The Funk-Radon transform for hyperplane sections through a common point

The Funk-Radon transform, also known as the spherical Radon transform, assigns to a function on the sphere 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 hyperplanes containing a common point inside the sphere. If this point is the origin, this is the same as the aforementioned 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.


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-Funk transform or spherical 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 where V (C ξ z ) denotes the (d − 2)-dimensional volume of C ξ z . The spherical transform U z on the two-dimensional sphere S 2 was first investigated by Salman [24] in 2016. He showed the injectivity of this transform U z for smooth functions f ∈ C ∞ (S 2 ) that are supported inside the spherical cap {ξ ∈ S 2 | ξ 3 < 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 ∈ C 1 (S d ). A different approach was taken in [19], where a relation with the classical Funk-Radon transform U 0 was established and also used for a characterization of the nullspace and range of U z 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 U z with the Funk-Radon transform F = U 0 in order to characterize its nullspace. However, the description of the range requires some more effort, because the operator U z 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 S d−1 or great circles on S 2 . We obtain the classical Funk-Radon transform F = U 0 . The case z = 1 corresponds to the subspheres containing the north pole (0, . . . , 0, 1) . This case is known as the spherical slice transform U 1 and has been investigated since the early 1990s in [1,4,9]. However, unlike U z for z < 1, the spherical slice transform U 1 is injective for f ∈ L ∞ (S d−1 ), see [22]. The main tool to derive this injectivity result of U 1 is the stereographic projection, which turns the subspheres of S d−1 through the north pole into hyperplanes in R d−1 and thus connecting the spherical slice transform U 1 to the Radon transform on R d−1 . 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 S 2 , 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 U z as well as for z = 0 the sections with subspheres having constant radius previously considered by Schneider [26] in 1969.
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 H s (S d−1 ) can be imagined as the space of functions f : S d−1 → C whose derivatives up to order s are square-integrable, and we denote by H s even (S d−1 ) its restriction to even functions f (ξ) = f (−ξ). A thorough definition of H s (S d−1 ) is given in Section 5.2. The behavior of the Funk-Radon transform was investigated by Strichartz [27], who found that the Funk-Radon transform F : is continuous and bijective. In Theorem 6.3, we show that basically the same holds for the generalized transform U z , i.e., the spherical transform U z : is bounded and bijective and its inverse is also bounded, where H s z (S d−1 ) contains functions that are symmetric with respect to the point reflection in (0, . . . , 0, z) . This paper is structured as follows. In Section 2, we give a short introduction to smooth manifolds and review the required notation on the sphere. In Section 3, we show the relation of U z with the classical Funk-Radon transform F = U 0 . With the help of that factorization, we characterize the nullspace of the spherical transform U z in Section 4. In Section 5, we consider Sobolev spaces on the sphere and show the continuity of certain multiplication and composition operators. Then, in Section 6, we prove the continuity of the spherical transform U z in Sobolev spaces. Finally, we include a geometric interpetation of our factorization result in Section 7.

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 Euclidean space R d equipped with the scalar product ξ, η = d i=1 ξ i η i and the norm ξ = ξ, ξ 1/2 . We denote the unit vectors in R d with i , i.e. i j = δ i,j , where δ is the Kronecker delta. Every vector ξ ∈ R d can be written as ξ = d i=1 ξ i i . A diffeomorphism is a bijective, smooth mapping R n → R n whose inverse is also smooth. We say a function is smooth if it has derivatives of arbitrary order. An ndimensional smooth manifold M without boundary is a subset of R d such that for every ξ ∈ M there exists an open neighborhood N (ξ) ⊂ R d containing ξ, an open set U ⊂ R d , and a diffeomorphism m : U → N (ξ) such that A k-form ω on M is a family (ω ξ ) ξ∈M of antisymmetric k-linear functionals where (T ξ M ) k = T ξ M × · · · × T ξ M . 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) on M that is defined for any v 1 , . . . , v k ∈ T ξ M by where γ i : [0, 1] → M are smooth paths satisfying γ i (0) = ξ and γ i (0) = v i . If the smooth function f : R d → R d extends to the surrounding space, the pullback can be expressed as where J f denotes the Jacobian matrix of f . An atlas where the latter is the standard volume integral dR n on V i ⊂ R n and the functions c i : V i → R are uniquely determined by the condition m * i (ϕ i · ω) = c i dR n . 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, for ξ ∈ S d−1 . We define an orientation on S d−1 by saying that a basis [

The spherical transform
where ξ ∈ S d−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 S(ξ, t) by saying that a basis [ In the following, we consider subspheres of S d−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 ξ d axis. For z ∈ [0, 1), we consider the point For a continuous function f : which computes the mean values of f along the subspheres C ξ z .
Remark 2.1. The subspheres C ξ z , along which we integrate, can also be imagined in the following way. The centers of the subspheres C ξ z are located on a sphere that contains the origin and z d and is rotationally symmetric about the North-South axis. This can be seen as follows. The center of the sphere C ξ z is given by zξ d ξ. Then the distance of zξ d ξ to the point z

Two mappings on the sphere
Let z ∈ (−1, 1). We define the transformations h z , g z : and Remark 3.1. The definitions of both h z and g z rely only on the d-th coordinate. The values in the other coordinates are just multiplied with the same factor in order to make the vectors stay on the sphere. Furthermore, the transformations h z and g z are bijective with their respective inverses given by and The computation of the inverses is straightforward and therefore omitted here.
The following lemma shows that the inverse of h z applied to the subsphere C ξ z yields a maximal subsphere of S d−1 with normal vector g z (ξ).
After subtracting the right-hand side from the last equation, we have which is equivalent to η, g z (ξ) = 0, so we obtain that η ∈ C the volume forms on the manifolds C ξ z and C g z (ξ) 0 , respectively. Then the following relation between the pullback of the volume form dC ξ z over h z and dC (3.6) Furthermore, we have for the volume form dS d−1 on the sphere (3.7) Proof. We compute the Jacobian J hz of h z , which comprises the partial derivatives of h z . For all l, m ∈ {1, . . . , d − 1}, we have be an orthonormal basis of the tangent space T η C g z (ξ) 0 . Then J hz e i ∈ T hz(η) C ξ z for i = 1, . . . , d − 1 is given by Hence, we have for all i, j ∈ {1, . . . , d − 2} Expanding the sum, we obtain Since the vectors e i and e j are elements of an orthonormal basis, we have e i , e j = d l=1 e i l e j l = δ i,j . Furthermore, we know that e i , η = e j , η = 0 because e i and e j are in the tangent space T η C g(ξ) 0 ⊂ T η S d−1 , and also η 2 = 1. Hence, we have The above computation shows that the vectors {J hz e i } d i=1 are orthogonal with length J hz e i = √ 1 − z 2 /(1 + zη d ). By the definition of the pullback in (2.1) and the fact that the volume form dC ξ z is a multilinear (d − 2)-form, we obtain (3.9) If we set [e i ] d−1 i=1 as a basis of the tangent space T η (S d−1 ) 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).
Finally, we prove that the basis J hz e 1 , . . . , J hz e d of T hz(η) C ξ z is oriented positively, i.e., that

Factorization
Let z ∈ (−1, 1) and f ∈ C(S d−1 ). We define the two transformations M z , N z : and Remark 3.4. The transformations M z and N z are inverted by and respectively. Now we are able to prove our main theorem about the factorization of the spherical transform U z . Proof. Let f ∈ C(S 2 ) and ξ ∈ S d−1 . By the definition of U z in (2.3), we have Then we have by the substitution rule (2.2) By (3.6) and (3.5), we obtain By the definition of M z in (3.10), we see that

The definition of the Funk-Radon transform (2.4) shows that
which implies (3.14).
The factorization theorem 3.5 enables us to investigate the properties of the spherical transform U z . Because the operators M z and N z 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 U z .

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 U z .
where r z : S d−1 → S d−1 is given by Proof. Let f ∈ C(S d−1 ). Since the operator N z is bijecive by Remark 3.1, we see that By the definition of M z in (3.10), we have We substitute ω = h z (η) and obtain In order to show that We have provided all denominators are nonzero.

Function spaces on the sphere
Before we can state the range of the spherical transform U z , we have to introduce some function spaces on the sphere S d−1 . The Hilbert space L 2 (S d−1 ) comprises all squareintegrable functions with the inner product of two functions f, g :

5.1
The space C s (S d−1 ) and differential operators on the sphere For brevity, we denote by ∂ i = ∂ ∂x i the partial derivative with respect to the i-th variable. We extend a function f : 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 : In a similar manner, the restriction of the Laplacian to the sphere is known as the Laplace-Beltrami operator [16, ( §14.20)] For a multi-index α = (α 1 , . . . , α d ) ∈ N d 0 , we define its norm α 1 = d i=1 |α i | and the differential operator D α = ∂ α 1 1 · · · ∂ α d d . Let s ∈ N 0 . We denote by C s (S d−1 ) the space of functions f : S d−1 → C whose extension f • has continuous derivatives up to the order s with the norm f C s (S d−1 ) = max We see that for f ∈ C s+1 (S d−1 )

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 P n,d of degree n ∈ N 0 and in dimension d by [3, (2.70)] For f ∈ L 2 (S d−1 ) and n ∈ N 0 , we define the projection operator Note that P n,d is the L 2 (S d−1 )-orthogonal projection onto the pairwise orthogonal spaces P n,d (L 2 (S d−1 )) of harmonic polynomials that are homogeneous of degree n restricted to the sphere S d−1 . Every function f ∈ L 2 (S d−1 ) can be written as the Laplace series We define the Sobolev space H s (S d−1 ) of smoothness s ≥ 0 as the space of all functions f ∈ L 2 (S d−1 ) with finite Sobolev norm [3, (3.98)] .
We have for s ∈ N 0 Then the Green-Beltrami identity [16, §14, Lemma 1] Since the gradient ∇ • and the Laplacian ∆ • commute by Schwarz's theorem, we obtain the recursion

Sobolev spaces as interpolation spaces
The norm of a bounded linear operator A : X → Y between two Banach spaces X and Y with norms · X and · Y , respectively, is defined as The following proposition shows that the boundedness of linear operators in Sobolev spaces H s (S 2 ) can be interpolated with respect to the smoothness parameter s. This result is derived from a more general interpolation theorem in [28].
Hence, L 2 (I; w) ∼ = H s θ (S d−1 ). The assertion is a property of the interpolation space.

Multiplication and composition operators
The following two theorems show that multiplication and composition with a smooth function are continuous operators in spherical Sobolev spaces H s (S d−1 ).
almost everywhere, where the point reflection r z about the point z d is given in (4.2).
Then the spherical transform is continuous and bijective and its inverse operator is also continuous.
Proof. In Theorem 3.5, we obtained the decomposition We are going to look at the parts of this decomposition separately. By Theorem 6.2, we obtain that is continuous and bijective. The same holds for the restriction is continuous and bijective. The continuity of the inverse operator of U z follows from the open mapping theorem. Theorem 6.3 is a generalization of Proposition 6.1 for the Funk-Radon transform F; the main difference is that the space H s even (S d−1 ) is replaced by H s z (S d−1 ), which contains functions that satisfy the symmetry condition (6.2) with respect to the point reflection in z d . Furthermore, the spherical transform U z is smoothing of degree d−2 2 , which comes from the fact that U z takes the integrals along (d − 2)-dimensional submanifolds.

Geometric interpretation
We give geometric interpretations of the mappings g z and h z : S d−1 → S d−1 that were defined in Section 3.1. The mapping g z consists of a scaling with the factor √ 1 − z 2 along the ξ d axis, which maps the sphere to an ellipsoid, which is symmetric with respect to rotations about the ξ d axis. Then a central projection maps this ellipsoid onto the sphere again.
In order give a description of the mapping h z , we define the stereographic projection The following corollary states that, via stereographic projection, the map h z on the sphere S d−1 corresponds to a uniform scaling in the equatorial hyperplane R d−1 with the scaling factor 1+z 1−z .
Proof. We are going to show that π(h z (ξ)) = 1 + z 1 − z π(ξ) holds. We have on the one hand The assertion follows by canceling √ 1 − z.