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 of
some of my work concerning the Radon transform. My interest in the
subject was actually evoked during a train trip from New York to Boston
once during the Spring 1955.
2. Some old times.
Back in 1955, I worked on extending the mean value theorem of Leifur
Ásgeirsson [1937] for the ultrahyperbolic equation on Rn × Rn to Riemannian homogeneous spaces G/K × G/K. I was motivated by Godement’s generalization [1952] of the mean value theorem for Laplace’s
equation Lu = 0 to the system Du = 0 for all G-invariant differential
operators D (annihilating the constants) on G/K. At the time (Spring
1955) I visited Leifur in New Rochelle where he was living in the house
of Fritz John (then on leave from NYU). They had both been students
of Courant in Göttingen in the 1930’s. Since John’s book [1955] treats
Ásgeirsson’s theorem in some detail, Leifur lent me a copy of it (in page
proofs) 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. In
John’s notation, let J(ω, p) denote the integral of f over the hyperplane
1991 Mathematics Subject Classification. Mathematics Subject Classification Primary 43A85, 53 C35, 22E46, 44A12; Secondary 53 C65, 14 M17, 22 F30.
c
0000
(copyright holder)
1
2
SIGURDUR HELGASON
hω, xi = p (p ∈ R, ω a unit vector), dω the area element on Sn−1 and L
the Laplacian. Then
Z
1−n
(n−1)/2
1
(Lx )
(2.1) f (x) = 2 (2πi)
J(ω, hω, xi) dw , n odd.
S n−1
−n
(2.2) f (x) = (2πi)
(n−2)/2
(Lx )
Z
dω
Sn−1
Z
dJ(ω, p)
, n even.
p − hω, xi
R
I was surprised at never having seen such formulas before. Radon’s
paper [1917] was very little known, being published in a journal that
was hard to find. The paper includes some suggestions by Herglotz in
Leipzig and John learned of it from lectures by Herglotz in Göttingen.
I did not see Radon’s paper until several years after the appearance of
John’s book but it has now been reproduced in several books about
the 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 of
their integrals over great circles. Funk’s papers are in turn related to a
paper 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
L
x
8π 2
Z
S2
J(ω, hω, xi dω)
it struck me that the formula involves two dual integrations, J the integral 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 → fb,
∨
ϕ → ϕ defined as follows:
For functions f on R3 , ϕ on P3 (the space of 2-planes in R3 ) put
Z
b
(3.2)
f (ξ) =
f (x) dm(x) ,
ξ ∈ P3 ,
ξ
(3.3)
∨
ϕ (x) =
Z
ϕ(ξ) dµ(ξ) ,
x ∈ R3 ,
ξ∋x
where dm is the Lebesgue measure and dµ the average over all hyperplanes containing x. Then (3.1) can be rewritten
(3.4)
f = − 21 L((fb)∨ ) .
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)/Z2 M(2) .
∨
The operators f → fb, 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
Z
Z
∨
b
ϕ(gkH) dkL .
f
(γhK)
dh
,
ϕ
(gK)
=
(3.6)
f (γH) =
L
K/L
H/L
Here G is an arbitrary locally compact group, K and H closed subgroups,
L = K ∩ H, and dhL and dkL the essentially unique invariant measure
on H/L and K/L, respectively. This is the abstract Radon transform for
the double fibration:
G/L
✡
✢✡
(3.7)
✡
❏
X = G/K
❏❏
❫
Ξ = G/H
∨
The operators f → fb, ϕ → ϕ map functions on X to functions on Ξ
and vice-versa. These geometrically dual operators also turned out to
be adjoint operators relative to the invariant measures dx and dξ,
Z
Z
∨
f (x)ϕ (x) dx = fb(ξ)ϕ(ξ) dξ .
(3.8)
X
Ξ
This suggests natural extensions to suitable distributions T on X, Σ on
Ξ, as follows:
∨
∨
Tb(ϕ) = T (ϕ ) , Σ (f ) = Σ(fb) .
Formulas (2.1) and (2.2) have another interesting feature. As functions of x the integrands are plane waves, i.e. constant on each hyperplane L perpendicular to ω. Such a function is really just a function of a
single variable so (2.1) and (2.2) can be viewed as a decomposition of an
n-variable function into one-variable functions. This feature enters into
the work of Herglotz and John [1955]. I have found some applications
of an analog of this principle for invariant differential equations on symmetric spaces ([1963], §7, [2008],Ch. V§1, No. 4), where parallel planes
are replaced by parallel horocycles.
4
SIGURDUR HELGASON
The setup (3.5) and (3.6) above has of course an unlimited supply of
examples. 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 the
isotropy groups of the origin O and a k-plane ξ at distance p from O.
Then the second transform in (3.6) becomes
Z
∨
ϕ(ξ) dµ(ξ)
(3.10)
ϕp (x) =
d(x,ξ)=p
and we get another inversion formula (cf. [1990], [2011]) of f → fb
involving (fb)∨
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 (fb)∨
p (x) is the average of the integrals of f
over the geodesics at distance p from x.
∨
The principal problems for the operators f → fb, ϕ → ϕ would be
A. Injectivity.
B. Inversion formulas.
C. Ranges and kernels for specific function spaces of X and on
Ξ.
D. Support problems (does compact support of fb imply compact
support of f ?)
These problems are treated for a number of old and new examples
in [2011]. Some unexpected analogies emerge, for example a complete
parallel between the Poisson integral in the unit disk and the X-ray
transform in R3 , see pp. 86–89, loc. cit..
My first example for (3.5) and (3.6) was for X a Riemannian manifold of constant sectional curvature c and dimension n and Ξ the set
of k-dimensional totally geodesic submanifolds of X. The solution to
Problem 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
(3.11)
Qk (L)((fb)∨ ) = γf if X is noncompact.
(3.12)
Qk (L)((fb)∨ ) = γ(f + f ◦ A) if X is compact.
PERSONAL REMARKS
5
In the latter case X = Sn and A the antipodal mapping. The constant
γ is given by
n n − k Γ
/Γ
(−4π)k/2 .
2
2
The proof used the generalization of Ásgeirsson’s theorem. At that time
I had also proved an inversion formula for k odd and the method used
in the proof of the support theorem for Hn ([1964, Theorem 5.2]) but
not published until [1990] since the formula seemed unreasonably complicated in comparison to (3.11), (3.12) and since the case k = 1 when
f → fb is the ”X-ray transform”, had not reached its distinction through
tomography.
For k odd the inversion formula is a combination of fb and the analog
of (3.10).
For X = Rn , Ξ = Pn , problems C–D are dealt with in [1965a]. This
paper also solves problem B for X any compact two-point homogeneous
space 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 integrals
of f over translates of conjugacy classes in G. This did not fit into the
framework (3.5)(3.6) so using the Iwasawa decomposition G = KAN
(K compact, A abelian, N nilpotent) I replaced the conjugacy classes
by their “projections” in the symmetric space G/K, and this leads to
the orbits of the conjugates gN g−1 in G/K. These orbits are the horocycles in G/K. They occur in classical non-Euclidean geometry (where
they carry a flat metric) and for G complex are extensively discussed in
Gelfand–Graev [1964].
For a general semisimple G, the action of G on the symmetric space
G/K turned out to permute the horocycle transitively with isotropy
group M N , where M is the centralizer of A in K [1963]. This leads
(3.5) and (3.6) to the double fibration
G/M
✡
✢✡
(4.1)
✡
X = G/K
❏
❏❏
❫
Ξ = G/M N
and for functions f on X, ϕ on Ξ, fb(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
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SIGURDUR HELGASON
papers [1963], [1964a], [1970] are devoted to a geometric examination of
this duality and its implications for analysis, differential equations and
representation theory. Thus we have double coset space representations
(4.2)
K\G/K ≈ A/W ,
M N \G/M N ≈ W × A
based on the Cartan and Bruhat decomposition of G , W denoting the
Weyl group.
The finite-dimensional irreducible representations with a K-fixed vector turn out to be the same as those with an M N -fixed vector. This leads
to simultaneous imbeddings of X and Ξ into the same vector space and
the horocycles are certain plane sections with X in analogy with their
flatness for Hn [2008,II,§4]. The set of highest restricted weights of these
representations is the dual of the lattice Σi Z+ βi where β1 , . . . , βℓ is the
basis 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 isomorphisms
(4.3)
D(X) ≈ D(A)/W ,
D(Ξ) ≈ D(A) .
The first is a reformulation of Harish-Chandra’s homomorphism, the
second comes from the fact that the G-action on the fibration of Ξ over
K/M is fiber preserving and generates a translation on each (vector)
∨
fiber. The transforms f → fb, ϕ → ϕ intertwine the members of D(X)
and D(Ξ). In particular, when the operator f → fb is specialized to
K-invariant functions on X it furnishes a simultaneous transmutation
operator between D(X) and the set of W -invariants in D(A) [1964a,
§2]. This property, combined with a surjectivity result of Hörmander
[1958] and Lojasiewicz [1958] for tempered distributions on Rn , yields
the result that each D ∈ D(X) has a fundamental solution [1964a].
A more technical support theorem in [1973] for f → fb on G/K then
implies the existence theorem that each D ∈ D(X) is surjective on
E(X) i.e. (C∞ (X)). It is also surjective on the space D′0 (X) of Kfinite distributions on X ([1976]) and on the space S′ (X) of tempered
distributions on X ([1973a]). The surjectivity on all of D′ (X) however
seems as yet unproved.
The method of [1973] also leads to a Paley–Wiener type Theorem
for the horocycle transform f → fb, that is an internal description of
the range D(X)b. The formulation is quite different from the analogous
result for D(Rn )b. Having proved the latter result in the summer of
1963, I always remember when I presented it in a Fall class, because
immediately afterwards I heard about John Kennedy’s assassination.
PERSONAL REMARKS
7
In analogy with (3.4), (3.11), the horocycle transform has an inversion
formula. The parity difference in (2.1), (2.2) and (3.11), (3.12) now takes
another form ([1964], [1965b]):
If G has all its Cartan subgroups conjugate then
(4.4)
f = ((fb)∨ ) ,
where is an explicit operator in D(X). Although this remains “formally valid” for general G with replaced by a certain pseudo-differential
operator, a better form is
(4.5)
f = (Λfb)∨ ,
with Λ a certain pseudo-differential operator on Ξ. These operators are
constructed from Harish-Chandra’s c-function for G. For G complex a
formula related to (4.5) is stated in Gelfand and Graev [1964], §5.
As mentioned, f → fb is injective and the range D(X)b 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 eigenfunctions of the operators in D(X). By analogy we define conical distribution on Ξ to be the M N -invariant eigendistributions of the operators
in D(Ξ). While Harish-Chandra’s formula for the spherical functions
parametrizes 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 spherical
functions are given by
Z
ϕλ (gK) = e(iλ−ρ)(H(gk)) dk ,
K
where g = k exp H(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 group
M N A on Ξ divides it into |W | orbits Ξs and for λ ∈ a∗c a conical distribution is constructed with support in the closure of Ξs . The construction is
done by a specific holomorphic continuaton. The identification in (4.7)
from [1970] is complete except for certain singular eigenvalues. For the
case of G/K of rank one the full identification of (4.7) was completed
by Men-Cheng Hu in his MIT thesis [1973]. Operating as convolutions
on K/M the conical distributions in (4.7) furnish intertwining operators
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SIGURDUR HELGASON
in the spherical principal series [1970, Ch. III, Theorem 6.1]. See also
Schiffmann [1971], Théorème 2.4 and Knapp-Stein [1971].
5. A Fourier transform on X.
Writing fb(ω, p) for John’s J(ω, p) in (2.1) the Fourier transform fe on
Rn can be written
Z
Z
−irhx,ωi
e
(5.1)
f (rω) =
f (x)e
dx = fb(ω, p)e−irp dp ,
Rn
R
which is the one-dimensional Fourier transform of the Radon transform.
The horocycle duality would call for an analogous Fourier transform on
X.
The standard representation–theoretic Fourier transform on G,
Z
e
F (π) = F (x)π(x) dx
G
is unsuitable here because it assigns to F a family of operators in different
Hilbert spaces. However, the inner product hx, ωi in (5.1) has a certain
vector–valued analog for G/K, namely
(5.2)
A(gK, kM ) = A(k−1 g) ,
where exp A(g) is the A-component in the Iwasawa decomposition G =
N AK. 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,
Z
(5.4)
fe(λ, b) = f (x)e−λ,b (x) dx .
X
The analog of (5.1) is then
fe(λ, kM ) =
Z
A
fb(kaM N )e(−iλ+ρ)(log a) da .
The main theorems of the Fourier transform on Rn , the inversion formula, 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
Z
−1
(5.5)
ϕλ (g h) = e(iλ+ρ)(A(kh)) e(−iλ+ρ)(A(kg)) dk .
K
PERSONAL REMARKS
9
Some results have richer variations, like the range theorems for the
various Schwartz spaces Sp (X) ⊂ Lp (X) (Eguchi [1979]).
The analog of (5.4) for the compact dual symmetric space U/K was
developed by Sherman [1977], [1990] on the basis of (5.5).
6. Joint Eigenspaces.
The Harish-Chandra formula for spherical functions can be written in
the form
Z
(6.1)
ϕλ (x) = e(iλ+ρ)(A(x,b)) db
B
a∗c
with λ ∈
given mod W -invariance. These are the K-invariant joint
eigenfunctions of the algebra D(X). The spaces
Z
(6.2)
Eλ (X) = {f ∈ E(X) : f (gk · x) dk = f (g · o)ϕλ (x)}
K
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, for
λ ∈ a∗c the space D′λ (Ξ) of distributions Ψ on Ξ given by
Z Z
ϕ(kaM N )e(iλ+ρ)(log a) da dS(kM )
(6.3)
Ψ(ϕ) =
K/M
A
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) into
Eλ (X). In terms of S this dual mapping amounts to the Poisson transform Pλ given by
Z
(6.4)
Pλ S(x) = e(iλ+ρ)(A(x,b)) dS(b) .
B
By definition the Gamma function of X, ΓX (λ), is the denominator
in the formula for c(λ)c(−λ) where c(λ) is the c-function of HarishChandra, Gindikin and Karpelevič. While ΓX (λ) is a product over all
indivisible 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
Z
(6.5)
e(iλ+ρ)(A(x,b)) F (b) db
B
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 eigenfunctions
of the Laplacian have the form (6.5) with F (b) db replaced by an analytic
functional (hyperfunction). This was a bit of a surprise since this concept
was in very little use at the time. The proof yielded the same result for
all X of rank one provided eigenvalue is ≥ −hρ, ρi. In particular, all
harmonic functions on X have the form
Z
(6.6)
u(x) = e2ρ(A(x,b)) dS(b) ,
B
where S is a hyperfunction on B.
For X of arbitrary rank it was proved by Kashiwara, Kowata, Minomura, 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) for
a suitable hyperfunction S on B. The image under Pλ of various other
spaces on B has been widely investigated, we just mention Furstenberg
[1963], Karpelevič [1963], Lewis [1978], Oshima–Sekiguchi [1980], Wallach [1983], Ban–Schlichtkrull [1987], Okamoto [1971], Yang [1998].
For the compact dual symmetric space U/K the eigenspace representations are all irreducible and each joint eigenfunction is of the form
(6.5) (cf. Helgason [1984] Ch. V, §4, in particular p. 542). Again this
relies on (5.5).
7. The X-ray transform.
The X-ray transform f → fb on a complete Riemannian manifold X is
given by
Z
b
(7.1)
f (γ) = f (x) dm(x) , γ a geodesic,
γ
f being a function on X. For the symmetric space X = G/K from §4, I
showed in [1980] the injectivity and support theorem for (7.1) (problems
A and D in §3). In [2006], Rouvière proved an explicit inversion formula
for (7.1).
For a compact symmetric space X = U/K we assume X irreducible
and simply connected. Here we modify (7.1) by restricting γ to be
a closed geodesic of minimal length, and call the transform the Funk
transform. 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 by
Klein, Thorbergsson and Verhóczki [2009]; an inversion formula and a
support theorem by the author [2007]. To each x ∈ X is associated the
midpoint locus Ax (the set of midpoints of minimal geodesics through
x) as well as a corresponding “equator” Ex . Both of these are acted on
transitively by the isotropy group of x. The inversion formula involves
integrals over both Ax and Ex .
For a closed subgroup H ⊂ G, invariant under the Cartan involution
θ of G (with fixed group K) Ishikawa [2003] investigated the double
fibration (3.7). The orbit HK is a totally geodesic submanifold of X
so this generalizes the X-ray transform. For many cases of H, this new
transform 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 has
been rather narrow and group-theory oriented. A satisfactory account
of progress on Problems A, B, C, D in §3 would be rather overwhelming.
My book [2011] with its bibliographic notes and references is a modest
attempt in this direction.
Here I restrict myself to the listing of topics in the field — followed by
a bibliography, hoping the titles will serve as a suggestive guide to the
literature. Some representative samples are mentioned. These samples
are just meant to be suggestive, but I must apologize for the limited
exhaustiveness.
(i) Topological properties of the Radon transform. Quinto [1981],
Hertle [1984a].
(ii) Range questions for a variety of examples of X and Ξ. The
first paper in this category is John [1938] treating the X-ray
transform in R3 . For X the set of k-planes in Rn the final version, 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 an
explicit 4th degree differential operator. Enormous progress has
been made for many examples. Remarkable analogies have emerged,
Berenstein, Kurusa, Casadio Tarabusi [1997]. For Grassmann
manifolds and spheres see e.g. Kakehi [1993], Gonzalez and
Kakehi [2004]. Also Oshima [1996], Ishikawa [1997].
12
SIGURDUR HELGASON
(iii) Inversion formulas. Here a great variety exists even within
a single pair (X, Ξ). Examples are Grassmann manifolds, compact and noncompact, X-ray transform on symmetric spaces
(compact and noncompact). Antipodal manifolds on compact
symmetric spaces. See e.g. Gonzalez [1984], Grinberg and Rubin [2004], Rouvière [2006], Helgason [1965a], [2007], Ishikawa
[2003]. Techniques of fractional integrals. Injectivity sets. Admissible families. Goncharov [1989]. The kappa operator. See
Rubin [1998], Gelfand, Gindikin and Graev [2003], Agranovsky
and Quinto [1996], Grinberg [1994] and Rouvière [2008b].
(iv) Spherical transforms (spherical integrals with centers restricted to specific sets). Range and support theorems. Use
of microlocal analysis, Boman and Quinto[1987], Greenleaf and
Uhlmann [1989], Frigyik, Stefanov and Uhlmann [2008]. Mean
Value operator. Agranovsky, Kuchment and Quinto [2007], Agranovsky, Finch and Kuchment [2009], Rouvière [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. Selfridge and Straus [1958], Bolker [1987], Abouelaz and Ihsane
[2008].
Hopefully the titles in the following bibliography will furnish helpful
contact with topics listed above.
PERSONAL REMARKS
Bibliography
Abouelaz, A. and Fourchi, O.E.
2001
Horocyclic Radon Transform on Damek-Ricci spaces, Bull. Polish
Acad. Sci. 49 (2001), 107-140.
Abouelaz, A. and Ihsane, A.
2008
Diophantine Integral Geometry, Mediterr. J. Math. 5 (2008), 77–99.
Abouelaz, A. and Rouvière, 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. Imaging
3 (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 systems 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’Analyse
Math. 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.
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