The Funk– for sections through a common point

Michael Quellmalz∗

The Funk–Radon transform, also known as the spherical Radon transform, assigns to a function on the its mean values along all great circles. Since its invention by Paul Funk in 1911, the Funk–Radon transform has been generalized to other families of circles as well as to higher dimensions. We are particularly interested in the following generalization: we consider the intersections of the sphere with containing a common point inside the sphere. If this point is the origin, this is the same as the afore- mentioned Funk–Radon transform. We give an injectivity result and a range characterization of this generalized Radon transform by finding a relation with the classical Funk–Radon transform. Math Subject Classifications. 45Q05, 44A12. Keywords and Phrases. Radon transform, spherical means, Funk–Radon transform.

1 Introduction

The reconstruction of a function from its integrals along certain submanifolds is a key task in various mathematical fields, including the modeling of imaging modalities, cf. [17]. One of the first works was in 1911 by Funk [6]. He considered what later became known by the names Funk–Radon transform, Minkowski– or spherical arXiv:1810.08105v1 [math.NA] 18 Oct 2018 Radon transform. Here, the function is defined on the unit sphere and we know its mean values along all great circles. Another famous example is the Radon transform [21], where a function on the plane is assigned to its mean values along all lines. Over

∗Chemnitz University of Technology, Faculty of , D-09107 Chemnitz, Germany. E-mail: [email protected]

1 the last more than 100 years, the Funk–Radon transform has been generalized to other families of circles as well as to higher dimensions. In this article, we draw our attention to integrals along certain subspheres of the (d−1)-dimensional unit sphere Sd−1 = {ξ ∈ Rd | kξk = 1}. Any subsphere of Sd−1 is the intersection of the sphere with a hyperplane. In particular, we consider the subspheres of Sd−1 whose hyperplanes have the common point (0,..., 0, z)> strictly inside the sphere d−1 for z ∈ [0, 1). We define the spherical transform Uzf of a function f ∈ C(S ) by Z 1 ξ d−1 Uzf(ξ) = f dCz , ξ ∈ S , ξ ξ V (Cz ) Cz which computes the mean values of f along the subspheres

ξ d−1 Cz = {η ∈ S | hη, ξi = zξd}, z ∈ [0, 1),

ξ ξ where V (Cz ) denotes the (d − 2)-dimensional volume of Cz . 2 The spherical transform Uz on the two-dimensional sphere S was first investigated by Salman [24] in 2016. He showed the injectivity of this transform Uz for smooth ∞ 2 2 functions f ∈ C (S ) that are supported inside the spherical cap {ξ ∈ S | ξ3 < z} as well as an inversion formula. This result was extended to the d-dimensional sphere in [25], where also the smoothness requirement was lowered to f ∈ C1(Sd). A different approach was taken in [19], where a relation with the classical Funk–Radon transform U0 was established and also used for a characterization of the nullspace and range of Uz for z < 1. In the present paper, we extend the approach of [19] to the d-dimensional case. We use a similar change of variables to connect the spherical transform Uz with the Funk– Radon transform F = U0 in order to characterize its nullspace. However, the description of the range requires some more effort, because the operator Uz is smoothing of degree (d − 2)/2. In the case z = 0, we have sections of the sphere with hyperplanes through the origin, which are also called maximal subspheres of Sd−1 or great circles on S2. We obtain the classical Funk–Radon transform F = U0. The case z = 1 corresponds to the subspheres containing the north pole (0,..., 0, 1)>. This case is known as the spherical slice transform U1 and has been investigated since the early 1990s in [1,4,9]. However, ∞ d−1 unlike Uz for z < 1, the spherical slice transform U1 is injective for f ∈ L (S ), see [22]. The main tool to derive this injectivity result of U1 is the stereographic projection, which turns the subspheres of Sd−1 through the north pole into hyperplanes in Rd−1 d−1 and thus connecting the spherical slice transform U1 to the Radon transform on R . For z → ∞, we obtain vertical slices of the sphere, i.e., sections of the sphere with hyperplanes that are parallel to the north–south axis. This case is well-known for S2, see [7, 29, 11, 23]. In 2016, Palamodov [18, Section 5.2] published an inversion formula for a certain nongeodesic Funk transform, which considers sections of the sphere with hyperplanes that have a fixed distance to the point (0,..., 0, z)>. This contains both the transform Uz as well as for z = 0 the sections with subspheres having constant radius previously considered by Schneider [26] in 1969.

2 A key tool for analyzing the stability of inverse problems are Sobolev spaces, cf. [14] for the Radon transform and [10] for the Funk–Radon transform. The Sobolev space Hs(Sd−1) can be imagined as the space of functions f : Sd−1 → C whose derivatives up s d−1 to order s are square-integrable, and we denote by Heven(S ) its restriction to even functions f(ξ) = f(−ξ). A thorough definition of Hs(Sd−1) is given in Section 5.2. The behavior of the Funk–Radon transform was investigated by Strichartz [27], who found s d−1 s+(d−2)/2 d−1 that the Funk–Radon transform F : Heven(S ) → Heven (S ) is continuous and bijective. In Theorem 6.3, we show that basically the same holds for the generalized s d−1 s+(d−2)/2 d−1 transform Uz, i.e., the spherical transform Uz : Hz (S ) → Heven (S ) is bounded s d−1 and bijective and its inverse is also bounded, where Hz (S ) contains functions that are symmetric with respect to the point reflection in (0,..., 0, z)>. This paper is structured as follows. In Section2, we give a short introduction to smooth manifolds and review the required notation on the sphere. In Section3, we show the relation of Uz with the classical Funk–Radon transform F = U0. With the help of that factorization, we characterize the nullspace of the spherical transform Uz in Section4. In Section5, we consider Sobolev spaces on the sphere and show the continuity of certain multiplication and composition operators. Then, in Section6, we prove the continuity of the spherical transform Uz in Sobolev spaces. Finally, we include a geometric interpetation of our factorization result in Section7.

2 Preliminaries and definitions

2.1 Manifolds We give a brief introduction to smooth manifolds, more can be found in [2]. We denote by R and C the real and complex numbers, respectively. We define the d-dimensional d Pd R equipped with the scalar product hξ, ηi = i=1 ξiηi and the norm 1/2 d i i kξk = hξ, ξi . We denote the unit vectors in R with  , i.e. j = δi,j, where δ is the d Pd i Kronecker delta. Every vector ξ ∈ R can be written as ξ = i=1 ξi . A diffeomorphism is a bijective, smooth mapping Rn → Rn whose inverse is also smooth. We say a function is smooth if it has derivatives of arbitrary order. An n- dimensional smooth manifold M without boundary is a subset of Rd such that for every ξ ∈ M there exists an open neighborhood N(ξ) ⊂ Rd containing ξ, an open set U ⊂ Rd, and a diffeomorphism m: U → N(ξ) such that

n d−n m(U ∩ (R × {0} )) = M ∩ N(ξ).

We call m: V → M a map of the manifold M, where V = U ∩ (Rn × {0}d−n). An atlas of M is a finite family of maps mi : Vi → M, i = 1, . . . , l, such that the sets mi(Vi) cover d M. We define the tangent space TξM of M at ξ ∈ M as the set of vectors x ∈ R for which there exists a smooth path γ : [0, 1] → M satisfying γ(0) = ξ and γ0(0) = x. A k-form ω on M is a family (ωξ)ξ∈M of antisymmetric k-linear functionals

k ωξ :(TξM) → R,

3 k where (TξM) = TξM × · · · × TξM. Let f : M → N be a smooth mapping between the manifolds M and N and let ω be a k-form on N. The pullback of ω is the k-form f ∗(ω) 1 k on M that is defined for any v ,..., v ∈ TξM by

 k ! ∗ i k d f (ω)([v ]i=1) = ω f ◦ γi(t) , dt t=0 i=1

0 i where γi : [0, 1] → M are smooth paths satisfying γi(0) = ξ and γi(0) = v . If the smooth function f : Rd → Rd extends to the surrounding space, the pullback can be expressed as ∗ i k i k f (ω)([v ]i=1) = ω([Jf v ]i=1), (2.1) where Jf denotes the Jacobian matrix of f. l An atlas {mi : Vi → M}i=1 is called orientation of M if for each i, j the determinant of −1 1 n the Jacobian of mi ◦ mj is positive wherever it exists. A basis e ,..., e of the tangent space TξM is oriented positively if for some i with ξ ∈ mi(Vi) the determinant of 1 n the matrix [J −1 e ,...,J −1 e ] is positive. Up to the multiplication of a constant real mi mi number depending only on ξ, there exists only one n-form on an n-dimensional manifold. 1 n Let e ,..., e be a positively oriented, orthonormal basis of the tangent space TξM. Then the volume form dM is the unique n-form on M satisfying dM(e1,..., en) = 1. l ∞ A set {ϕi}i=1 of functions ϕi ∈ C (M) is a partition of unity of the manifold M with l Pl respect to an atlas {mi : Vi → M}i=1 if supp(ϕi) ⊂ mi(Vi) for all i and i=1 ϕi ≡ 1 on M. Then the integral of a k-form ω on M is defined as

Z l Z l Z X ∗ X n ω = mi (ϕi · ω) = ci dR , M i=1 Vi i=1 Vi

n n where the latter is the standard volume integral dR on Vi ⊂ R and the functions ∗ n ci : Vi → R are uniquely determined by the condition mi (ϕi · ω) = ci dR . Let f : M → N = f(M) be a diffeomorphism between the n-dimensional manifolds M and f(M), and let ω be an n-form on N. Then the substitution rule [13, p. 94] holds, Z Z ω = f ∗(ω). (2.2) f(M) M

2.2 The sphere Let d ≥ 3. The (d − 1)-dimensional unit sphere

d−1 d S = {ξ ∈ R | kξk = 1} is a (d − 1)-dimensional manifold in Rd with tangent space

d−1 d TξS = {x ∈ R | hξ, xi = 0}

4 d−1 d−1 i d−1 d−1 for ξ ∈ S . We define an orientation on S by saying that a basis [x ]i=1 of TξS is oriented positively if the determinant det[ξ, x1,..., xd−1] > 0. We denote the volume of the (d − 1)-dimensional unit sphere Sd−1 with

Z d/2 d−1 d−1 2π S = dS = . Sd−1 Γ(d/2)

2.2.1 The spherical transform

Every (d − 2)-dimensional subsphere of the sphere Sd−1 is the intersection of Sd−1 with a hyperplane, i.e.  d−1 S(ξ, t) = η ∈ S | hξ, ηi = t , where ξ ∈ Sd−1 is the normal vector of the hyperplane and t ∈ [−1, 1] is the signed distance of the hyperplane to the origin. We define an orientation on the subsphere i d−2 S(ξ, t) by saying that a basis [e ]i=1 of the tangent space TηS(ξ, t) is oriented positively if det η, ξ, e1,..., ed−2 > 0. In the following, we consider subspheres of Sd−1 whose hyperplanes have a common point located in the interior of the unit ball. Because of the rotational symmetry, we can assume that this point lies on the positive ξd axis. For z ∈ [0, 1), we consider the point zd = (0,..., 0, z)>. The (d − 2)-dimensional subsphere that is located in one hyperplane together with zd d−1 d can be described by S(ξ, t) with ξ ∈ S and t = ξ, z = zξd. We define the subsphere ξ Cz = S(ξ, zξd). ξ p 2 2 The (d − 2)-dimensional sphere Cz has radius 1 − z ξd and volume

ξ d−2 2 2(d−2)/2 V (Cz ) = S 1 − z ξd .

d−1 For a f : S → C, we define its spherical transform Uzf by Z 1 ξ d−1 Uzf(ξ) = f dCz , ξ ∈ S , (2.3) ξ ξ V (Cz ) Cz

ξ which computes the mean values of f along the subspheres Cz .

ξ Remark 2.1. The subspheres Cz , along which we integrate, can also be imagined in the ξ following way. The centers of the subspheres Cz are located on a sphere that contains the origin and zd and is rotationally symmetric about the North–South axis. This can

5 ξ be seen as follows. The center of the sphere Cz is given by zξdξ. Then the distance of z d zξdξ to the point 2  reads

d−1 z 2 X  z 2 zξ ξ − d = ξ2z2ξ2 + zξ2 − d 2 i d d 2 i=1  z 2 = (1 − ξ2)z2ξ2 + zξ2 − d d d 2 z 2 = , 2

ξ z d which is independent of ξ. So, the centers of Cz are located on a sphere with center 2  z and radius 2 .

2.2.2 The Funk–Radon transform

Setting the parameter z = 0, the point 0d = (0,..., 0)> is the center of the sphere Sd−1. Hence, the spherical transform U0 integrates along all maximal subspheres of the sphere Sd−1. This special case is called the Funk–Radon transform Z 1 ξ d−1 Ff(ξ) = U0f(ξ) = f dC0 , ξ ∈ S , (2.4) | d−2| ξ S C0 which is also known by the terms Funk transform, Minkowski–Funk transform or spheri- cal Radon transform, where the latter term occasionally also refers to means over (d − 1)- dimensional in Rd, cf. [20].

3 Relation with the Funk–Radon transform

In this section, we show a connection between the spherical transform Uz and the Funk– Radon transform F.

3.1 Two mappings on the sphere

d−1 d−1 Let z ∈ (−1, 1). We define the transformations hz, gz : S → S by √ d−1 2 X 1 − z z + ηd h (η) = η i + d (3.1) z 1 + zη i 1 + zη i=1 d d and d−1 √ ! 1 X i 2 d g (ξ) = ξi + 1 − z ξd . (3.2) z p 2 2 1 − z ξd i=1

Remark 3.1. The definitions of both hz and gz rely only on the d-th coordinate. The values in the other coordinates are just multiplied with the same factor in order to make

6 the vectors stay on the sphere. Furthermore, the transformations hz and gz are bijective with their respective inverses given by √ d−1 2 X 1 − z ωd − z h−1(ω) = h (ω) = ω i + d (3.3) z −z 1 − zω i 1 − zω i=1 d d and d−1 √ ! −1 1 X 2 i d gz (ω) = g √ iz (ω) = p 1 − z ωi + ωd . (3.4) 1−z2 2 2 2 1 − z + z ωd i=1 The computation of the inverses is straightforward and therefore omitted here. ξ The following lemma shows that the inverse of hz applied to the subsphere Cz yields d−1 a maximal subsphere of S with normal vector gz(ξ). Lemma 3.2. Let z ∈ (−1, 1) and ξ ∈ Sd−1. Then

−1 ξ gz(ξ) hz (Cz ) = C0 . (3.5) d−1 −1 ξ ξ Proof. Let η ∈ S . Then η lies in hz (Cz ) if and only if hz(η) ∈ Cz , i.e.,

hhz(η), ξi = zξd.

By the definition of hz in (3.1), we have √ d−1 2 X 1 − z z + ηd η ξ + ξ = zξ . 1 + zη i i 1 + zη d d i=1 d d After subtracting the right-hand side from the last equation, we have

d−1 √ X 1 − z2 1 − z2 η ξ + η ξ = 0. 1 + zη i i 1 + zη d d i=1 d d 2 −1/2 2 2 −1/2 Multiplication with (1 + zηd)(1 − z ) (1 − z ξd) yields d−1 √ X 1 1 − z2 ηiξi + ηdξd = 0, p 2 2 p 2 2 i=1 1 − z ξd 1 − z ξd

gz(ξ) which is equivalent to hη, gz(ξ)i = 0, so we obtain that η ∈ C0 .

d−1 ξ gz(ξ) Lemma 3.3. Let z ∈ (−1, 1) and ξ ∈ S . Denote by dCz and dC0 the volume ξ gz(ξ) forms on the manifolds Cz and C0 , respectively. Then the following relation between ξ gz(ξ) gz(ξ) the pullback of the volume form dCz over hz and dC0 holds. For η ∈ C0 , we have √ d−2  2  ∗ ξ 1 − z g(ξ) hz(dCz ) = dC0 . (3.6) 1 + zηd Furthermore, we have for the volume form dSd−1 on the sphere √ d−1  2  ∗ d−1 1 − z d−1 hz(dS ) = dS . (3.7) 1 + zηd

7 Proof. We compute the Jacobian Jhz of hz, which comprises the partial derivatives of hz. For all l, m ∈ {1, . . . , d − 1}, we have √ √ ∂[h ] 1 − z2 ∂[h ] z 1 − z2 z l = δ , z l = −η , ∂η 1 + zη l,m ∂η l (1 + zη )2 m d d d (3.8) 2 ∂[hz]d ∂[hz]d 1 − z = 0, = 2 . ∂ηm ∂ηd (1 + zηd)

i d−2 gz(ξ) i ξ Let [e ]i=1 be an orthonormal basis of the tangent space TηC0 . Then Jhz e ∈ Thz(η)Cz for i = 1, . . . , d − 1 is given by √ √ d−1  2 2  2 i X 1 − z i z 1 − z i l 1 − z i d Jhz e = el − ηl 2 ed  + 2 ed . 1 + zηd (1 + zηd) (1 + zηd) l=1 Hence, we have for all i, j ∈ {1, . . . , d − 2}

d−1  2 2 i j X 1 − z i j z(1 − z ) i j j i  Jhz e ,Jhz e = 2 elel − 3 ηl eled + el ed (1 + zηd) (1 + zηd) l=1 2 2  2 2 z (1 − z ) 2 i j (1 − z ) i j + 4 ηl eded + 4 eded. (1 + zηd) (1 + zηd) Expanding the sum, we obtain

2 d−1 2 d−1 d−1 ! i j 1 − z X i j z(1 − z ) j X i i X j Jhz e ,Jhz e = 2 elel − 3 ed ηlel + ed ηlel (1 + zηd) (1 + zηd) l=1 l=1 l=1 2 2 d−1 2 2 z (1 − z ) i j X 2 (1 − z ) i j + 4 eded ηl + 4 eded. (1 + zηd) (1 + zηd) l=1 Since the vectors ei and ej are elements of an orthonormal basis, we have hei, eji = Pd i j i j i j l=1 elel = δi,j. Furthermore, we know that he , ηi = he , ηi = 0 because e and e are g(ξ) d−1 2 in the tangent space TηC0 ⊂ TηS , and also kηk = 1. Hence, we have i j Jhz e ,Jhz e 2 2 1 − z i j z(1 − z ) i j = 2 (δi,j − eded) + 2 3 ηdeded (1 + zηd) (1 + zηd) 2 2 2 2 z (1 − z ) i j 2 (1 − z ) i j + 4 eded(1 − ηd) + 4 eded (1 + zηd) (1 + zηd) 2 1 − z i j 2 2 2 2 = 4 eded −(1 + zηd) + 2zηd(1 + zηd) + z (1 − ηd) + 1 − z (1 + zηd) 1 − z2 + 2 δi,j (1 + zηd) 1 − z2 = 2 δi,j. (1 + zηd)

8 The above computation shows that the vectors {J ei}d are orthogonal with length √ hz i=1 i 2 kJhz e k = 1 − z /(1 + zηd). By the definition of the pullback in (2.1) and the fact ξ that the volume form dCz is a multilinear (d − 2)-form, we obtain

√ d−2  2  ∗ ξ i d−2 ξ i d−2 1 − z hz(dCz )([e ]i=1 ) = dCz ([Jhz e ]i=1 ) = . (3.9) 1 + zηd

i d−1 d−1 If we set [e ]i=1 as a basis of the tangent space Tη(S ) in order to obtain (3.7), the previous calculations still hold except that the exponent d − 2 is replaced by d − 1 in equation (3.9).  1 d ξ Finally, we prove that the basis Jhz e ,...,Jhz e of Thz(η)Cz is oriented positively, i.e., that 1 d d(z) := det hz(η), ξ,Jhz e ,...,Jhz e > 0.

By the formula (3.8) of Jhz , the function d: [0, 1) → R is continuous, and it satisfies

1 d−2 1 d−2 d(0) = det h0(η), ξ,Jh0 e ,...,Jh0 e = det η, g0(ξ), e ,..., e > 0 since both h0 and g0 are equal to the identity map and we assumed the orthonormal  1 d−2 basis e ,..., e be oriented positively. By the orthogonality of the vectors ξ, hz(η) i and Jhz e , we see that d(z) vanishes nowhere and, hence, we obtain that d(z) > 0 for gz(ξ) all z ∈ [0, 1). The assertion follows by the uniqueness of the volume form dC0 .

3.2 Factorization

d−1 d−1 Let z ∈ (−1, 1) and f ∈ C(S ). We define the two transformations Mz, Nz : C(S ) → C(Sd−1) by √ d−2  2  1 − z d−1 Mzf(ξ) = f ◦ hz(ξ), ξ ∈ S (3.10) 1 + zξd and d−2 2 2 − 2 d−1 Nzf(ξ) = (1 − z ξd) f ◦ gz(ξ), ξ ∈ S . (3.11)

Remark 3.4. The transformations Mz and Nz are inverted by

√ d−2  2  1 − z −1 f(η) = Mzf(hz (η)) = M−zMzf(η), (3.12) 1 − zηd and d−2  2  2 1 − z −1 d−1 f(η) = 2 2 Nz(gz (η)), η ∈ S , (3.13) 1 − (1 − z )ηd respectively.

Now we are able to prove our main theorem about the factorization of the spherical transform Uz.

9 Theorem 3.5. Let z ∈ [0, 1). Then the factorization of the spherical transform

Uz = NzFMz (3.14) holds, where F is the Funk–Radon transform (2.4). 2 d−1 Proof. Let f ∈ C(S ) and ξ ∈ S . By the definition of Uz in (2.3), we have

d−2 Z d−2 2 2 2 ξ S (1 − z ξd) Uzf(ξ) = f dCz . (3.15) ξ Cz Then we have by the substitution rule (2.2) Z Z ξ ∗ ξ f dCz = (f ◦ hz) hz(dCz ). ξ −1 ξ Cz hz (Cz ) By (3.6) and (3.5), we obtain √ d−2 Z Z  2  ξ 1 − z gz(ξ) f dCz = f(hz(η)) dC0 (η). ξ gz(ξ) 1 + zη Cz C0 d

By the definition of Mz in (3.10), we see that Z Z ξ gz(ξ) f dCz = Mzf dC0 . ξ gz(ξ) Cz C0 The definition of the Funk–Radon transform (2.4) shows that Z ξ f dCz = FMzf(gz(ξ)), ξ Cz which implies (3.14). The factorization theorem 3.5 enables us to investigate the properties of the spherical transform Uz. Because the operators Mz and Nz are relatively simple, we can transfer many properties from the Funk–Radon transform, which has been studied by many authors already, to the spherical transform Uz.

4 Nullspace

With the help of the factorization (3.14) obtained in the previous section, we obtain the following characterization of the nullspace of the spherical transform Uz. d−1 Theorem 4.1. Let z ∈ [0, 1) and f ∈ C(S ). Then Uzf = 0 if and only if  2 d−2 1 − z d−1 f(ω) = − 2 f ◦ rz(ω), ω ∈ S , (4.1) 1 − 2zωd + z d−1 d−1 where rz : S → S is given by

d−1 2 2 X z − 1 2z − z ωd − ωd r (ω) = ω i + d. (4.2) z 1 + z2 − 2zω i 1 + z2 − 2zω i=1 d d

10 d−1 Proof. Let f ∈ C(S ). Since the operator Nz is bijecive by Remark 3.1, we see that Uzf = NzFMzf = 0 if and only if FMzf = 0. The nullspace of the Funk–Radon transform F consists of the odd functions, cf. [8, Proposition 3.4.12], so we obtain

d−1 Mzf(η) = −Mzf(−η), η ∈ S .

By the definition of Mz in (3.10), we have

√ d−2 √ d−2  1 − z2   1 − z2  f ◦ hz(η) = − f ◦ hz(−η). 1 + zηd 1 − zηd

We substitute ω = hz(η) and obtain

d−2 1 + z ωd−z ! 1−zωd −1 f(ω) = − f ◦ hz(−h (ω)) 1 − z ωd−z z 1−zωd  d−2 1 − zωd + z(ωd − z) −1 = − f ◦ hz(−hz (ω)) 1 − zωd − z(ωd − z)  2 d−2 1 − z −1 = − 2 f ◦ hz(−hz (ω)). 1 − 2zωd + z

−1 In order to show that rz = hz(−hz ), we compute the d-th component

z−ω z + d z − z2ω + z − ω 2z − z2ω − ω [h (−h−1(ω))] = 1−zωd = d d = d d . z z d z−ωd 2 2 1 + z 1 − zωd + z − zωd 1 − 2zωd + z 1−zωd For i ∈ {1, . . . , d − 1}, we have √ √ 1 − z2 − 1 − z2 z2 − 1 [h (−h−1(ω))] = ω = ω . z z i ωd−z i 2 i 1 − z 1 − zωd 1 − 2zωd + z 1−zωd

d−1 Remark 4.2. The map rz from (4.2) is the point reflection of the sphere S about the point zd. This can be seen as follows. Let ω ∈ Sd−1. The vectors ω − zd and d rz(ω) − z are parallel if for all i ∈ {1, . . . , d} [r (ω)] [r (ω)] − z z i = z d . ωi ωd − z We have

2 2 ωi [rz(ω)]d − z 2z − z ωd − ωd − z(1 + z − 2zωd) = 2 [rz(ω)]i ωd − z (z − 1)(ωd − z) 2 3 z + z ωd − ωd − z = 2 = 1, (z − 1)(ωd − z) provided all denominators are nonzero.

11 5 Function spaces on the sphere

Before we can state the range of the spherical transform Uz, we have to introduce some function spaces on the sphere Sd−1. The Hilbert space L2(Sd−1) comprises all square- integrable functions with the inner product of two functions f, g : Sd−1 → C Z hf, gi = f(ξ) g(ξ) d d−1(ξ) L2(Sd−1) S Sd−1 1/2 and the norm kfk 2 d−1 = hf, fi . L (S ) L2(Sd−1)

5.1 The space Cs(Sd−1) and differential operators on the sphere ∂ For brevity, we denote by ∂i = the partial derivative with respect to the i-th variable. ∂xi We extend a function f : Sd−1 → C to the surrounding space Rd \{0} by setting  x  f •(x) = f , x ∈ d \{0}. kxk R The surface gradient ∇• on the sphere is the orthogonal projection of the gradient > ∇ = (∂1, . . . , ∂d) onto the tangent space of the sphere. For a differentiable function f : Sd−1 → C, we have • • d−1 ∇ f(ξ) = ∇f (ξ), ξ ∈ S . In a similar manner, the restriction of the Laplacian

2 2 ∆ = ∂1 + ··· + ∂d to the sphere is known as the Laplace–Beltrami operator [16,(§14.20)]

• • d−1 ∆ f(ξ) = ∆f (ξ), ξ ∈ S . d Pd For a multi-index α = (α1, . . . , αd) ∈ N0, we define its norm kαk1 = i=1 |αi| and the α α1 αd s d−1 differential operator D = ∂1 ··· ∂d . Let s ∈ N0. We denote by C (S ) the space of functions f : Sd−1 → C whose extension f • has continuous derivatives up to the order s with the norm α • kfkCs( d−1) = max sup |D f (ξ)| . S kαk ≤s 1 ξ∈Sd−1 The space C0(Sd−1) = C(Sd−1) is the space of continuous functions with the uniform norm. The definition implies for f ∈ Cs+1(Sd−1) kfk ≤ kfk . (5.1) Cs(Sd−1) Cs+1(Sd−1) We define the space Cs(Sd−1 → Rd) of vector fields f : Sd−1 → Rd with the norm as the d Euclidean norm over its component functions, i.e., for f(ξ) = [fi(ξ)]i=1 we set v u d uX 2 kfk s d−1 d = t kf k s d−1 . C (S →R ) i C (S ) i=1

12 We see that for f ∈ Cs+1(Sd−1)

d d 2 X 2 X 2 2 k∇•fk = k∂ f •k ≤ kfk = d kfk . (5.2) Cs(Sd−1→Rd) i Cs(Sd−1) Cs+1(Sd−1) Cs+1(Sd−1) i=1 i=1

5.2 Sobolev spaces We give a short introduction to Sobolev spaces on the sphere based on [3] (see also [5, 15]). We define the Legendre polynomial Pn,d of degree n ∈ N0 and in dimension d by [3, (2.70)]  n n (d − 3)!! 2 3−d d 2 n+ d−3 P (t) = (−1) (1 − t ) 2 (1 − t ) 2 , t ∈ [−1, 1]. n,d (2n + d − 3)!! dt

2 d−1 For f ∈ L (S ) and n ∈ N0, we define the projection operator Z Nn,d d d−1 Pn,df(ξ) = d−1 f(η) Pn,d(hξ, ηi) dS (η), ξ ∈ S , |S | Sd−1 where (2n + d − 2)(n + d − 3)! N = dim(P (L2( d−1))) = . n,d n,d S n!(d − 2)! 2 d−1 Note that Pn,d is the L (S )-orthogonal projection onto the pairwise orthogonal spaces 2 d−1 Pn,d(L (S )) of harmonic polynomials that are homogeneous of degree n restricted to the sphere Sd−1. Every function f ∈ L2(Sd−1) can be written as the Laplace series

∞ X f = Pn,df. n=0

We define the Sobolev space Hs(Sd−1) of smoothness s ≥ 0 as the space of all functions f ∈ L2(Sd−1) with finite Sobolev norm [3, (3.98)] v u ∞  2s uX d − 2 2 kfk s d−1 = t n + kPn,dfk 2 d−1 (5.3) H (S ) 2 L (S ) n=0 s/2  • (d−2)2  = −∆ + 4 f . (5.4) L2(Sd−1)

Similarly to (5.1), we define the Sobolev norm of a vector field f : Sd−1 → Rd as the Euclidean norm over its component functions f(ξ) = (f1(ξ), . . . , fd(ξ)), i.e.,

d 2 X 2 kfk = kf k . (5.5) Hs(Sd−1→Rd) i Hs(Sd−1) i=1

13 Since the negative Laplace–Beltrami operator −∆• is self-adjoint, we can write the Sobolev norm (5.4) as

D 2 s E kfk2 = −∆• + (d−2) f, f . Hs(Sd−1) 4 L2(Sd−1)

We have for s ∈ N0

Z  2 s+1  kfk2 = −∆• + (d−2) f(ξ) f(ξ) d d−1(ξ) Hs+1(Sd−1) 4 S Sd−1 Z s  • (d−2)2   • (d−2)2   d−1 = −∆ + 4 −∆ + 4 f(ξ) f(ξ) dS (ξ). Sd−1 Then the Green–Beltrami identity [16, §14, Lemma 1] Z Z • d−1 • • d−1 − f(ξ) ∆ g(ξ) dS (ξ) = h∇ f(ξ), ∇ g(ξ)i dS (ξ), Sd−1 Sd−1 (5.6) 2 d−1 1 d−1 f ∈ C (S ), g ∈ C (S ) implies that

Z D  2 s E kfk2 = ∇•f(ξ), ∇• −∆• + (d−1) f(ξ) d d−1(ξ) Hs+1(Sd−1) 4 S Sd−1 2 Z s (d − 2)  • (d−2)2   d−1 + −∆ + 4 f(ξ) f(ξ) dS (ξ). 4 Sd−1 Since the gradient ∇• and the Laplacian ∆• commute by Schwarz’s theorem, we obtain the recursion

2 kfk2 = k∇•fk2 + (d−2) kfk2 . (5.7) Hs+1(Sd−1) Hs(Sd−1→Rd) 4 Hs(Sd−1)

5.3 Sobolev spaces as interpolation spaces The norm of a bounded linear operator A: X → Y between two Banach spaces X and

Y with norms k·kX and k·kY , respectively, is defined as

kAxkY kAkX→Y = sup . x∈X\{0} kxkX The following proposition shows that the boundedness of linear operators in Sobolev spaces Hs(S2) can be interpolated with respect to the smoothness parameter s. This result is derived from a more general interpolation theorem in [28].

s1 d−1 s1 d−1 Proposition 5.1. Let 0 ≤ s0 ≤ s1, and let A: H (S ) → H (S ) be a bounded linear operator such that its restriction to Hs0 (Sd−1) is also bounded. For θ ∈ [0, 1], we sθ d−1 set sθ = (1 − θ)s0 + θs1. Then the restriction of A to H (S ) is bounded with 1−θ θ kAk s s ≤ kAk s s kAk s s . H θ (Sd−1)→H θ (Sd−1) H 0 (Sd−1)→H 0 (Sd−1) H 1 (Sd−1)→H 1 (Sd−1)

14 k 2 d−1 Proof. For n ∈ N0, let {Yn,d | k = 1,...,Nn,d} be an orthonormal basis of Pn,d(L (S )), and let f ∈ L2(Sd−1). We write f as the Fourier series

∞ Nn,d X X f = f, Y k Y k . n,d L2(Sd−1) n,d n=0 k=1 On the index set I = {(n, k) | n ∈ N0, k = 1,...,Nn,d}, we define for s ≥ 0 the weight function

d−2 2s ws(n, k) = (n + 2 ) .

Then the Sobolev space Hs(Sd−1) is isometrically isomorphic to the weighted L2-space    2  2 ˆ ˆ 2 X ˆ L (I; ws) = f : I → C kfk 2 = f(n, k) ws(n, k) < ∞ L (I;ws)  (n,k)∈I  that consists of the Fourier coefficients

fˆ(n, k) = f, Y k n,d L2(Sd−1) on the set I with the counting . By [28, Theorem 1.18.5], the complex interpo- 2 ∼ s0 d−1 2 ∼ s1 d−1 lation space between L (I; ws0 ) = H (S ) and L (I; ws1 ) = H (S ) is  2 2  2 L (I; ws0 ),L (I; ws1 ) θ = L (I; w), where

1−θ θ d−2 2((1−θ)s+θt) w(n, k) = (ws0 (n, k)) (ws1 (n, k)) = n + 2 = wsθ (n, k).

2 ∼ sθ d−1 Hence, L (I; w) = H (S ). The assertion is a property of the interpolation space.

5.4 Multiplication and composition operators The following two theorems show that multiplication and composition with a smooth function are continuous operators in spherical Sobolev spaces Hs(Sd−1).

Theorem 5.2. Let s ∈ N0. The multiplication operator

s d−1 s d−1 s d−1 H (S ) × C (S ) → H (S ), (f, v) 7→ fv is continuous. In particular, for all f ∈ Hs(Sd−1) and v ∈ Cs(Sd−1), we have kfvk ≤ cs kfk kvk , (5.8) Hs(Sd−1) d Hs(Sd−1) Cs(Sd−1) where √ cd = 2d + 2.

15 Proof. We use induction over s ∈ N0. For s = 0, we have Z kfvk2 = |f(ξ) v(ξ)|2 dξ ≤ kfk2 kvk2 . L2(Sd−1) L2(Sd) C(Sd−1) Sd−1

s+1 d−1 Let the claimed equation (5.8) hold for s ∈ N0, and let f ∈ H (S ) and v ∈ Cs+1(Sd−1). Then the decomposition (5.7) of the Sobolev norm yields

2 kfvk2 = k∇•(fv)k2 + (d−2) kfvk2 Hs+1(Sd−1) Hs(Sd−1→Rd) 4 Hs(Sd−1) 2 = kf∇•v + v∇•fk2 + (d−2) kfvk2 . Hs(Sd−1→Rd) 4 Hs(Sd−1)

By the triangle inequality and since (a + b)2 ≤ 2(a2 + b2) for all a, b ∈ R, we obtain

2 kfvk2 ≤ 2 kf∇•vk2 + 2 kv∇•fk2 + (d−2) kfvk2 . Hs+1(Sd−1) Hs(Sd−1→Rd) Hs(Sd−1)→Rd 4 Hs(Sd−1) By the induction hypothesis, we have

c−2s kfvk2 ≤ 2 kfk2 k∇•vk2 + 2 k∇•fk2 kvk2 d Hd+1(Sd−1) Hs(Sd−1) Cs(Sd−1→Rd) Hs(Sd−1→Rd) Cs(Sd−1) 2 + (d−2) kfk2 kvk2 . 4 Hs(Sd−1) Cs(Sd−1) = 2 kfk2 k∇•vk2 + k∇•fk2 kvk2 Hs(Sd−1) Cs(Sd−1→Rd) Hs(Sd−1→Rd) Cs(Sd−1) + kfk2 kvk2 , Hs+1(Sd−1) Cs(Sd−1) where we made use of the decomposition (5.7) of the Sobolev norm. Furthermore, we apply (5.2) and obtain

c−2s kfvk2 ≤ 2d kfk2 kvk2 + kfk2 kvk2 d Hd+1(Sd−1) Hs(Sd−1) Cs+1(Sd−1) Hs+1(Sd−1) Cs(Sd−1) + kfk2 kvk2 . Hs+1(Sd−1) Cs(Sd−1) Because the involved norms are non-decreasing with respect to s, we see that √ kfvk ≤ cs 2d + 2 kfk kvk , Hs+1(Sd−1) d Hs+1(Sd−1) Cs+1(Sd−1) which shows (5.8).

d−1 d−1 s d−1 Theorem 5.3. Let s ∈ N0, and let v : S → S be bijective with v ∈ C (S → d−1 −1 1 d−1 d−1 S ) and v ∈ C (S → S ). Then there exists a constant bd,s(v) such that for all f ∈ Hs(Sd−1), we have kf ◦ vk ≤ b (v) kfk . Hs(Sd−1) d,s Hs(Sd−1) Proof. We have for s = 0 Z kf ◦ vk2 = |f(v(ξ))|2 d d−1(ξ). L2(Sd−1) S Sd−1

16 The substitution η = v(ξ) yields with the substitution rule (2.2) Z kf ◦ vk2 = |f(η)|2 (v−1)∗(d d−1) (η). L2(Sd−1) S Sd−1 Since v−1 ∈ C1(Sd−1 → Sd−1), there exists a continuous function ν : Sd−1 → R such that the pullback satisfies (v−1)∗(dSd−1) = ν dSd−1. Hence, we have kf ◦ vk2 ≤ kfk2 kνk , L2(Sd−1) L2(Sd−1) C(Sd−1) which shows the claim for s = 0. We use induction on s ∈ N0. By the decomposition (5.7) of the Sobolev norm, we have 2 kf ◦ vk2 = k∇•(f ◦ v)k2 + (d−2) kf ◦ vk2 . (5.9) Hs+1(Sd−1) Hs(Sd−1→Rd) 4 Hs(Sd−1) By the induction hypothesis, the second summand of (5.9) is bounded by kf ◦ vk ≤ b (v) kfk . (5.10) Hs(Sd) d,s Hs(Sd) Furthermore, by (5.5) and the chain rule, we have for the first summand of (5.9) k∇•(f ◦ v)k2 = k∇(f ◦ v)•k2 Hs(Sd−1→Rd) Hs(Sd−1→Rd) d X 2 = k∂ (f ◦ v)•k i Hs(Sd−1) i=1 2 d d X X • • • = ((∂jf ) ◦ v ) ∂ivj . i=1 j=1 Hs(Sd−1)

Pd 2 Applying the triangle inequality for the sum over j and Jensen’s inequality ( j=1 xj) ≤ Pd 2 d j=1 xj , we obtain

d d • 2 X X • • • 2 k∇ (f ◦ v)k s d−1 d ≤ d ((∂jf ) ◦ v ) ∂iv s d−1 . H (S →R ) j H (S ) i=1 j=1

By Theorem 5.2, we have

d d • 2 2s X • • 2 X • 2 k∇ (f ◦ v)k s d−1 d ≤ d c k(∂jf ) ◦ v k s d−1 ∂iv s d−1 H (S →R ) d H (S ) j C (S ) j=1 i=1 d X 2 2 ≤ d2 c2s k(∂ f •) ◦ v•k kv k , d j Hs(Sd−1) j Cs+1(Sd−1) j=1 where the last line follows from (5.2). By the induction hypothesis, we see that

d 2 X 2 2 k∇•(f ◦ v)k ≤ d2 c2s b (v)2 k∂ f •k kv k . Hs(Sd−1→Rd) d d,s j Hs(Sd−1) j Cs+1(Sd−1) j=1

17 By (5.5) and the fact that kv k2 ≤ kvk2 for all j = 1, . . . , d, we j Cs+1(Sd−1) Cs+1(Sd−1→Rd) obtain

k∇•(f ◦ v)k2 ≤ d2 c2s b (v)2 k∇•fk2 kvk2 . Hs(Sd−1→Rd) d d,s Hs(Sd−1→Rd) Cs+1(Sd−1→Rd) Inserting the last equation and (5.10) into (5.9), we obtain

kf ◦ vk2 Hs+1(Sd−1) 2 = k∇•(f ◦ v)k2 + (d−2) kf ◦ vk2 Hs(Sd−1→Rd) 4 Hs(Sd−1)  2  ≤ b (v)2 d2 c2s k∇•fk2 kvk2 + (d−2) kfk2 d,s d Hs(Sd−1→Rd) Cs+1(Sd−1→Rd) 4 Hs(Sd)    = b (v)2 d2 c2s kvk2 − 1 k∇•fk2 + kfk2 d,s d Cs+1(Sd−1→Rd) Hs(Sd−1→Rd) Hs+1(Sd−1) ≤ b (v)2d2 c2s kvk2 kfk2 , d,s d Cs+1(Sd−1→Rd) Hs+1(Sd−1) where we have made use of (5.7). Remark 5.4. The last theorem resembles a similar result found in [12, Theorem 1.2]: d Let M be a smooth, closed and oriented d-dimensional manifold and, for s > 2 +1, let ϕ ∈ Hs(M → M) be an orientation-preserving C1-diffeomorphism. Then the composition map Hs(M) → Hs(M), f 7→ f ◦ ϕ is continuous. However, Theorem 5.3 is not a special case of this result because Theorem 5.3 requires only s ≥ 0 but imposes stronger assumptions on ϕ.

6 Range

In this section, we show that for s ≥ 0 the spherical transform

s d−1 s+(d−2)/2 d−1 Uz : H (S ) → H (S ) is continuous. To this end, we seperately investigate the parts of the decomposition obtained in Theorem 3.5, Uz = NzFMz.

6.1 Continuity of F The following property of the Funk–Radon transform in Sobolev spaces was shown by Strichartz [27, Lemma 4.3] using an asymptotic analysis of its eigenvalues.

s d−1 s d−1 Proposition 6.1. Denote by Heven(S ) the restriction of the Sobolev space H (S ) to even functions f(ξ) = f(−ξ). The Funk–Radon transform

d−2 s d−1 s+ 2 d−1 F : Heven(S ) → Heven (S ) is continuous and bijective.

18 6.2 Continuity of Mz and Nz Theorem 6.2. Let z ∈ [0, 1) and s ∈ R with s ≥ 0. The operators

s d−1 s d−1 Mz : H (S ) → H (S ) and s d−1 s d−1 Nz : H (S ) → H (S ), as defined in (3.10) and (3.11), are continuous and open.

Proof. We first perform the proof for Mz. Initially, we consider only the situation s d−1 s ∈ N0. Let f ∈ H (S ) and z ∈ (−1, 1). We write

d−1 Mzf(ξ) = wz(ξ)[f ◦ hz](ξ), ξ ∈ S , where √ d−2  2  d−1 1 − z wz : S → R, wz(ξ) = 1 + zξd and hz is given in (3.1). We see that the extension

  √ d−2 • x 2 kxk d wz (x) = wz = 1 − z , x ∈ R \{0}, kxk kxk + zxd

• ∞ d ∞ d−1 is smooth except in the origin, i.e., wz ∈ C (R \{0}). Hence, wz ∈ C (S ). Then Theorem 5.2 implies that

kw (f ◦ h )k ≤ cs kw k kf ◦ h k . (6.1) z z Hs(Sd−1) d z Cs(Sd−1) z Hs(Sd−1)

Moreover, the extension of hz, √ d−1 2 X 1 − z z kxk + xd h• : d \{0} → d, h•(x) = x i + d, z R R z kxk + zx i kxk + zx i=1 d d

∞ d−1 d−1 −1 is also smooth, so hz ∈ C (S → S ). This implies that also the inverse hz = h−z, see (3.3), is smooth. So hz is a diffeomorphism and Theorem 5.3 together with (6.1) implies that

kM fk ≤ cs kw k kf ◦ h k z Hs(Sd−1) d z Cs(Sd−1) z Hs(Sd−1) ≤ cs kw k b (h ) kfk . d z Cs(Sd−1) d,s z Hs(Sd−1)

s d−1 s d−1 Thus, the operator Mz : H (S ) → H (S ) is continuous. Now let s ∈ R with s ≥ 0. The above proof shows that both the restrictions of Mz to Hbsc and to Hbsc+1 are continuous, where bsc denotes the largest integer that is smaller s d−1 than or equal to s. The continuity of Mz on H (S ) follows by the interpolation result Proposition 5.1.

19 −1 In order to prove the openness of Mz, we show that the inverse Mz restricted to s d−1 −1 H (S ) is continuous. However, we have already done this because Mz = M−z by (3.12). The same argumentation as above also works for the operator Nz as follows. Let z ∈ [0, 1) and s ∈ N0. We write

d−1 Nzf(ξ) = vz(ξ)[f ◦ gz](ξ), ξ ∈ S , where d−2 2 2 − 2 d−1 vz(ξ) = (1 − z ξd) , ξ ∈ S . We see that the extension

d−2 ! 2 kxk2 v•(x) = , x ∈ d \{0}, z 2 2 2 R kxk − z xd

s d is smooth and hence vz ∈ C (S ). Theorem 5.2 yields kv (f ◦ g )k ≤ cs kv k kf ◦ g k . z z Hs(Sd−1) d z Cs(Sd−1) z Hs(Sd−1) Since the extensions of both

  d−1 √ ! • x 1 X i 2 d d gz(x) = gz = q xi + 1 − z xd , x ∈ R \{0}, kxk 2 2 2 kxk − z xd i=1 and its inverse (3.4)

√ d−1 ! −1 • 1 2 X i d d [(gz ) ](x) = q 1 − z ωi + xd , x ∈ R \{0}, 2 2 2 2 kxk − z + z xd i=1

d s d−1 are smooth functions on R \{0}, we see that gz is a smooth diffeomorphism in C (S ). By Theorem 5.3, there exists a constant bd,s(gz) independent of f such that

kf ◦ g k ≤ b (g ) kfk . z Hs(Sd−1) d,s z Hs(Sd−1)

s d−1 Hence, Nz is a bounded operator on H (S ). An analogue computation shows that the inverse operator (3.13)

d−2 −1  2  2 −1 f(gz (η)) 1 − z −1 d−1 Nz f(η) = −1 = 2 2 2 (f ◦ gz )(η), η ∈ S vz(gz (η)) 1 − z + z ηd is also bounded on Hs(Sd−1). The assertion for general s follows by the same interpola- tion argument as for Mz.

20 6.3 Continuity of Uz s d−1 Theorem 6.3. Let z ∈ (0, 1) and s ∈ R with s ≥ 0. We set Hz (S ) as the subspace of all functions f ∈ Hs(Sd−1) that satisfy

 2 d−2 1 − z d−1 f(ω) = 2 f ◦ rz(ω), ω ∈ S , (6.2) 1 − 2zωd + z

d almost everywhere, where the point reflection rz about the point z is given in (4.2). Then the spherical transform

d−2 s d−1 s+ 2 d−1 Uz : Hz (S ) → Heven (S ) is continuous and bijective and its inverse operator is also continuous.

Proof. In Theorem 3.5, we obtained the decomposition

Uz = NzFMz.

We are going to look at the parts of this decomposition separately. By Theorem 6.2, we obtain that s d−1 s d−1 Mz : H (S ) → H (S ) is continuous and bijective. The same holds for the restriction

s d−1 s d−1 Mz : Hz (S ) → Heven(S ), which follows from the characterization of the nullspace in Theorem 4.1. By Proposition 6.1, the Funk–Radon transform

d−2 s d−1 s+ 2 d−1 F : Heven(S ) → Heven (S ) is continuous and bijective. Finally, Theorem 6.2 and the observation that any function d−1 f : S → C is even if and only if Nzf is even show that

d−2 d−2 s+ 2 d−1 s+ 2 d−1 Nz : Heven (S ) → Heven (S ) is continuous and bijective. The continuity of the inverse operator of Uz follows from the open mapping theorem. Theorem 6.3 is a generalization of Proposition 6.1 for the Funk–Radon transform F; s d−1 s d−1 the main difference is that the space Heven(S ) is replaced by Hz (S ), which contains functions that satisfy the symmetry condition (6.2) with respect to the point reflection in d d−2 z  . Furthermore, the spherical transform Uz is smoothing of degree 2 , which comes from the fact that Uz takes the integrals along (d − 2)-dimensional submanifolds.

21 7 Geometric interpretation

d−1 d−1 We give geometric interpretations of the mappings gz and hz : S → S that√ were 2 defined in Section 3.1. The mapping gz consists of a scaling with the factor 1 − z along the ξd axis, which maps the sphere to an ellipsoid, which is symmetric with respect to rotations about the ξd axis. Then a central projection maps this ellipsoid onto the sphere again. In order give a description of the mapping hz, we define the stereographic projection

d−1 X ξi π : d−1 \{d} → d−1, ξ 7→ i S R 1 − ξ i=1 d and its inverse

2 d −1 d−1 d−1 d 2x + (kxk − 1)  π : R → S \{ }, x 7→ . 1 + kxk2

The following corollary states that, via stereographic projection, the map hz on the sphere Sd−1 corresponds to a uniform scaling in the equatorial hyperplane Rd−1 with the q 1+z scaling factor 1−z .

Corollary 7.1. Let ξ ∈ Sd−1 and z ∈ (0, 1). Then we have ! r1 + z h (ξ) = π−1 π(ξ) . z 1 − z

Proof. We are going to show that

r1 + z π(h (ξ)) = π(ξ) z 1 − z holds. We have on the one hand

r r d−1 1 + z 1 + z X ξi π(ξ) = i 1 − z 1 − z 1 − ξ i=1 d and the other hand √ 1−z2 ξ √ d−1 i d−1 2 X 1+zξd i X 1 − z ξi i π(hz(ξ)) = z+ξ  =  1 − d 1 + zξd − (z + ξd) i=1 1+zξd i=1 √ d−1 2 X 1 − z ξi = i. (1 − z) (1 − ξ ) i=1 d √ The assertion follows by canceling 1 − z.

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