is the result of averaging the junction 'R" J over the horospheres equidistant from the
point w; see (6.20). COROLLARY.
The junction eT F( w. r) is
etre1I
as a juncticm oJ r,
eTF(w,r) =e-rF(w,-r). Substituting bere the expression for the following symmetry relation:
Fgiven by (6.20), we see that 'Rh I satisfies
er J'Rh f(e r / 2{l,e r / 2{2)6(p(w;{»du
(6.24)
= e- r f'Rh J(e- r / 2(1,e- r / 2(2)cS(P(W;(»dv.
,3
6.6. Inversion formula for the horospherical transform In another model of the hyperbonc space. Let us represent the byperbolic space as the upper sheet, Xl > 0, of the hyperboloid Ix. x} == ~ - ~ - ~ - ~ = 1 in R4. In this model, the horospheres are the sectiODS of the ~ by the byperplalle8
le. x] ==e1 Xl -
(2%2
-(aX3 -{.X4
= 1.
when> { is a point of the upper uappe K, Xl > 0, of the cone giwn by [(,{j The borospberical transform is given by the following formula:
'Rh J({)
(6.25) where U
= ( J(x) 6«e. x] lc.s
1) U,
= xII U2u3 dx4 is an invariant measure on
=
o.
~ E K,
'3.
THEOREM 6.3. A smooth compactly .rupported function f on ,3 am be ~ed in tenns of the horospherical transform 'Rh J by the following inversion formula:
(6.26)
lex) =
-s!,2 L'Rh J«()&H([(,X] -1)~,
~ ={11c1Qd(3d(4,
where 6" (.) is the second denvative 0/ the delta function cm R.
3. INTEGRAL GEOMETRY IN
102
(;2
AND
(;s
Formulas (6.25) and (6.26) are similar to the corresponding formulas of Chapter 1 for the Radon transform and, as in the case of Radon transform, it suffices to prove the inversion formula (6.26) for functions on C3 depending only on the distance from the point xO (1,0,0,0). One can readily see that, for these functions I, the inversion formula (6.26) coincides with that proven above.
=
8.'1. Integral tr8DSform related to completely geodesic surfaces in C3 • Here we consider another integral transform on C3; it takes functions on C3 to their integrals over the completely geodesic surfaces in C3. We denote this transform by
'R' .
e
We represent the hyperbolic space 3 as the upper sheet of the hyperboloid (6.7) in R4. In this model, the complete geodesics are the sections of this hyperboloid by the planes {(, x)
(6.27)
= O. = -1. The integral transform
== (IXl + (2X2 + (aX3 + ~X4
where ( ranges over the hyperboloid of one sheet [(, () 'R' is given by the following formula:
'R'I«()
= 1(;2 ( I(x) 6( {(, x»
d:r2 d:r 3 d:r 4 , Xl
where 6(·) stands for the delta function on R; d. the definition for the hyperbolic plane in § 5. AB in the case of hyperbolic plane, the inversion formula for the integral tr&lllr form in question can be obtained as a consequence of the inversion formula for the Radon transform 'R.. on the projective space; see § 8 of Chapter 1. Namely, to a function I e C3 we assign the Coo function F on R4 \ 0 satisfying the homogeneity condition F(.\:r) = 1~1-3 F(x) and the conditions F Ilz,z)=1 = I and F = 0 for
[x, xl < o.
It follows from the definition of the operator 'R that 'R F«() = 'R'/«() for ((,(] = -1 and 'R..F(e) = 0 for le,e) > O. Therefore, the following assertion is an immediate coro1lary of the inversion formula (8.6) of Chapter 1 for the integral transform'R (see page 32). THEOREM 6.4. A smooth compactly supported junction I on C3 can be expressed in terms 01 the translorm 'R..' I rel4tetl to the completely geodesic sur/aces by the lollowing inversion formula:
(6.28)
I(x)
= _(211')-2
(
'R' J(e) 6"({e.
1[(.{)=-1
x» w(e),
where 6"(·) stands lor the second deritJative 01 the delta junction on R. For the meaning of this integral; see § 8 of Chapter 1. We emphasize that the inversion formula (6.28) is local, in contrast to the case of hyperbolic plane.
Appendix. HOl'08pherica1 tr8DSform for the hyperbolic space of an arbitrary dimension. The hyperbolic space en 01 an arbitrary dimension n is the upper sheet of the hyperboloid in Rn+1 given by
(6.29)
[x. x] == :rf - x~ - ... -
X!+l
= I,
Xl
> 0,
7. ANALOG OF THE FOURlER TRANSFORM
103
in which the motions are the linear transformations of the space Rn+! that preserve this hypersurface. By the horospheres we mean the sections of the hyperboloid by the hyperplanes
[{, x] := {I Xl
(6.30)
- ••• - ~n+!
Xn+l
= I,
where { is a point of the upper nappe K ({I> 0) of the cone [~,{] = O. The horospheriool translorm is given by 'R" I({)
(6.31)
= f I(x) 6([(, x] It!ft
1) dz,
(
e K,
where dz = xII dz2 ... dzn+l is an invariant measure on £n. The structure of the inversion formula depends on the parity of n. We present this formula without proof. If n = 2m + I, then
I(x) = 2\~!~;m
(6.32) where de then
(6.33)
= (1 1 de2'"
L
'Rh I«()
6(2m)([(, X] - 1) de,
den+l is an invariant measure on the cone K. If n
lex) = (-I)m(2m -1)1 f 'Rh I«()([e xl _1}-2m de (21r)2m
lK
'
= 2m,
.
Thus, as in the case of Radon transform, the inversion formula is local for odd n and nonlocal for even n.
7. Analog of the Fourler tramsform in the hyperboUc space, and its relation to the horospherical transform 7.1. DefInition of the Fourier transform. We define the Fourier integral for the hyperbolic space £3 in the same way as in the case of hyperbolic plane. Consider the Laplace-Beltrami operator 6 on £3,
2(fP fP fP) 0 OX2 + 8y2 + 1Jy2 -%8z'
A=z
It is known that the operator 6 on L2(£3} has continuous Lebesgue spectrum coinciding with the interval (-00, -I). Let Z be the unipotent subgroup of complex matrices of the form ( 1 By the Fourier integral on £3 we mean the expansion of a function / e L2(£ } in certain special functions related to the operator A, namely, in the eigenfunctions that are invariant with respect to the subgroup Z, and their shifts. Every eigenspace of the operator 6 corresponds to an eigenvalue of the form (8 e R) and contains two functions invariant with respect to the subgroup Z, namely, the functions z~. Take one of them, say, za;uj any translation sends it to a function of the form
J)'
4±l
(7.1)
•• (Wj() = (zlwe2 - (11- 2a;u )
= exp (2+iS --2- P(w;() ) ,
where p(w;~} is the distance &om the point W = X + yi + zj to the hor08phere h({). We c&ll the functions za;u the zonal horospherical /unctions, and their shifts (7.1) &re simply referred to &8 horospherical functions. If and are proportional, then the corresponding functions •• differ by factors only, and for 8 > 0, there are no other relations among the functions .,. If 8 ranges over the positive re&ls and ( over a set containing exactly one representative of every complex line in C2 \ 0
e
e'
3. INTEGRAL GEOMETRY IN
104
t:.2 AND t:.3
passing through the origin 0, then these functions form a complete set of functions in which every function in L2(£3) can be expanded.
I
DEFINITION. The Fourier translorm of a smooth compactly supported function on the hyperbolic space £3 is defined by the relation
:F 1(1.;8)
(7.2)
= Jt:.f 3/(W)+-.(WiF.) tb:dydz z3
(cf. the definition of :F I for £2 in (3.6».
+.
Since is homogeneous with respect to (, it follows that:F I is also a homogeneous function of
e,
:F I(~; 8) =
(7.3)
1.\li . -2 :F 1(1.; 8).
7.2. Inversion formula. The completeness of the family of horospherical functions +. means that an inversion formula for the Fourier transform :F must exist. We construct this formula by repeating the argument carried out in the case of hyperbolic plane. First, 88 in the case of hyperbolic plane, there is a simple relation between the Fourier transform :F I and the horospherical transform 'R,h I, namely,
:F I(e; 8) =
(7.4)
1-00+00
'R,h
l(e-),/2e)e¥), d.\.
Hence, by the inversion formula for the ontHlimensional Fourier transform we obtain (7.5)
'R,h 1(1.)
= (211")-1
1-00+00
:F 1(1.; 8) ds.
The inversion formula for the Fourier transform :F can be obtained from the inversion formula (6.21) for the horospherical transform 'R,h by substituting the expression for 'R,h I in terms of :F I (see (7.5» into (6.21). This gives (7.6)
I(w)
1 = -16,.-4
LC2 1+-00
00
L 2:F I(ei 8) 6(p(Wie» dsdv,
where
8
L = 1.1 8e1
8
-8
-8
+ e2 8F.2 + e1 8(1 + 1.2 81.2 + 2.
We simplify this expression. It follows from the homogeneity condition (7.3) for
:F I that L 2 :F I(e; 8) = -82 :F I(e; 8). Further, we pass in (7.6) from to the new coordinates F. = ~/, where .\ e C \ 0 and e' ranges over a surface r c C2 \ 0 intersecting once almost every complex line passing through the origin O. We obtain
e
I(w) =
1~ (~)2L:00 B2(ifcl.\I··:FI(eiB)6(2Iogl.\l+p(Wie» x d.\dl(e1 tIe2 - e2 tIe1)(e1 tIe2 - 1.2 tiel) ) ds.
Integrating with respect to .\, we finally obtain the following assertion.
8. REPRESENTATION THEORY FOR THE GROUP SL(2.C}
THEOREM
in tenns
7.1. A smooth compactly supported/u.nction / on (,3 am he ezpressed
0/ the Fouricr tmnsform F f
(7.7)
105
i loo i
1 . few) = lfur3
+oc
xcxp
= U:1T3 ~
by the following mversion formula:
s2 F f«(; 8)
2+iS) (~1~2-e2~1)(el~2-e2~1)d8 - - -( -~p(w;()
1: i 00
s2 F
f({;s)~"(W;{)({l ~2 -
{-
2~1)({1 ~ - (2~1)d8.
7.S. Symmetry relation and the Plancherel formula. Similarly to the case of hyperbolic plane. the shifts of the zonal horospberical funct.ions z:~ and z ¥ can be expressed in terms of each other,
zlT = c(8)~ where c(s)
=
*.
l 0 only,
(7.8) I(w)
t
fee;
= 8~3 ~ LOO a2 F I({i s)~.({; S)({l de2 -
(2del)((i tlf.2 - (.2 de0dJl.
REMARK. As in the case of hyperbolic plane, the symmetry relation for F can be obtained from the synametry relation (6.24) for 'Rh f.
The next assertion follows from tilt- definition of the Fburier transform F from the inversion formula (7.5). THEOREM 7.2. The following PlanchereJ fonnula holds pactly supported function I on c3 :
f
lea
(7.9)
for
I
I
and
tm1J smooth com-
l/(w)12dzdydz %3
=
~ ~ 100 82 (i 1F f(t.; 8)1 2(el ~2 - (2 ~l )(el ~2 - (2 ~l») ds
(cf. the Plancherel formula (3.14) Cor C2). Using the Planchercl formula, one can t.'Xtend the Fourier transform F f to all functlous f E L2(CJ) by the classical
scheme. 8. Relation to the representation theory for the group SL(2,C) By analogy with the case oC hyperbolic plane, we establish a relation between the results of § 7 and the represe.ntation theory for the group SL(2, C). We haw the HiJbert space L2(C3) and a unitary representation T of the group SL(2.C) ill this space.
Tgf(w)
= I(w 0 g),
9 e SL(2,C),
3. INTEGRAL GEOMETRY IN
106
e2 AND CS
where W ...... wog is the motion (of the hyperbolic space) corresponding to g. The problem of representation theory (harmonic analysis for the hyperbolic space) is to decompose L2(£,2) into the irreducible invariant subspaces. One must first describe the irreducible invariant subspaces into which the space L2(£,2) can be decomposed. Consider the eigenspace of the Laplace-Beltrami operator 4 on £,3 corresponding to an eigenvalue This subspace is spanned by the distributions •• (Wj() (see (4.1» and the elements of the subspace can be written 88
¥.
(8.1)
I.(w)
£
= ~ u.(~)•• (Zi e)(el fie2 -
(2 fiel)(el fie2 -
e2 fiel),
where u. satisfies the homogeneity condition u.(~)
(8.2)
= 1..\1,·-2u.(e),
..\eC\o,
and the integral is taken over an arbitrary surface r c C2 \ 0 that intersects once almost every complex line passing through the origin O. We equip the space of functions I. with the norm (8.3)
11/.112 =
i
lu.(e)1 2(el fie2 -
e2 fiel)(el fie2 - e2 fiel),
consider the space L2 with this norm, and denote the latter space by H.. The motions on £3 define a unitary representation T. of the group SL(2,C) in H •. In terms of functions u.(e), this representation is of the form
g=(~ ~). It is known that this representation is irreducible [8]. We introduce the operator p. : L 2 (£3) _ H. by the rule p.1 = I., where I. is given by formula (8.1) in which u.(~) = F I(e). This operator commutes with the action of the group SL(2, C) on the spaces L2(£3) and H •. Finally, using (7.8) and (7.9), we obtain the expansion of a function I e L2(£3) with respect to the irreducible suhspaces, 1 (00 I. = p./, (8.4) I 811'3 10 8 2 /. ds,
=
and the Plancherel formula (8.5) where
(
lea
lI(w)12dxd1ldz z3
= _13
(+00 s211/.11 2ds,
811' 10
11/.11 is given by (8.3).
9. Wave equation for the hyperbolic plane and hyperbollc space, and the Huygens principle 9.1. Two-dimensional cue. R.ecaIl that the Laplace-Beltrami operator on the hyperbolic plane £2 is 4 2 = y2 + and the spectrum of this operator in L2(£2) is the interval (-00, -1). Consider the following wave equation on £2:
(-£ I/;s),
(9.1)
lPu =. (lPu + lPu) + 41.1· 1 &2
.2 11
8%2
8y2
107
9. WAVE EQUATION
The term 1u is added for the spectrum of the operator on the right-hand side to coincide with the half-line (-00,0). We show how one can solve the following Cauchy problem for equation (9.1): u~(O,%)=/(%)
u(O,%) =0.
(9.2)
by using the horocycle transform; for simplicity we assume that / E V(£2). The argument is parallel to the process of solution of the Cauchy problem for the wave equation on Euclidean plane (see §3 of Chapter 1). Our solution uses the method of horocycle waves based on the following remarkable fact.
PRoposmON 9.1. 1/ !p(s) iI a smooth function
0/ one voriable,
then the func-
tion (9.3)
~(t, %)
= e:i:t/2!p(p(%; e:i:t/2(» = e:i:t/2!p(p(%; () ± t),
where p(%;() = log(1I- 11(2% - (112) iI the diltance from % to the horocycle h«(), a solution 0/ the wave equation (9.1) for each ( E R2.
u
The proof can be obtained by a direct verification. The functions (9.3) are called horocycle tlJtJve8. The method 0/ horocycle UHWe8 consists in representing an arbitrary solution of (9.1) as asuperposition ofborocycle
waves. We now proceed with the construction of a solution of the Cauchy problem (9.2). Let 'Rh / be the horocycle transform of a function / and let F(z;r) be the result of averaging the function 'Rh / over the borocycles equidistant from a point % E £2; see (2.11). Consider the function
(t ) u,z
(9.4)
= -~1+00 e~F(%,r+t)~_ 4
11' -00
It follows from (2.11) that
e~ F(%, r + t) = !e-~ 11'
J.
R2
. hr sln:l
UT.
'Rh f«() 6(P(%;() - r - t) d(l d(2,
and thus u( t, %) is a solution of (9.1) as a superposition of horocycle waves. We show that u is the desired solution of the Cauchy problem (9.2). By the symmetry relation (2.16), r ...... er / 2 F(z, r) is an even function of r; hence,
1 1+00 er / 2 F(%;r) u(O,z) = --4 sinb!: dr = O. 11' -00 2 Further, by the inversion formula (2.10) for the borocycle transform we have
.!..1+
00 (er / 2 F(z, r»~ dr = I( ) . hr Z. 11' -00 SID:I We want to retum in (9.4) to the original function /. To do this, we use the Asgeirsson relation (2.15) for the hyperbolic plane. From (2.15), for t > 0 we obtain '(0,z ) = _ 4
Ut
u(t,%) (9.5)
= _~
1+00
[00 F(z,s)sdsdr 211' -00 12181nh ~ 1 sinb !: ·/s2 - 4 sinb2 !:±! 2V'
1 [00 [
2
F(z,s)sdrds
= - 211' 10 12181nh i 1
(lIinh-l(z - Cl) -aiDh-l(Z+ a»dz -1" ainhClcoehzdz v'ainh2 6 - sinh2 Z -I> (ainh2 z - ainh2 a)v'ainh2 6 - ainh2 z .
Therefore, by elementary manipulatloDl we get J-
[
IDU
- elDU
(t2
-
ainhadt 11 sinhadt ainh2 a)v'ainh2 6 _ e2 - -1 (t 2 ainh2 6 - aiDh2 CI)v'f='i!
= aiDha 1"/2
tit
-71/2 ainh2 6ain3 t - aiDh2"
The Iaal integral can be obtained dlrectly: P. V. J
" > 6 (see formula 3.613.1 in (:l5».
=-1"/2
tit
-"/2 aiDh6ain t + ainh,,'
=0 for Cl < 6, IUKl J = - Jlialai a-slab " i for I>
9. WAVE EQUATION
and the spectrum of this operator in following wave equation on
.c3:
(9.8)
lPu lJt2
109
L2(.c3 ) is the interval (-00, -1). Consider the
lPu) = z2 (lPu ox2 + lPu 1Jy2 + OZ2
- Z
lJu
oz + u.
.c
AB in the case of 2 , the term u is added for the spectrum of the operator on the right-hand side to coincide with the half-line (-00,0). We solve the following Cauchy problem for (9.8) by the method of horospherical waves: (9.9)
u(O, w) = 0,
u:(O, w) = I(w)
(w = x
+ ui + zj).
The method uses the following property. PROPOSITION 9.2. 11 !p(s)
is a smooth function 01 one tJari4ble, then the func-
tion 4)~(t,w)
(9.10)
= e:i:c!p(p(w;{) ± t),
where p(w;{) = log(z- 1Iw{2 - {112) is the distance from a point w to a horosphere h({). is a solution 01 the wave equation (9.8) lor etJCh E C2.
e
The proof reduces to a direct verification. We refer to the solutions of the form (9.10) as horospherico.l waves. Let us proceed with solving the Cauchy problem (9.9). Let 'Rh 1 be the hornspherical transform of the function 1 and let F(w; r) be the result of averaging the function 'Rh lover the horospheres equidistant from the point w E .c3 ; see (6.20). Consider the function 1 d ,(9.11) u(t, w) = - 27f dt (e F(w, t». It follows from (6.20) that
e'F(w,t) = 7f- 2e-'
lea Rh 1({)cS(p(w;{»dv.
Since the function u( t, w) is a superposition of horospherical waves, it is a solution of the wave equation (9.8). We prove that u is the desired solution of the Cauchy problem (9.9). It follows from the symmetry relation (6.23) that e' F(w, t) is an even function of t; hence,
u(O, w)
= dtd (e.-F(w, t»I.=o = O.
FUrther, using the inversion formula (6.19), we see for the horospherical transform that
uao, w) =
-
2~ ~ (et F(w, t»I.=o = I(w).
We wish to return in (9.11) to the original function I. To do this, we use the ABgeirsson relation (6.22) for the hyperbolic space. From (6.22), for t > 0 we obtain
u(t,w) = sinhtF(w,2sinh ~),
.c3
where F( w, s) is the result of averaging the function 1 over the spheres in centered at the point W; see (6.23). Using (6.23), we obtain the following result.
3. INTEGRAL GEOMETRY IN
110
,2
AND "
THEOREM 9.2. The solution of tJ&e Cauch1l problem (9.9) for the wave equation (9.8) is of tJ&e form
u(t,w) =
8~ sinht
1,
2
sinh- 1 d(W w')
2
x 6 (4sinh2 d(W w') _ 4sinh2
I
= _1_ tanh-1 ! f (w')6 1611" 2 "
~) few') dx'~~:dzl
(sinh dew,2w') _ sinh!) dx'dy'dz' . 2 z,3
The formula thus obtained is similar to the Kirchhofl" fonnula in § 3 of Chapter 1 for the wave equation in Euclidean space. As in the Euclidean case, the value of the solution of the wave equation at a point w at a moment t depends only on the initial data at the points w' that are at the non-Euclidean distance t from the point w (the Huygens principle).
CHAPTER 4
Integral Geometry and Harmonic Analysis on the Group G = SL(2,C) In Chapter 3 we constructed integral geometry and harmonic aoaIysis on the hyperbolic spare equipped with the natural action of the group G = 5L(2, C), i.e., of the group of complex matrices 9 = (~ ~) with unit determinant, a6 -/h = 1. In this chapter \\"e present similar constructions OD the group G itself. The theory thus obtained naturally includes the theory constructed in Chapter 3 for the hyperbolic space. We regard the group G as a homogeneous space of the group G)( G, and from this point of view, we can simply treat the group G as the hyperboloid a6 -/h = 1 in et equipped with the action of the subgroup of linear transformations of C' that preserve this byperboloid.
1. Geometry
OD
the group
a
1.1. Group G as a homogeneous space. We regard G as a homogeneous space of the group G x G acting OD G by left and right translations,
9"'" g1"l g92 ,
(9t.!h)
e G)( G.
Since the isotropy subgroup of the identity element e e G is the diagonal subgroup, diag(G x G) ~ G, it follo\\'S that this homogeneous space is isomorphic to G x G/G, where Gc G x G is represented as the subgroup of elements (g,g),
geG. REMARK. The homogeneous space G is symmetric because the isotropy sul> group diag( a x a) is the subgroup of elements invariant under t,be involution 8(91,92)"'" (!h,g). The space G is not Riemaunian, but it is pseucio-Rimnannian, i.e., the metric OD this space is nondegenerate but not positive definite. This fact is of importance because integral geometry and harmonic analysis for symmetric spaces can be constructed in a more complete form than those for arbitrary h0mogeneous spaces. Nou> that the action of G x GOD G is not effective, and the inefFective kernel, i.e., the subgroup preserving all element!! of a, cousists of two eJements (E, E) and -(E, E). Therefore, the quotient group G x G/{±{E, E)} effectively acts on G. k, WDtI already said, in what follows, "ileD. considering G, we ignore the group structure and view G only as the hyperboloid in C4 with coordinates (o,P,-y,6) such that Q~-fJ'"Y=l.
The group of linear transformations of C3 that preserve the quadratic form o:~ - Ih. i.e., the group 50(4, C). act.s OD G effectively and transitively. The transformations corresponding to the elements of G )( G obviously preserve the III
112
4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP
En
form ocS - (J"Y. and we obtain an embedding of G x G/{±(E, in SO(4,C). It follows from the dimension considerations (the dimensions of these groups coincide) that we have in fact an isomorphism
SL(2,C) x SL(2,C)/{±(E.E)} ~ SO(4,C). Under the action of SO(4. C) on G, the isotropy subgroup coincides with SO(3. C), and another classical isomorphism arises,
SO(3,C)
~
SL(2,C)/{±E}.
1.2. Plane sectloDS of the hyperboloid G. We intend to relate the action of the group G x G on the hyperboloid G to the geometry of this hyperboloid. Namely, we study the sections of G by byperplanes in Col as orbits of some subgroups of GxG. It is convenient to present hyperplanes in C4 by equations of the form
(go,g)
(1.1)
= c,
90
= (oo,{Jo,"Yo,cSo),
where ( , ) is the polarization of the quadratic form ocS In the detailed notation, 1 2(600 - "Yot3 - Pu"Y + oocS) = c.
ih.
Thus, each hyperplane is given by a pair go, c. where 90 E C4 \ 0 and c E C, and this pair is defined up to a common factor. The group G x G naturally acts on the set of hyperplanes; namely, an element (91092) E G x G sends a hyperplane (go,g) c to the hyperplane (gllgo 92 ,g) c. We first describe the orbits of G x G in the set of hyperplanes. In general position «(go, go) :F 0) one can normalize go by the condition (go, go) = 1. Obviously, for any chosen number c, the hyperplanes (1.1) with (go,go) = 1 form an orbit of G )( G isomorphic to the original homogeneous space G. The remaining subset of hyperplanes (for which (go, go) = 0) is partitioned into two orbits of G x G. namely, the family of hyperplanes
=
(1.2)
=
(90,9)
= 0,
(go, go)
= 0,
tangent to the asymptotic cone K given by ocS - /3"Y = 0, and the family of hyperplanes (1.3)
(go,g) = 1,
(go,go) = 0,
parallel to the previous hyperplanes. Obviously, the second orbit is isomorphic (as a homogeneous space of the group G x G) to the asymptotic cone K with deleted vertex; the isotropy subgroup of this orbit consists of the pairs (6z 1 ,6- 1 z2 ), where %1 and %2 are matrices of the form ( Af ) and 6 is a diagonal matrix. The first orbit is isomorphic to the set of generatrices of the asymptotic cone K and has a stabilizer which is the subgroup of all pairs (6IZJ,~Z2) with ZI and Z2 as above and diagonal
cSl
and~.
Thus, the set of plane sections of the hyperboloid G is decomposed into the following orbits of the group G x G: the one-parameter family of orbits isomorphic to the homogeneous space G, the orbit isomorphic to the asymptotic cone K with deleted vertex, and the orbit isomorphic to the set of generatrices of the cone K. We now study the structure of plane sections of the hyperboloid G. It follows from the invariance considerations that it suffices to consider a single representative
1. GEOMETRY ON TKE GROUP G
113
on every orbit of G x G in the set of plane sections. For the repre5('.ntatives on the orbits in general position \\'e take the sections of G by hyperplanes of the form
(1.4)
0'+6 =c.
~ ±2. then the byperplane (1.4) intersects G along a t\\'O-dimensional hyperboloid. The hyperplane (1.4) is tangent to G in the special case c =< ±2, and the intenlection of this hyperplane with G is a cone with the vertex at the tangency
If c
point. The sections of G by the hyperplanes (1.4) have a simple group-theoretic sense.
Namely, since 0' + 6 = 1'1' 9 for 9 = (~~), the section of G by a hypcrplane + 6 = c is a conjugacy class; if c ~ ±2, then the conjugacy class is generic, and for c = ±2, the conjugacy class consist.s of elements of Jordan type (the vertex of the cone is assumed to be deleted). For representatiws of the two special orbits of G x G one can take the sections of G by the byperpJanes ..., = 0 and "'f = I, respectively. Obviously, the first section is a cylinder and the other is a paraboloid. Thus, there are four types of sections of G by hypp.rplanes in C', namely, hyperboloids, cones, paraboloids, and cylinders. Naturally, the scctioIUI of parabolic and cylindrical types are obtained by passing to the limit from scctioll.'f of hyperbolic and conic types. Every plane section of the hyperboloid G can be equipped with t.he structure of a homogeneous space. Let us first collsider the sections by plaue8 of the fonn 0' +6 = c. The subgroup diag( G x G) e! G transith'(>ly acts on each of these sections (in the case of conic sect-ion it. is assumed that the \"eI'tex of the cone is deleted); thus, 811 these sections, and hence all their shifts as well, are homogeneous spaces of G. Oil{' can readily 5CC that the group D of diagonal matrices is a stabilizer of any section of hYJ)(>.rbolic type, and the group ±Z, where Z is the subgroup of unipotent matrices (A l), is a stabilizer of every section of conic type. We now consider the sections of G by the hyperplanes 'l = 1 and 'Y = O. The subgroup Z x Z ~ freely and tral18itively on the 6rst section; thus, this section is equipped with the structure of the homogeneous space Z )( Zj{E}. The group Z x Z acts on tbe other section 88 well, but t.his action is no longer traDSitive. One can readily sce that the subgroup U C Z x Z of all pairs (6z 1,6- 1 Z2), where 6 E D and Z1,.4:2 E Z. 8oI."ts on the section t.raw.;t.il-ely. and the subgroup diag(±'Z.±Z) of pairs (±z, ±z), :z: E Z, which LOS isomorphic to ±Z, is a stabilizer ofthls scctl.on. Thus, the sectioILO"f of G of hyperbolk, conic, parabolic, and cylindrical type arc equipped \\;th the structures ofthe homogeneous spaces G/D, G/±Z, Z)( Z/{E} , and U I diag(±Z, ±Z), respecth-ely. Q
1.3. Manifold of horospheres. Consider the set of gener&trices of the bypcrboloid G. In our theory these lines plA\v the role similar to that of horospheres in tht! hyperbolic space, and we n!fer to these objects 88 horospberes by analogy with the hyperbolic space. Note that the horospheres are the generatrices of all plant> sections of the hyperboloid G. The group G x G naturally acts on the set of Iwrospheres H. auc:I this action is transitive (to prove this assertion, it suffices to consider the set of horospheres passing through an arbitrarily chosen point 9 E G and to show that the stabilizer of this point transitively acts on this set of horospheres). The horospheres on G have a simple group-theoretic interpretation. Namely. the subgroup Z c G of matrices of the fono (A f ) is one of the horoolpberes. Since
4. INTBOIlAL GEOMETRY ON THE LORENTZ GROUP
114
all horospheres are obtained from this horosphere by translations, it follows that the set of horospheres H coincides with the set of double cosets 91 1ZI/2. Therefore, HE!! (G x G)/V,
where V is the subgroup preserving the horosphere Z. One can readily see that the elements of V are all possible pairs (6z1,6z2), where Zl,%2 E Z, 6 E D. Since we obviously have dim V = 3, it follows that dim H = 3, i.e., the dimension of the space of horospheres is equal to that of G. We introduce a coordinate system on H. To this end, we first consider the set Ho of the generatrices of the asymptotic cone K given by et6 - P"1 = O. The cone K admits two one-parameter families of tWCHlimensional planes, namely, the family elet + e2"1 0, (1.5)
=
elP+e26 =0
and the family
"1 6 - 'l2"1 = 0,
(1.6)
-"lP + 'l2et = O.
In the matrix notation, this gives the family of planes eg = 0 and the family of planes g"J. = 0, where = Every generatrix of the cone K can be obtained as the intersection of a pair of planes (1.5), (1.6) and is thus given by a pair of vectors E C 2 \ 0, each defined up to a factor; we denote this generatrix by l(e, ,,). The manifold of horospheres H forms a natural bundle over Ho, with the fiber over l(e,,,) being the set of all horospheres parallel to l(e, ,,). We claim that every horosphere parallel to l(e,,,) is given in C4 by the system of equations
r L":J
e,,,
e~ et +
eh = ,,~,
eiP + e~6 = rh, ,,~6 - rh"1 = e~.
(1.7)
-fliP + rh et = e~,
e',,,'
e =
where E C2 \ 0 are vectors proportional to and ", respectively. Indeed, consider the first two and the last two equations of (1.7). On G we can represent them in the form e'g fI', ",g-1 e'; hence, these two systems of equations are equivalent on G. Hence, system (1.7) defines a line in C', and this line belongs to G. Further, it is clear that this line is parallel to l(e,,,).
=
REMARK. The horospheres parallel to the line l(e,,,) are generatrices of the cylinder obtained 88 the section of G by the hyperplane which is tangent to the asymptotic cone K along the line lee. ,,). The equation of this hyperplane is (1.8)
'l2(elet + e2"1) - '11 (elP + (26) = 0,
or e9'lJ. = 0 in the matrix form. Thus, every horosphere on G can be given by a system of equations i.e., elet + e2"1
= 'It.
F.IP + F.26 = 'l2,
eg = ",
1. GF..oMETRY ON THE GROUP G
where
115
e.." E (:2 \ 0, or by the cquh-aleut system 719- 1 =~, i.e., fl 16 - '12, = {I.
-fl18
+ rn o = {2.
Denote this horosphere by h(~,,,) and regard~. '1 as paraDlCters on the manifold of horospherM H. Since \\"e obviou.'!ly have h(~ •.\tl) = h({.fl) for each .\ #; 0, it is natural to refer to fI 311 homogeneous coordifuJtes on H. The action of an element (91.92) E G x G on H is given in the homogeneous coordinates 8.'1 follows:
e.
e,,,
h({. fI) ...... 1&(e9lt TJ92).
Note that the borosphere Z has the homogeneous coordinates ~ = '10 = (0.1); hence, every horosphere g1 1 Z92 has the homogeneous coordinates = eog}, fI = {092· It ilf of prime impurtanc(! in what follows that. along with the action of the group G x G, the manifold of horospheres H is equipped with an action of ex; namely, corresponding to each number ~ E ex is the transformation
e
h(e, ,,) ..... h({, A'I)
which takes every horosphelt' to a parallel horosphere. These transformations are called left translatimu on H. Obviously. they commute with the action of the group
GxG. We now describe the families of generatrices (horospheros) on the plalle sections r of the hyperboloid G in the homogeneou8 coordinates By virtue of the aforesaid, ea hyperboloid a6 - 8-t = 1 to the (',oordinate 8ubsp8CC C 3 with the coordinates (a, (J. 6). The image of the bypcrboloid u. the entire space C3 except for the pomt-s (a, 0, 6) such that ct6 f:. 1. \Vhen projecting the hyperboloid G to Cl, the generfttrices of G that are not parallel to the hyperplam.> {J = 0 are taktm to the lines int.erscct.ing the hyperbola (:J = O. 0'6 = 1 and not belonging to the plane fJ = O. Thus, with the manifold of hOlUlllbP.res on the hyperboloid G we 8HSociatoo a. tJtroo.dbnensional manifold K of lines in C3. In difW.n>..ntial geometry, three-dimemriollal manifolds of lines in C' arc usualJy caUed line complexes. Only one point oCtbc hypcrboloid lies over any point (0,,8,6) e C 3 On the other hand, IM.'.r any point. such that P = 0 alld 0'6 = 1, lies the entire line. Thus, the hyperboloid is obtained by pasting lines, by means of a blow-up process, in place of points of the hyperbola given by the equations f1 = O. 06 = 1. The lines of our complex are just the lines t.bat can be lift.ed to the hyperboloid as 8 result of this blow-up process. REMARK.
with {j
of: O.
1.6. Manifold of paraboloids. Now wt' describe another \"enUOIl of borospheres ill G. Consider paraboloid." in G, i.e., t~lSionaJ surfaces Cllt out in G by il:;utropic hY]:lCrpl.anel in C' not pa8lring through 0, (!Io. g)
= p,
geG,
where 90 is a nonzero point of the asymptotic cone
(go. go)
E
det(go)
= 0,
and p of: 0 is an arbitrary nOD7.CJ'O number. 1 We introduce coordinates in the space IP' of all paraboloids; these coordinates an- similar to the homogeneous coordinates in the spare of borospberes H. Set (..l = for each 2-vcctor ( = ('1'~)' Thus, ~ set of matrices 9 E G satisfying the relation "g = or. equivalently. the relation = g'
it -ax' respectlveiy.
We fioa1ly
(AT' -6 8A-I) ,A dAd.\.. 0 -
The desired inversion formula follows by changing the variable, x formulate the final result.
= ¥.
We now
THEOREM 2.1. Let /(o,tJ,,,() be a compactly supported Coo function on CS that vanishes Jor the sufficiently smalllJfJlUe8 of fj, and let cp = R.o / be the integral fronsform of f given by (2.6). Then the foUoUling inversion formu14 holds:
(2.14)
f
/(a,f~.6) = ~2 ~ .p~.X (x, ~::~ ,px + 6) dz~.
Note that the differentiation with respect to A in this formula is the differentiation with J'e!ipect to the set of parallelUnes intersecting the h.l-perbola fj = 0, 06 = 1, and thl" integral is taken over thE- set of lines intc.-rsecting this hyperbola and passing through the point (0, p, 6). As in the C8lie of the hyperbolic space, the inversion formula thus obtained is local, namely, to reconstruct the ~'8lue of a function / at a point. (a, p,~) one must know only the integrals of this function over the lines intersecting the hyperbola fJ = O. 06 = 1 that are infuUtely close to this point. REMARK. The line complex K is admissible in the sense of Chapter 2, §5. Therefore. the inversion formula (2.13) can he obtained using the operator It.
2.4. Inversion formula for the horospherical trausform. Assume that leg), 9 = (~~). is a compactly supported CtXl function on G. The problem is to express f in terms ohlle horospberical transform 'R. f(~,,1) defined by (2.3). We temporarily assume that / = 0 for all sufficiently small values of p. In tu case, the function /, viev.'ed as a function of 0, /1. 6, can be extended to a compactly support.ed COQ flmction on the entire space C 3 , and this ext.ension vanishes for all sufficiently small values of p. Hence, the extended function can be expressed in terms of cp = 'Ro / by the inversion formula (2.14). Hence, to obtain the desired result, it suffices to use the expression for 'Ro / in terms of 'R /; see {2.9}. We thus obtain (2.15)
J(9)
= 8r.12 2i /
LL'R/(x, l.crx + "(,{Jx + 6)dxdX,
4. INTEGRAL GEOMETRY ON THE LORENTZ GROllP
124
where L = (ox + '})~ + (/3x + 6)~ + 1. Using the homogeneity condition (2.4) for 'RI, one can write the inversion formula in a more symmetric form,
I(g)
(2.16)
= 8~ ~
1
L"L" 'R I(~.,,)
I,,=~g w(~)w(~).
"11rn
where L" = + 7121.;; + 1, w(~) = ~1 ~2 - ~2 ~J, and the integral is taken over an arbitrary surface '} C C 2 \ 0 which intersects once alma&1: every line passing through the origin O. Since the horospherical transform 'R is invariant with respect to the action of G x G, it follows that the initial assumption on f is unessential: namely, if the inversion formula (2.16) holds for the functions vanishing for all sufficiently small values of /3, then this formula remains valid for any compactly supported Coo function on G. This proves the following assertion. THEOREM 2.2. Let I be an arbitrary compactly supported ex> function on G. If 'R I is the horospherical transfoma of I defined by (2.3), then the inversion /omaula (2.16), which expresses I in temas o/'RI, holds.
In formula (2.16), the integral is taken over the set of horospheres h(~. ~g), i.e., the horospheres passing through a point g E G. The operator L" = 7111.,. +712/.;; + 1 occurring in the formula can be given as follows:
L"cp(e.71) = :>. (>. cp(~. >.,,»
1.\=1 .
Recall that h(~,,,) 1-+ h(eo, >.,,) is the left translation taking every horosphere to a paraUel horosphere; see 1.3. Thus, L" is the operator of the infinitesimal left translation. REMARK.
It follows from the homogeneity condition (2.4) for 'Rf that
+ L,,)'RI = (L~ + L,,)'RI = O. = el ili + ~2 Jl; + 1. Therefore, the operators L" and L" in the inversion (L~
where L~
formula (2.16) can be replaced with L L, respectively.
= ~ (e1 ~ + e2 Jl; - '1l-k - 712 ~)
and
2.5. Inversion formula for the horosphericaI transform on the hyperbolic space £3. The inversion formula for the horospherical transform 'R on G implies an inversion formula for the horospherical transform 'Rh on the hyperbolic space £3 (for the definition of 'Rh ,see Chapter 3). Namely, since C3 = U\G. where U = SU(2), it follows that each function f on £3 can be lifted to G, i.e., I can be viewed as a function 1on G satisfying the condition l(ug)
= l(g) for each u E U.
Since U is compact, it follows that the image j of a compactly supported function f on C3 is a compactly supported function on G: therefore, the space of compactly supported functions on £3 is embedded in the space of compactly supported functions on G. FUrther, one can readily see that, under the mapping G - U \ G = £3, the horospheres on G are taken to horospheres on £3 and the integrals 'R j of a function lover the horospheres on G are equal (under an appropriate agreement between the measures on the horospheres) to the integrals
:2 I"TF.GRAL GEOMETRY ON THE GROUP
a-
SL(2,C)
126
'Rh I of the function
I over the images of these horospheres. Thus, to obtain an inversion formula for 'Rh. one must only find an explicit expression for 'RI in terms of 'Rh .f and substitute this expression into tbe inversion formula (2.16) for'R.. Let us state several assertions readily following from the definitions of the borospbcres and the horospberical transforms OD G and r.3. 1. Uocler the mapping G ~ U \ G = £3. t!''ery horosphere h«(, '1) on G goos to the borosphere h(lI) on £3. 2. The transform 'R as a function of depends on]y on the variable
l.el = (I~J p~ + 1~213)1!2.
J(e, ,,),
e.
3. The functiom; 'R j and Rh I are related as follows: 'R/(-l,O;I'h.'l2)
= 'R 10 1(1I1,'l2). terms of 'Rh I follows
Tbt' desired expression for 'Ri in from tht> homogeneity condition for 'RI,
'Rj(~,7J) = 1{1-
(2.17)
4
'R/a
from 2) and 3) and
C~I)·
Substituting this expression into the inversion formula (2.16) for the borospherica1 tramd"onrl 'R on G for 9 = e, after elementary manipulations we obtain the followiog ioversion formula for the horospherical transform 'Rh 00 £3:
11
I ('IL'I) = 87r2
(2.18)
-h "Y~ L"I,,'I 'R. f(,,) w(,,) w(,,),
c.
where U'c,J e 3 is the point corresponding to the identity coset in K \ G. The expression for I at an arbitrary point of w e £3 is obtained from (2.18) by the group translation. The resulting inversion formula coincides with the in\wsion formula (6.17) of Chapter 3.
2.6. Integral transform related to paraboloids OD G. Recall (sce 1.6) that each paraboloid on G can be obtained as the section of G by a hyperplane in :R4 of the following fonn: (~lQ + ~"")'2 - (~1{:1 + ~26)'1
= p.
P'l= Ot
or, in the matrix fonn.
{g(1. =P.
(2.19)
p;i:O.
where ~ = (el. &) and , = «1. {2) are noozero vectors in C 2• and .,.,', where>. E C and 1l' E ,.". Since 'f'(>'''') = x(>')I>'I- 2'f'(1]) and d'1 = >. d>.w(,.,), it follows that we obtain (3.36) after (Tx(I)",)(f.)
integrating with respect to >.. COROLLARY. TIu! operator TX(f) has a trace. and this
(3.37)
Tr Tx(f)
= ~1
F
trace is equal to
I{~,~;x-l)w(f.)w(~).
"re
Using this result, we can evaluate the trace of a unitary operator Tx(g), 9 E G. Certainly, this operator hu no trace in the usual sense. However, it is natural to define Tr TX(g) U a distribution on the space of test functions on G, (Tr
T X {')'
f)
= Tr Tx(f)·
PROPOSITION 3.8.
(3.38) where >.~l) and >.~2) are the eigentJalues 01 the matriz g.
4. ANOTHER VERSION OF THE FOURJER TRANSFOR.'f ON G = S£{2.C)
PROOF. According to the definition of F
~
1 ~
F I({,e. X-l)W({)W(~)
=~
11 ~
Hence, if for ")'( Vt-e take the line {l =
1'1' T1«g)
f ~f
=~
6(to
=
6«s -
(3.39)
f, we have
6({2({1 0 +(2"') -
G
x
139
~1«(IP +e26»
X(ell3 ~ e~) l(g)dp{9)W(e)W(e).
t. e2 = I, then
+.., - t(tP + 6» X(fP + 6) dtdl ~~I»(s - ~~2»)X(s)dsdi
(the last equality is obtained by changing the variable immediately follows from (3.39).
tp + cS = s).
Fbrmula (3.38)
Since F I(e,,,; X-I) is the integral kernel of the operator T,,(J), we can modify the Plancherel formula. Namely, consider the operator 1',,(1)1';.(1) = fX(1. where r;,(f) is the operator conjugate to T')(/). By (3.18), the integral kernel of this operator is
r).
'flee,,,: x) = ~
1
F
1«(, (;x) F 1(". (. x)w«) wC,)·
"re
Hence,
Thus. the Plancherel formula (3.28) can be written (3.40)
f
II(g)1 2d#l(g) = 2- 6 71'-5
ko
88
c(x)Tr (1'1«(f)T;(/)) dX·
REMARK. The expansion (3.25) IIl&b8 sense not only for the test functions but also for the distributions on G. Note that it is natural. to view the delta function 6(g) on G 88 the character of the regular representation. Therefore, the inversion formula (3.25) (or 6(9) can be ?Titten as
c5(g)
(3.41)
= 2- 6 71'-5 f c(X) 1'1' TX(g) dX. Jxo
4. Another version of the Fourier transfOrm on G = SL(2, C) In § 3 we have const.ruct.ed the projections of functions on G to the eigenspaoes of the Laplace-Beltrami operators ~ and~. The projection was carried out by using the zonal. horoepherica1 functions ~'!«g) 6(')') x(o). However, one can do it in a different way, by using tht' zonal. functions (I:t given by
=
(4.1)
(I')((g)
= x- 1(1.) 1..,1- 2 ,
where 9
= (~ :)
(see 3.2). This leads to another version of the Fourier transform on G, which we now briefly describe.
4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP
140
4.1. Functions 'ir7(g;e, (). Define analogsofthe functions .,,(g;e, ,,) (see §3) by the formula (4.2)
where
e, ( E C2 \ 0 are given by the relations
e= eogl,
(4.3)
(= (092,
where
eo = (0 = (0,1).
One can readily see that the left-hand side of (4.2) does not depend OD the choice ofthe matrices 91192 E G satisfying (4.3). In particular, 'ir,,{g)
= 'ir,,(g; eo, (0).
Formulas (4.1) and (4.2) lead to the foUowing explicit expression for these functions: (4.4)
We note the foUowing two properties of 'ir 7( (which are immediate consequences of the definitions): (4.5) for any g1l!12 E G; 'ir,,{g; ~le, ~2() = X-1{~1~2) 1~1~21-2'ir,,(g;
(4.6)
e, ()
for any ~t. ~2 E C \ O. THEOREM 4.1. The functions 'ir" are related to the horospheriaJl functions ." (see § 3) as follows:
'ir,,(g;
(4.7)
e, () = ~
1
.,,{g;
e. ,,) x- {,,(.!.) 1J1(.!.1- w(,,) we,,); l
2
'l"
(4.8)
.,,{g;
e, ,,) = (2'11")-4 c{x) ~ 1'ir,,(g; e, () X{,,(.!. ) 1J1(J.1- w«) w{(), 2
'le
where c{x)
= jl + m 2 for x(t) = t.l'ei
m
• The integration is carried out over arbitrary two-dimensional real surfaces 'Y" and 'Ye in C2 \ 0 intersecting once almost every complex line passing through the origin O.
Both integrals must be understood as integrals of distributions. We do not present rigorous definitions of these integrals and restrict ourselves to a formal proof of (4.7) and (4.8). We first Dote that it suffices to prove these relations for a chosen element g and for a chosen = By setting 9 = e and = (1,0), we obtain
e eo.
."{e,eo,,,) = 6{'12) X{"l); If in (4.7) with g
eo
'ir,,{e,eo,()
= X- l «2) 1(2r2.
= e and e= eo we take for 'Y" the line "1 = 1, then the integral
becomes
~ fc 6{'12) X- l {(2 -
'12(d 1(2 - '12(11- 2 d'12 dTi2'
Obviously, this integral is equal to X- l «2) 1(21-2, i.e., to 'ir,,(e,eo,().
... ANOTHER VERSION OF THE FOURIER TRANSFORM ON G ... SL(2.C)
Obviously, the distribution
"'('12) = ~
L
X(I- '12(1) 11
satisfies the homogeneity condition 'P{>''12)
141
- '12(11- 2 (/(1 del
= 1>'1-2 '1'('12). This implies that 'P('12) =
c6('12). We leave the evaluation of the factor c to the reader as a useful exercise integration of distributions.
G. DEFINITlON. We define the Founer tronsloTTn ex' function lonG by the rule 4.2. Fourier transform.
(4.9)
OD
OD
FI
of a compactly supported
FI«(.(;xJ = ("'~(·.e.(), I) = L/(gh- 1 «(g(i)legt;J.1- 2 dp(g).
There is a simple relation between the Fourier transform FI and the integral transform 'R." I in § 2 related to paraboloids on G. Namely, comparing the definition of FI with the de6nition of 'Rl'l (see (2.20)). we obtain the following assertion. THEOREM 4.2.
The operators F and'R,P are related as lollows:
F/(e,(:x) = ~
(4.10)
f
'R,P l(e,(;p)x-l(p)lpl-2dpdp.
Thus, FI is obtained from 'R,1' I by using the Me1lin transform with respect to p; cr. the relation between F I and 'R, I in 3.4. Hence, applying the i.nvenion formula for the Mellin transform, we obtain the following assertion. COROLLARY. The
junction 'R,P I can be upressed in tenns 01 Ff as follows:
'R,P I({,(,p)
(4.11)
= (211')-2
Ix
FI«(,(,X)X(P)dX,
where the integral is taken over the set X of unitary chamcters.
4.3. Relation between the above two versions of the Fourier transform. It follows from the relations between the borospberical functions tt ~ and .x (see 4.1) that the functions Ff and F I satisfy the same relations, where F is the Fourier operator introduced in § 3. THEOREM 4.3.
The Fourier operators F and F ~ related as /ollouzs:
F/({, (; X) = ~
(4.12)
1
F I({, 71a)x.-l(flC.l.) l«il-2 W(71) w('I);
'Y.,
(4.13)
F
I(~, 'I; X) = (211')-4c(X)~
1FI(e,(; 'Y(
where c(x) =,; + m 2 for x(t)
= t~l"i'" .
x) x( reoonstmcted from the rest.riction of the function to The reconstruction of f from llr is carried out by the general method of the operator 1'1':. which was first applied in [1]. This method was explained in Chapter 2 using, as an example, the John transform in In the case considered in chapter. for any point x EX, we assign to a function a closed differential form Kz! of degree n on the manifold of plane sections. The integrals of this form over various cycles are equal to the value of the original function! at the point x. This is a rare case where the method of the operator" is developed for a nonlocal problem of real integral geometry. Many fomlUlas of this chapter can be written for arbitrary quadrks; however, the choice of appropriate cycles giving the inversion formulas is diJlicult and often impossible. For this reason ...-e restrict oun!elves to the case of a byperboloid 01 two sheets and a sphere (which are projectively equivalent). The case of a byperboloid of two sheets gives another proof of the inversion formulas in problems of integral geometry related to the hyperbolic space (see § 3). The case of a sphere is equivalent to the problem of integral geometry related to spheres in Rft. The operator K for quadrics was COWItrUcted by Gindikin (15, 16] and Goncharov [24). The problem of sections of the quadric can be interpreted as 8 conformally invariant extension of the Radon transform. In this interpretation the Radon transform (Chapter 1) and the horospherical hyperbolic transform are conformally equivalent.
lof
1 r.
cs.
1
tu
1. Integral transform related to the byperpIane sectloDs of a byperboloid of two sheets In RII+1 1.1. Deftnition. Let en be the upper sheet of the hyperboloid of two sheets in Rn+l given by
= ~ - ~ -'" - x~ = 1, Xo > O. Consider the sections of en by the h}1)erplanes [~. xl = pin RM·l, where (~.x] = ~oXo - ~lXl - ••• - enxn· (1.1)
[x.x]
148
146
5. INTEGRAL GEOMETRY ON QUADRJCS
We recall that en is one of the models of the n-dimensional hyperbolic space. and corresponding to the hyperplane sections of en are various spheres in the hyperbolic space and their limit cases, namely, the geodesic hypersurfaces and the horospheres. Generic spheres occur as sections of en by the hyperplanes
[e,X]
= p,
where p ::/= 0 and le,e] ::/= O. The sections related to the limit cases p = 0 and le, el = 0 are the geodesic hypersurfaces and the hor08pheres, respectively. Thus, the manifold of spheres is (n + 1)-dimensional, and the manifolds of horospheres and of the geodesic hypersurfaces are n-dimensional. Note that the horospheres and the geodesic hypersurfaces are the only ndimensional families (in the manifold of all hyperplane sections) that are invariant with respect to the motion group 50(1, n) of the hyperbolic space. We introduce a differential n-form #leX) on Rn+! satisfying the formula
d([x, x]) " #leX)
= dx,
where dx
= dxo""'" dxn
(i.e., #leX) is an interior product ofthe differential forms dx and d([x, xJ)). Such a form on Rn+l is not unique; for instance, it can be given by the relation 1
#leX)
= 2 le, (I
n+!
. L(-I)'ei
,=0
However, the restriction of this form to on en, this restriction is
(\dei'
);i,
en is uniquely defined.
In the coordinates
Xl, •.• ,Xn
#leX)
... " = dx. " 2.xo
dx n
•
We introduce the integral transform 'R taking a function I on en to the integrals of lover the sections of en by the hyperplanes [e,x] = pin Rn+l, (1.2)
'RI«(,p) =
1 t:. ..
I(x) 6(1(, xl - p) #leX).
Assume that the functions I are compactly supported and infinitely differentiable on en. Obviously, 'RI(e,p) = 0 if the hyperplane le.x) = p is disjoint
from en.
REMARK. The integral transform 'R can be regarded as the Radon transform ofthe distribution I(x) 6([x, xl -1) on Rn+!.
It immediately follows from (1.2) that satisfying the homogeneity condition (1.3) and the differential equation
(1.4) Note that (1.4) is the wave equation.
I{J
= 'RI is a function
,\ ER, 0,
011
Rn+2 , 0
1. INTEGRAL TRANSFORM RELATED TO HVPEItPLANE SECTIONS
147
1.2. Admissible submanlfolds in the manifold of hyperplane sections of en. An n-dimensional 8ubmanifold r of the manifold of hyperplane sections of en is said to be admissible if any function I on en can be reconstructed from the restriction of the function tp = RI to r. In Chapter 3 we obtained inversion formulas (for n = 2 and n = 3) reconstructing a function I on the hyperbolic space e" from the integrals of this function over the horospheres or over the geodesics. In particular, we have established that the manifold of horospheres and the manifold of geodesics are admissible submanifolds of the manifold of all hyperplane sections. We now present, for an arbitrary n, three examples of admissible submanifolds in the space of hyperplane sections of e". In each of these examples, the proof of the inversion formula is reduced to the inversion formula of Chapter 1 for the Radon transform in the space Rn. EXAMPLE
1.
r
is the manifold of sections of e" by the hyperplanes
[e,x]
= P.
where
eo = O.
Denote by ,,(e1,'" ,(",p) the restriction of the function
f{J
= RI to r, i.e.,
"«(1, .. ',(n,P) = Jr." f 12(X) 6«(lX1 + ... + {nX" + p) dx 1 /\ '" Xo
/\ dx".
IJ!;!
We see that" is the Radon transform of the function viewed as a function of Xl, ... , Xn on Rn. Hence, by using the inversion formula for the Radon transform (see 2.3 of Chapter 1). we obtain the desired inversion formula:
(1.5) 2xo
11
00
.,
-00
"«(1, ... ,(n,p) «(IX1 + .. ·+enXn +p-iO)-n dp/\a«() = en I(x),
where en = iJr:l~)" a«() = E:=t (-I)i-1(i I\#i de:l, and 'Y C R" is an arbitrary surface intersecting once every ray issuing from the origin O. In particular, we have proved that the manifold r is admissible. EXAMPLE
2.
r
is the manifold of sections of
[(.x] Denote by
= P.
where (0
en by the hyperplanes = (1.
"(e1 .... ,(n,P) the restriction of the function
"«(1 •...• (n,P)
= f{J«(lt(I •... ,(n,p) = f I(x) 6«(1 (XI Jr."
xo) +
(2X2
f{J
= RI to r, i.e.•
+ ... + (nXn + p) p(x).
We find the inversion formula reconstructing I from the function ". To this end, we introduce a new system of coordinates on
en.
111
= Xl -
Xo.
Y2
= X2 • •••• 1/n = X n •
and denote by 11 (fI) the original function I in these coordinates. It follows from the assumptions imposed on I that the function 11 is defined on the domain 111 < 0 and is compactly supported in this domain. Assume that 11 is extended to the entire space Rn by setting l1(y) = 0 for Y1 ~ O. In the coordinates fI. the differential form I' becomes
p(y) =
148
S. INTEGRAL GEOMETRY ON QUADRICS
and hence (1.6)
1/1(~1o ••. '~n'P) = - f
la..
121 (y) J/I
6(~IYI + ... + (nXn + p) dYI A. ••• A. dJ/no
Thus, 1/1 is the Radon transform of the function F(y) = _/~~!). Therefore, it foUows from the inversion formula for the Radon transform that
-2YI
11
00
..,
-00
tJ1(~I' ... '~n'P) «(iYI + ... + ~nYn + P -
io)-n dp A. w(~)
= en I(x) •
where en, w(~), and "'1 are defined in Example 1. Returning from Y to the original variables x, we obtain the desired inversion formula (1.7)
2 (xo - XI)
11°C tJ1(~I' ..,
... ,(n,P)
-00
x (ei(XI - xo) + ~2X2 + ... + ~nxn + P - io)-n dp A. w(~)
= en I(x).
REMARK. The manifolds r in Examples 1 and 2 have a simple geometric description; namely, they consist of the sections of en by the hyperplanes passing through the points at infinity with the homogeneous coordinates (1,0, ... ,0) and (1,1,0, ... ,0), respectively. The first point does not belong to the projectivization of the hyperboloid, whereas the other point belongs to the projectivization. EXAMPLE 3. r is the manifold of geodesic hypersurfaces in en, i.e., the submanifold of sections by the hyperplanes passing through the origin, I~, x] = O. Denote by tJ1(eo, ... '~n) the restriction of the function r.p = 'RI to r, i.e.,
1
t/J«(o, ... , en) = -2
1
fez) cS([e. zl>
"..
dz l A. ••• A. dz n
.
Xo
One can view I(x) as the restriction to en of a homogeneous function on Rn+l \ 0 defined by the relations I(>.x) = I.\I-n I(x) for any .\ :/: 0 and x e en and I(x) = 0 for Ix.x] o. We also note that the differential form (xo)-ldz 1 A. ••• A. dz n in the integrand is the restriction to en ofthe differential form Wn +1 (x) = E:'::o(-l)iXiAj",idzJ on Rn+l. Hence (see 8.8 in Chapter 1). the function 2t/J«(o, ... ,en) is the projective Radon transform ofthe homogeneous function I(x). Thus, in view of the inversion formula for the projective Radon transform (see 8.8 in Chapter 1), we obtain
:s
(1.8) where
2 it/J(~o •... '~n)(le,X]-io)-nwn+l(~)=en/(X),
en = ijr:~~)!·
1.3. Operator ,,~. Let us proceed with constructing a universal inversion formula for the integral transform r.p = 'RI. For any point x e en, we construct an operator ,,~ from a function space on the manifold of hyperplane sections to the space of differential forms of degree n on this manifold. If r.p = 'RI, then the dif£erential form "~r.p is closed. We also claim the following inversion formula for various cycles r : if r.p = 'RI, then
i "~r.p =
c/(x).
1. INTEGRAL TRANSFORM RELATED TO HYPERPLANE SECTIONS
149
We first introduce the following notation which is useful in what follows. We denote by 100, ... , Bn+ll the determinant of order n + 2 with columns ai some of which consist of 1-forms. We expand the determinant from left to right and assume that multiplication of I-forms is the exterior multiplication. Such a determinant is symmetric with respect to the columns consisting of I-forms and may contain identical columns of 1-forms without being equal to zero. For instance, under this notation, the differential form ,,+1
w(e)
= L(-I)~ei A~j ;.Fi
i=O
becomes
1
wee) = (n + 1),I{,~, ... , ~I, or, briefly,
wee) = {n+1 1)1' le ~("+1)1 , where ~("+1) stands for the sequence of (n + 1) equal columns~. For convenience, in what follows we write e,,+l instead of p and 'P{e) instead of 'P(e,p), where e = (eo, .. . ,e,,+1) E R,,+2 \ O. Let 'P{e) be an arbitrary infinitely differentiable function on R,,+2 \ 0 satisfying the homogeneity condition 'P(.\e) = I~I-l 'P{e) for ~ ~ D. To the function 'P and to an arbitrary point % E e" we assign the follOWing distributions on Rn+2:
%i 'P(e)
(1.9)
.,(e, %) = ([e, %] - e"+1 - iD)"
+ _1_ n- 1
Ei'P(, (e) . , aO),,-l i = 0,1, ... , n + I,
{le, %] - e"+1 -
=
where EO = I, Ei = -1 for i > 0, and %,,+1 1. Recall that the distributions ([{,%] - e"+1 - iO)-n (see [10]) were introduced in Chapter 1 and used there in the inversion formulas for the Radon transform in the affine and projective ndimensional spaces; see 2.3 and 8.8 in Chapter 1. Note that %) can be given as
.,(e,
(1.10)
.,«(,%) = Ei n~ 1 {([e,%1 -e"+1- iO)-,,+l,'P(e) }i'
where the braces stand for the Poisson brackets
{'PI, 'P2h
8'P2
= 'PI 8ei
/Jtpl - 'P2 8ei .
en
To every point % E we assign the following operator I£z from the space of functions 'P(e) to the space of differential n-forms whose coefficients are distributions:
(1.11) where .((,%) is the column with the elements .i({,%) defined by (1.9). Expanding the determinant (1.11) in the elements of the 6rst column, we obtain the representation of I£z'P in the form ,,+1
(1.l2)
I£z'P =
L .i(e,%)Wi({), i=O
11. INTEGRAL GEOMETRY ON QUADRlCS
150
where n+l
WO(e)
= E(-l)'-I~i
"
d{;.
j~O.i
i=1
and the subsequent forms Wi are obtained from WO by a cyclic permutation of the coordinates ~n+l. and the sign depends on the parity ofthe corresponding permutation. It follows from the definition that the differential form "ztp is homogeneous of degree zero; moreover. this form is orthogonal to the fibers of the bundle Rn+2\0 5"+l. where 5"+l is the (n + I)-dimensional sphere. Hence. this form is defined on the manifold 5"+l of rays in Rn+2 issuing from the origin.
eo •..••
THEOREM 1.1. The differential/orm /Cztp is closed on Rn+2 for any point x E en and any function tp(e) on Rn+2 satisfying the homogeneity condition tp(~) = 1-'1- 1 tp(e) Uor -' :/: 0) and the differential equation
{fltp
Dtp
= a~~ -
{fltp
a~l
(fltp
- ... - ae~+l = o.
The proof of the theorem results from the following general lemma on determinants. LEMMA. [laCe) is a column o/length n+2 whose elements ao(~), ... , an+l(e) are homogeneous functions 0/ degree -n - 1, then
dla(e),e,~(n+1)] = __1_ n+ 1
(E aeJ
8aj (e»)
;=0
[e,~(n+1)].
This lemma can be applied to the determinant (1.11) because the elements of the column +(e.x) are homogeneous of degree -n - 1. Hence, by the lemma we have dtcztp
= __1_
(E
n+ 1 ;=0
M;(e, X») [e,d{(n+1»). a~
On the other hand, it follows from the definition of the functions +j(e. x) in the form of Poisson brackets that
E
Mj(e.x)
;=0
8e;
= ([e. x]- e"+1 Hence. d("ztp) = O. COROLLARY.
io}-n+1 Otp(e) - tp(e) o([e. x]- en+l - iO)-n+l
= O.
1/ a function tp belongs to the image 0/ the integral transform 'R.,
then the differential/arm /Cztp is closed.
In conclusion we present the proof of the lemma. It suffices to prove it for the case in which only one element of the column aCe) is nonzero. say. the element ao(e). Then
u ==
[aCe). e. d{(n)1 = n! ao(e) Wo(e).
I. INTEGRAL TRANSFORM ftELAnV TO HYPERPLANE SECTIONS
1st
where the form wo({) was defined above. Hcncc.
1 n+1 800({) -. du :;: ~ d{. /\ wo(~)
L
n.
.=0
v ...
+ ao({) /\ dwo({).
By the Euler theorem on homogeneous functions, ao«() Thus, since dwo«()
= __1_
n+ 1
E
8ao({) (1'
0
.=
8(;
= (n + 1)d{1/\ "'/\d{n+l, it follows that
ao(~)/\dwo(e):;: -
n+18a o«() L ~ei~I/\···/\den+l' ,=0 .,...
Substituting this expression into the formula for duo we obtain
..!.. du = OaO«() (d(o /\ '"-'G({) - eo d{1 /\ ... /\ d{n+d n!
8(0
+
n+l OaO(~)
t; ~(d{i
/\ wo«() -{.d(l/\"· 1\11(11+1)'
It remains to note that the coeffident of ~() is -rn:hJy[{,d«"+l)] and the coefficients of the other deri\'&tivcs vanish. REMARK. One can extend both the definition of the operator K-z and the proof that the differential foml Kzl{) is closed to any nondegenerate quadric in R"+2.
1.4. Local and nonlocal operators K.. The distl'ibution (t-io)-n occurring in the definition of the operator K-z cau be represented 88 a tinear combination of the distributions t- n and 6(n-l)(t), (t - iO)-n
(1.13)
= t- n + ur. ( -
l)n-l
6(n-l)(t)
(n. - I)! (see, for instance, 110». Moreover, depending on the parity of n, one of the terms is an ('"\"en distribution and the other is odd. Accordingly, the operator K-z can also be represented 88 the sum of loca.l and uonlocaJ part.s. Namt>Jy, dt-..6nt> the operators K~ and K; by (1.14)
K~'; :;:
n+l
L +:«(, X)Wi«(), ,=0
n+l
K~I{J
=L
+:'({,x)Wt(e),
i=O
where
, i",(-I),,-1 ( n-I .i(~' x) = (n _ I)! Xj !p(~) 6< )([e, xl - (,,+d
(US)
- Ei'PE< ({) 6("-2)([~.xl - (,.+I»); .:'(~,J:)
= x, r;({)([{,x]- {n+1)-n + Ej
