Contemporary Mathematics

Some Personal Remarks on the Radon

Transform.

Sigurdur Helgason

1. Introduction.

The editors have kindly asked me to write here a personal account ofsome of my work concerning the Radon transform. My interest in thesubject was actually evoked during a train trip from New York to Bostononce during the Spring 1955.

2. Some old times.

Back in 1955, I worked on extending the mean value theorem of LeifurAsgeirsson [1937] for the ultrahyperbolic equation on Rn ×Rn to Rie-mannian homogeneous spaces G/K ×G/K. I was motivated by Gode-ment’s generalization [1952] of the mean value theorem for Laplace’sequation Lu = 0 to the system Du = 0 for all G-invariant differentialoperators D (annihilating the constants) on G/K. At the time (Spring1955) I visited Leifur in New Rochelle where he was living in the houseof Fritz John (then on leave from NYU). They had both been studentsof Courant in Gottingen in the 1930’s. Since John’s book [1955] treats

Asgeirsson’s theorem in some detail, Leifur lent me a copy of it (in pageproofs) to look through on the train to Boston.

I was quickly enticed by Radon’s formulas (in John’s formulation)for a function f on Rn in terms of its integrals over hyperplanes. InJohn’s notation, let J(ω, p) denote the integral of f over the hyperplane

1991 Mathematics Subject Classification. Mathematics Subject Classification Pri-

mary 43A85, 53 C35, 22E46, 44A12; Secondary 53 C65, 14 M17, 22 F30.

c©0000 (copyright holder)

1

2 SIGURDUR HELGASON

〈ω, x〉 = p (p ∈ R, ω a unit vector), dω the area element on Sn−1 and Lthe Laplacian. Then

f(x) = 12(2πi)

1−n(Lx)(n−1)/2

∫

Sn−1

J(ω, 〈ω, x〉) dw , n odd.(2.1)

f(x) = (2πi)−n(Lx)(n−2)/2

∫

Sn−1

dω

∫

R

dJ(ω, p)

p− 〈ω, x〉, n even.(2.2)

I was surprised at never having seen such formulas before. Radon’spaper [1917] was very little known, being published in a journal thatwas hard to find. The paper includes some suggestions by Herglotz inLeipzig and John learned of it from lectures by Herglotz in Gottingen.I did not see Radon’s paper until several years after the appearance ofJohn’s book but it has now been reproduced in several books aboutthe Radon transform (terminology introduced by F. John). Actually,the paper is closely related to earlier papers by P. Funk [1913, 1916](quoted in Radon [1917]) which deal with functions on S2 in terms oftheir integrals over great circles. Funk’s papers are in turn related to apaper by Minkowski [1911] about surfaces of constant width.

3. General viewpoint. Double fibration transform.

Considering formula (2.1) for R3,

(3.1) f(x) = −1

8π2Lx

( ∫

S2

J(ω, 〈ω, x〉 dω))

it struck me that the formula involves two dual integrations, J the in-tegral over the set of points in a plane and then dω, the integral over

the set of planes through a point. This suggested the operators f → f ,

ϕ → ϕ∨defined as follows:

For functions f on R3, ϕ on P3 (the space of 2-planes in R3) put

f(ξ) =

∫

ξ

f(x) dm(x) , ξ ∈ P3 ,(3.2)

ϕ∨(x) =

∫

ξ∋x

ϕ(ξ) dµ(ξ) , x ∈ R3 ,(3.3)

where dm is the Lebesgue measure and dµ the average over all hyper-planes containing x. Then (3.1) can be rewritten

(3.4) f = −12L((f)

∨) .

PERSONAL REMARKS 3

The spaces R3 and P3 are homogeneous spaces of the same group M(3),the group of isometries of R3, in fact

R3 = M(3)/O(3) , P3 = M(3)/Z2M(2) .

The operators f → f , ϕ → ϕ∨

in (3.2)–(3.3) now generalize ([1965a],[1966]) to homogeneous spaces

(3.5) X = G/K , Ξ = G/H ,

f and ϕ being functions on X and Ξ, respectively, by

(3.6) f(γH) =

∫

H/L

f(γhK) dhL , ϕ∨(gK) =

∫

K/L

ϕ(gkH) dkL .

HereG is an arbitrary locally compact group,K andH closed subgroups,L = K ∩H, and dhL and dkL the essentially unique invariant measureon H/L and K/L, respectively. This is the abstract Radon transform forthe double fibration:

(3.7) X = G/K Ξ = G/H

G/L

✡✡✡✢

❏❏❏❫

The operators f → f , ϕ → ϕ∨map functions on X to functions on Ξ

and vice-versa. These geometrically dual operators also turned out tobe adjoint operators relative to the invariant measures dx and dξ,

(3.8)

∫

X

f(x)ϕ∨(x) dx =

∫

Ξ

f(ξ)ϕ(ξ) dξ .

This suggests natural extensions to suitable distributions T on X, Σ onΞ, as follows:

T (ϕ) = T (ϕ∨) , Σ

∨(f) = Σ(f) .

Formulas (2.1) and (2.2) have another interesting feature. As func-tions of x the integrands are plane waves, i.e. constant on each hyper-plane L perpendicular to ω. Such a function is really just a function of asingle variable so (2.1) and (2.2) can be viewed as a decomposition of ann-variable function into one-variable functions. This feature enters intothe work of Herglotz and John [1955]. I have found some applicationsof an analog of this principle for invariant differential equations on sym-metric spaces ([1963], §7, [2008],Ch. V§1, No. 4), where parallel planesare replaced by parallel horocycles.

4 SIGURDUR HELGASON

The setup (3.5) and (3.6) above has of course an unlimited supply ofexamples. Funk’s example

(3.9) X = S2 , Ξ = {great circles on S2}

fits in, both X and Ξ being homogeneous spaces of O(3).Note that with given X and Ξ there are several choices for K and

H. For example, if X = Rn we can take K and H, respectively, as theisotropy groups of the origin O and a k-plane ξ at distance p from O.Then the second transform in (3.6) becomes

(3.10) ϕ∨p (x) =

∫

d(x,ξ)=p

ϕ(ξ) dµ(ξ)

and we get another inversion formula (cf. [1990], [2011]) of f → f

involving (f)∨p (x), different from (3.4).Similarly, for X and Ξ in (3.9) we can take K as the isotropy group

of the North Pole N and H as the isotropy group of a great circle at

distance p from N . Then (f)∨p (x) is the average of the integrals of fover the geodesics at distance p from x.

The principal problems for the operators f → f , ϕ → ϕ∨would be

A. Injectivity.B. Inversion formulas.C. Ranges and kernels for specific function spaces of X and onΞ.D. Support problems (does compact support of f imply compactsupport of f?)

These problems are treated for a number of old and new examplesin [2011]. Some unexpected analogies emerge, for example a completeparallel between the Poisson integral in the unit disk and the X-raytransform in R3, see pp. 86–89, loc. cit..

My first example for (3.5) and (3.6) was for X a Riemannian man-ifold of constant sectional curvature c and dimension n and Ξ the setof k-dimensional totally geodesic submanifolds of X. The solution toProblem B is then given [1959] by the following result, analogous to(3.4):

For k even let Qk denote the polynomial

Qk(x) = [x− c(k−1)(n−k)][x− c(k−3)(n−k+2)] · · · [x− c(1)(n−2)] .

Then for a constant γ

Qk(L)((f)∨) = γf if X is noncompact.(3.11)

Qk(L)((f)∨) = γ(f + f ◦ A) if X is compact.(3.12)

PERSONAL REMARKS 5

In the latter case X = Sn and A the antipodal mapping. The constantγ is given by

Γ(n2

)/Γ

(n− k

2

)(−4π)k/2 .

The proof used the generalization of Asgeirsson’s theorem. At that timeI had also proved an inversion formula for k odd and the method usedin the proof of the support theorem for Hn ([1964, Theorem 5.2]) butnot published until [1990] since the formula seemed unreasonably com-plicated in comparison to (3.11), (3.12) and since the case k = 1 when

f → f is the ”X-ray transform”, had not reached its distinction throughtomography.

For k odd the inversion formula is a combination of f and the analogof (3.10).

For X = Rn, Ξ = Pn, problems C–D are dealt with in [1965a]. Thispaper also solves problem B for X any compact two-point homogeneousspace and Ξ the family of antipodal submanifolds.

4. Horocycle duality.

In the search of a Plancherel formula for simple Lie groups, Gelfand–Naimark [1957], Gelfand–Graev [1955] and Harish-Chandra [1954], [1957]showed that a function of f on G is explicitly determined by the integralsof f over translates of conjugacy classes in G. This did not fit into theframework (3.5)(3.6) so using the Iwasawa decomposition G = KAN(K compact, A abelian, N nilpotent) I replaced the conjugacy classesby their “projections” in the symmetric space G/K, and this leads tothe orbits of the conjugates gNg−1 in G/K. These orbits are the horo-cycles in G/K. They occur in classical non-Euclidean geometry (wherethey carry a flat metric) and for G complex are extensively discussed inGelfand–Graev [1964].

For a general semisimple G, the action of G on the symmetric spaceG/K turned out to permute the horocycle transitively with isotropygroup MN , where M is the centralizer of A in K [1963]. This leads(3.5) and (3.6) to the double fibration

(4.1) X = G/K Ξ = G/MN

G/M

✡✡✡✢

❏❏❏❫

and for functions f onX, ϕ on Ξ, f(ξ) is the integral of f over a horocycle

ξ and ϕ∨(x) is the average of ϕ over the set of horocycles through x. My

6 SIGURDUR HELGASON

papers [1963], [1964a], [1970] are devoted to a geometric examination ofthis duality and its implications for analysis, differential equations andrepresentation theory. Thus we have double coset space representations

(4.2) K\G/K ≈ A/W , MN\G/MN ≈ W ×A

based on the Cartan and Bruhat decomposition of G , W denoting theWeyl group.

The finite-dimensional irreducible representations with a K-fixed vec-tor turn out to be the same as those with anMN -fixed vector. This leadsto simultaneous imbeddings of X and Ξ into the same vector space andthe horocycles are certain plane sections with X in analogy with theirflatness for Hn [2008,II,§4]. The set of highest restricted weights of theserepresentations is the dual of the lattice ΣiZ

+βi where β1, . . . , βℓ is thebasis of the unmultipliable positive restricted roots.

For the algebras D(X),D(Ξ), respectively D(A), of G-invariant (resp.A-invariant) differential operators on X, Ξ and A we have the isomor-phisms

(4.3) D(X) ≈ D(A)/W , D(Ξ) ≈ D(A) .

The first is a reformulation of Harish-Chandra’s homomorphism, thesecond comes from the fact that the G-action on the fibration of Ξ overK/M is fiber preserving and generates a translation on each (vector)

fiber. The transforms f → f , ϕ → ϕ∨intertwine the members of D(X)

and D(Ξ). In particular, when the operator f → f is specialized toK-invariant functions on X it furnishes a simultaneous transmutationoperator between D(X) and the set of W -invariants in D(A) [1964a,§2]. This property, combined with a surjectivity result of Hormander[1958] and Lojasiewicz [1958] for tempered distributions on Rn, yieldsthe result that each D ∈ D(X) has a fundamental solution [1964a].

A more technical support theorem in [1973] for f → f on G/K thenimplies the existence theorem that each D ∈ D(X) is surjective onE(X) i.e. (C∞(X)). It is also surjective on the space D′

0(X) of K-finite distributions on X ([1976]) and on the space S′(X) of tempereddistributions on X ([1973a]). The surjectivity on all of D′(X) howeverseems as yet unproved.

The method of [1973] also leads to a Paley–Wiener type Theorem

for the horocycle transform f → f , that is an internal description ofthe range D(X) . The formulation is quite different from the analogousresult for D(Rn) . Having proved the latter result in the summer of1963, I always remember when I presented it in a Fall class, becauseimmediately afterwards I heard about John Kennedy’s assassination.

PERSONAL REMARKS 7

In analogy with (3.4), (3.11), the horocycle transform has an inversionformula. The parity difference in (2.1), (2.2) and (3.11), (3.12) now takesanother form ([1964], [1965b]):

If G has all its Cartan subgroups conjugate then

(4.4) f = �((f)∨) ,

where � is an explicit operator in D(X). Although this remains “for-mally valid” for generalG with� replaced by a certain pseudo-differentialoperator, a better form is

(4.5) f = (Λf)∨ ,

with Λ a certain pseudo-differential operator on Ξ. These operators areconstructed from Harish-Chandra’s c-function for G. For G complex aformula related to (4.5) is stated in Gelfand and Graev [1964], §5.

As mentioned, f → f is injective and the range D(X) is explicitly

determined. On the other hand ϕ → ϕ∨is surjective from C∞(Ξ) onto

C∞(X) but has a big describable kernel.By definition, the spherical functions on X are the K-invariant eigen-

functions of the operators in D(X). By analogy we define conical distri-bution on Ξ to be the MN -invariant eigendistributions of the operatorsin D(Ξ). While Harish-Chandra’s formula for the spherical functionsparametrizes the set F of spherical functions by

(4.6) F ≈ a∗c/W

(where a∗c is the complex dual of the Lie algebra of A) the space Φ of

conical distribution is “essentially” parametrized by

(4.7) Φ ≈ a∗c ×W .

Note the analogy of (4.6), (4.7), with (4.2). In more detail, the sphericalfunctions are given by

ϕλ(gK) =

∫

K

e(iλ−ρ)(H(gk)) dk ,

where g = k expH(g)n in the Iwasawa decomposition, ρ and λ as in(5.3) and λ unique mod W . On the other hand, the action of the groupMNA on Ξ divides it into |W | orbits Ξs and for λ ∈ a

∗c a conical distribu-

tion is constructed with support in the closure of Ξs. The construction isdone by a specific holomorphic continuaton. The identification in (4.7)from [1970] is complete except for certain singular eigenvalues. For thecase of G/K of rank one the full identification of (4.7) was completedby Men-Cheng Hu in his MIT thesis [1973]. Operating as convolutionson K/M the conical distributions in (4.7) furnish intertwining operators

8 SIGURDUR HELGASON

in the spherical principal series [1970, Ch. III, Theorem 6.1]. See alsoSchiffmann [1971], Theoreme 2.4 and Knapp-Stein [1971].

5. A Fourier transform on X.

Writing f(ω, p) for John’s J(ω, p) in (2.1) the Fourier transform f onRn can be written

(5.1) f(rω) =

∫

Rn

f(x)e−ir〈x,ω〉 dx =

∫

R

f(ω, p)e−irp dp ,

which is the one-dimensional Fourier transform of the Radon transform.The horocycle duality would call for an analogous Fourier transform onX.

The standard representation–theoretic Fourier transform on G,

F (π) =

∫

G

F (x)π(x) dx

is unsuitable here because it assigns to F a family of operators in differentHilbert spaces. However, the inner product 〈x, ω〉 in (5.1) has a certainvector–valued analog for G/K, namely

(5.2) A(gK, kM) = A(k−1g) ,

where expA(g) is the A-component in the Iwasawa decomposition G =NAK. Writing for x ∈ X, b ∈ B = K/M ,

(5.3) eλ,b(x) = e(iλ+ρ)(A(x,b)) , λ ∈ a∗c , ρ(H) = 1

2 Tr (ad H|n)

we define in [1965b] a Fourier transform,

(5.4) f(λ, b) =

∫

X

f(x)e−λ,b(x) dx .

The analog of (5.1) is then

f(λ, kM) =

∫

A

f(kaMN)e(−iλ+ρ)(log a) da .

The main theorems of the Fourier transform on Rn, the inversion for-mula, the Plancherel theorem (with range), the Paley–Wiener theorem,the Riemann–Lebesgue lemma, have analogs for this transform ([1965b],[1973], [2005], [2008]). The inversion formula is based on the new identity

(5.5) ϕλ(g−1h) =

∫

K

e(iλ+ρ)(A(kh))e(−iλ+ρ)(A(kg)) dk .

PERSONAL REMARKS 9

Some results have richer variations, like the range theorems for thevarious Schwartz spaces Sp(X) ⊂ Lp(X) (Eguchi [1979]).

The analog of (5.4) for the compact dual symmetric space U/K wasdeveloped by Sherman [1977], [1990] on the basis of (5.5).

6. Joint Eigenspaces.

The Harish-Chandra formula for spherical functions can be written inthe form

(6.1) ϕλ(x) =

∫

B

e(iλ+ρ)(A(x,b)) db

with λ ∈ a∗c given mod W -invariance. These are the K-invariant joint

eigenfunctions of the algebra D(X). The spaces

(6.2) Eλ(X) = {f ∈ E(X) :

∫

K

f(gk · x) dk = f(g · o)ϕλ(x)}

were in [1962] characterized as the joint eigenspaces of the algebra D(X).Let Tλ denote the natural representation of G on Eλ(X). Similarly, forλ ∈ a

∗c the space D′

λ(Ξ) of distributions Ψ on Ξ given by

(6.3) Ψ(ϕ) =

∫

K/M

(∫

A

ϕ(kaMN)e(iλ+ρ)(log a) da)dS(kM)

is the general joint distribution eigenspace for the algebra D(Ξ). Here

S runs through all of D′(B). The dual map Ψ → Ψ∨maps D′

λ(Ξ) intoEλ(X). In terms of S this dual mapping amounts to the Poisson trans-form Pλ given by

(6.4) PλS(x) =

∫

B

e(iλ+ρ)(A(x,b)) dS(b) .

By definition the Gamma function of X, ΓX(λ), is the denominatorin the formula for c(λ)c(−λ) where c(λ) is the c-function of Harish-Chandra, Gindikin and Karpelevic. While ΓX(λ) is a product over allindivisible roots, Γ+

X(λ) is the product over just the positive ones. See[2008], p. 284. In [1976] it is proved that for λ ∈ a

∗c ,

(i) Tλ is irreducible if and only if 1/ΓX(λ 6= 0).(ii) Pλ is injective if and only if 1/Γ+

X(λ) 6= 0.(iii) Each K-finite joint eigenfunction of D(X) has the form

10 SIGURDUR HELGASON

(6.5)

∫

B

e(iλ+ρ)(A(x,b))F (b) db

for some λ ∈ a∗c and some K-finite function F on B.

For X = Hn it was shown [1970, p.139, 1973b] that all eigenfunctionsof the Laplacian have the form (6.5) with F (b) db replaced by an analyticfunctional (hyperfunction). This was a bit of a surprise since this conceptwas in very little use at the time. The proof yielded the same result forall X of rank one provided eigenvalue is ≥ −〈ρ, ρ〉. In particular, allharmonic functions on X have the form

(6.6) u(x) =

∫

B

e2ρ(A(x,b)) dS(b) ,

where S is a hyperfunction on B.For X of arbitrary rank it was proved by Kashiwara, Kowata, Mino-

mura, Okamoto, Oshima and Tanaka that Pλ is surjective for 1/Γ+X (λ) 6=

0 [1978]. In particular, every joint eigenfunction has the form (6.4) fora suitable hyperfunction S on B. The image under Pλ of various otherspaces on B has been widely investigated, we just mention Furstenberg[1963], Karpelevic [1963], Lewis [1978], Oshima–Sekiguchi [1980], Wal-lach [1983], Ban–Schlichtkrull [1987], Okamoto [1971], Yang [1998].

For the compact dual symmetric space U/K the eigenspace represen-tations are all irreducible and each joint eigenfunction is of the form(6.5) (cf. Helgason [1984] Ch. V, §4, in particular p. 542). Again thisrelies on (5.5).

7. The X-ray transform.

The X-ray transform f → f on a complete Riemannian manifold X isgiven by

(7.1) f(γ) =

∫

γ

f(x) dm(x) , γ a geodesic,

f being a function on X. For the symmetric space X = G/K from §4, Ishowed in [1980] the injectivity and support theorem for (7.1) (problemsA and D in §3). In [2006], Rouviere proved an explicit inversion formulafor (7.1).

For a compact symmetric space X = U/K we assume X irreducibleand simply connected. Here we modify (7.1) by restricting γ to bea closed geodesic of minimal length, and call the transform the Funktransform. All such geodesics are conjugate under U (Helgason [1966a]so the family Ξ = {γ} has the form U/H and the Funk transform falls

PERSONAL REMARKS 11

in the framework (3.7). The injectivity (for X 6= Sn) was proved byKlein, Thorbergsson and Verhoczki [2009]; an inversion formula and asupport theorem by the author [2007]. To each x ∈ X is associated themidpoint locus Ax (the set of midpoints of minimal geodesics throughx) as well as a corresponding “equator” Ex. Both of these are acted ontransitively by the isotropy group of x. The inversion formula involvesintegrals over both Ax and Ex.

For a closed subgroup H ⊂ G, invariant under the Cartan involutionθ of G (with fixed group K) Ishikawa [2003] investigated the doublefibration (3.7). The orbit HK is a totally geodesic submanifold of Xso this generalizes the X-ray transform. For many cases of H, this newtransform was found to be injective and to satisfy a support theorem.

For one variation of these questions see Frigyik, Stefanov and Uhlmann[2008].

8. Concluding remarks.

For the sake of unity and coherence, the account in the sections above hasbeen rather narrow and group-theory oriented. A satisfactory accountof progress on Problems A, B, C, D in §3 would be rather overwhelming.My book [2011] with its bibliographic notes and references is a modestattempt in this direction.

Here I restrict myself to the listing of topics in the field — followed bya bibliography, hoping the titles will serve as a suggestive guide to theliterature. Some representative samples are mentioned. These samplesare just meant to be suggestive, but I must apologize for the limitedexhaustiveness.

(i) Topological properties of the Radon transform. Quinto [1981],Hertle [1984a].

(ii) Range questions for a variety of examples of X and Ξ. Thefirst paper in this category is John [1938] treating the X-raytransform in R3. For X the set of k-planes in Rn the final ver-sion, following intermediary results by Helgason [1980b], Gelfand,Gindikin and Graev [1982], Richter [1986], [1990], Kurusa [1991],is in Gonzalez [1990b] where the range is the null space of anexplicit 4th degree differential operator. Enormous progress hasbeen made for many examples. Remarkable analogies have emerged,Berenstein, Kurusa, Casadio Tarabusi [1997]. For Grassmannmanifolds and spheres see e.g. Kakehi [1993], Gonzalez andKakehi [2004]. Also Oshima [1996], Ishikawa [1997].

12 SIGURDUR HELGASON

(iii) Inversion formulas. Here a great variety exists even withina single pair (X,Ξ). Examples are Grassmann manifolds, com-pact and noncompact, X-ray transform on symmetric spaces(compact and noncompact). Antipodal manifolds on compactsymmetric spaces. See e.g. Gonzalez [1984], Grinberg and Ru-bin [2004], Rouviere [2006], Helgason [1965a], [2007], Ishikawa[2003]. Techniques of fractional integrals. Injectivity sets. Ad-missible families. Goncharov [1989]. The kappa operator. SeeRubin [1998], Gelfand, Gindikin and Graev [2003], Agranovskyand Quinto [1996], Grinberg [1994] and Rouviere [2008b].

(iv) Spherical transforms (spherical integrals with centers re-stricted to specific sets). Range and support theorems. Useof microlocal analysis, Boman and Quinto[1987], Greenleaf andUhlmann [1989], Frigyik, Stefanov and Uhlmann [2008]. MeanValue operator. Agranovsky, Kuchment and Quinto [2007], Agra-novsky, Finch and Kuchment [2009], Rouviere [2012], Lim [2012].

(v) Attenuated X-ray transform. Hertle [1984b], Palamodov[1996], Natterer [2001], Novikov [1992].

(vi) Extensions to forms and vector bundles. Okamoto [1971],Goldschmidt [1990].

(vii) Discrete Integral Geometry and Radon transforms. Sel-fridge and Straus [1958], Bolker [1987], Abouelaz and Ihsane[2008].

Hopefully the titles in the following bibliography will furnish helpfulcontact with topics listed above.

PERSONAL REMARKS 13

Bibliography

Abouelaz, A. and Fourchi, O.E.

2001 Horocyclic Radon Transform on Damek-Ricci spaces, Bull. PolishAcad. Sci. 49 (2001), 107-140.

Abouelaz, A. and Ihsane, A.

2008 Diophantine Integral Geometry, Mediterr. J. Math. 5 (2008), 77–99.

Abouelaz, A. and Rouviere, F.

2011 Radon transform on the torus, Mediterr. J. Math. 8 (2011), 463–471.

Agranovsky, M.L., Finch, D. and Kuchment, P.

2009 Range conditions for a spherical mean transform, Invrse Probl. Imaging3 (2009), 373–382.

Agranovsky, M.L., Kuchment, P. and Quinto, E.T.

2007 Range descriptions for the spherical mean Radon transform, J. Funct.Anal. 248 (2007), 344–386.

Agranovsky, M.L. and Quinto, E.T.

1996 Injectivity sets for the Radon transform over circles and complete sys-tems of radial functions, J. Funct. Anal. 139 (1996), 383–414.

Aguilar, V., Ehrenpreis, L., and Kuchment, P.

1996 Range conditions for the exponential Radon transform, J. d’AnalyseMath. 68 (1996), 1–13.

Ambartsoumian, G. and Kuchment, P.

2006 A range description for the planar circular Radon transform, SIAM J.Math. Anal. 38 (2006), 681–692.

Andersson, L.-E.

1988 On the determination of a function from spherical averages, SIAM J.Math. Anal., 19 (1988), 214–232.

Antipov, Y.A. and Rudin, B.

2012 The generalized Mader’s inversion formula for the Radon transform,Trans. Amer. Math. Soc., (to appear).

Asgeirsson, L.

1937 Uber eine Mittelwertseigenschaft von Losungen homogener linearer par-tieller Differentialgleichungen 2. Ordnung mit konstanten Koefficienten,Math. Ann. 113 (1937), 321–346.

Ban, van den, e.p. and Schlichtkrull, H.

1987 Asymptotic expansions and boundary values of eigenfunctions on aRiemannian symmetric space, J. Reine Angew. Math. 380 (1987), 108–165.

Berenstein, C.A., Kurusa, A., and Casadio Tarabusi, E.

1997 Radon transform on spaces of constant curvature, Proc. Amer. Math.Soc. 125 (1997), 455–461.

Berenstein, C.A. and Casadio Tarabusi, E.

1991 Inversion formulas for the k-dimensional Radon transform in real hy-perbolic spaces, Duke Math. J. 62 (1991), 613–631.

1992 Radon- and Riesz transform in real hyperbolic spaces, Contemp. Math.140 (1992), 1–18.

1993 Range of the k-dimensional Radon transform in real hyperbolic spaces,Forum Math. 5 (1993), 603–616.

1994 An inversion formula for the horocyclic Radon transform on the realhyperbolic space, Lectures in Appl. Math. 30 (1994), 1–6.

Bolker, E.

1987 The finite Radon transform, Contemp. Math. 63 (1987), 27–50.

14 SIGURDUR HELGASON

Boman, J.

1991 “Helgason’s support theorem for Radon transforms: A new proof anda generalization,” in: Mathematical Methods in Tomography, LectureNotes in Math. No. 1497, Springer-Verlag, Berlin and New York, 1991,1–5.

1992 Holmgren’s uniqueness theorem and support theorems for real analyticRadon transforms, Contemp. Math. 140 (1992), 23–30.

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1987 Support theorems for real-analytic Radon transforms, Duke Math. J.55 (1987), 943–948.

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1994 A bundle-valued Radon transform with applications to invariant waveequations, Quart. J. Math. Oxford 45 (1994), 429–461.

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2001 Helgason spheres of compact symmetric spaces of finite type, Bull.Austr. Math. Soc.63 (2001), 243–255.

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1983 Formule d’inversion en geometrie integrale Lagrangienne, C. R. Acad.Sci. Paris Ser. I Math. 296 (1983), 423–425.

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1979 Asymptotic expansions of Eisenstein integrals and Fourier transform ofsymmetric spaces, J. Funct. Anal. 34 (1979), 167–216.

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2007 Inversion and spherical means and the wave equation in even dimension,SIAM J. Appl. Math. 68 (2007), 392–412.

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1990 The Radon transform for symmetric forms on real symmetric spaces,Contemp. Math. 113 (1990), 81–96.

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16 SIGURDUR HELGASON

Gonzalez, F.

1984 Radon transforms on Grassmann manifolds, thesis, MIT, Cambridge,MA, 1984.

1988 Bi-invariant differential operators on the Euclidean motion group andapplications to generalized Radon transforms, Ark. Mat. 26 (1988),191–204.

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1990b Invariant differential operators and the range of the Radon d-planetransform, Math. Ann. 287 (1990), 627–635.

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2003 Pfaffian systems and Radon transforms on affine Grassmann manifoldsMath. Ann. 326 (2003), 237–273.

2004 Dual Radon transforms on affine Grassmann manifolds, Trans.Amer.Math. Soc. 356 (2004), 4161–4180.

2006 Invariant differential operators and the range of the matrix Radontransform, J. Funct. Anal. 241 (2006), 232–267.

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1994 Support theorems for Radon transforms on higher rank symmetricspaces, Proc. Amer. Math. Soc. 122 (1994), 1045–1052.

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1989 Non-local inversion formulas for the X-ray transform, Duke Math. J.58 (1989), 205–240.

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1966 Spharische Mittelwerte in kompakten harmonischen RiemannschenMannigfaltigkeiten, Math. Ann. 165 (1966), 281–296.

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1959 Differential Operators on homogeneous spaces, Acta Math. 102 (1959),239–299.

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1964 A duality in integral geometry: some generalizations of the Radon trans-form, Bull. Amer. Math. Soc. 70 (1964), 435–446.

1964a Fundamental solutions of invariant diffferential operators on symmetricspaces, Amer. J. Math. 86 (1964), 565–601.

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1980b The Radon Transform, Birkhasuer 1980, 2nd Ed. 1999.

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1984 Groups and Geometric Analysis. Integral Geometry, Invariant Differ-ential Operators and Spherical functions, Academic Press, N.Y., 1984.(Now published by American Mathematical Society, 2000.)

1990 The totally geodesic Radon transform on constant curvature spaces,Contemp. Math. 113 (1990), 141–149.

18 SIGURDUR HELGASON

1992 The flat horocycle transform for a symmetric space, Adv. in Math. 91(1992), 232–251.

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1973 Determination of the conical distributions for rank one symmetricspaces, Thesis, MIT, Cambridge, MA, 1973.

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1997 The range characterization of the totally geodesic Radon transform onthe real hyperbolic space, Duke Math. J. 90 (1997), 149–203.

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1993 On Radon transform for Minkowski space, J. Fac. Gen. Ed. TattoriUniv. 27 (1993), 139–157.

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1991 A characterization of the Radon transform’s range by a system ofPDE’s, J. Math. Anal. Appl. 161 (1991), 218–226.

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1971 Harmonic analysis on homogeneous vector bundles, Lecture Notes inMath. 266 (1971), 255–271.

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1981 Topological restrictions on double fibrations and Radon transforms,Proc. Amer. Math. Soc. 81 (1981), 570–574.

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Massachusetts Institute of Technology, Cambridge, MA USA

E-mail address: helgason@mit.edu