On Two Families of Funk-Type Transforms

TL;DR

This paper studies two types of Funk-like spherical transforms, exploring their relationships, invertibility, and connections to Radon transforms, thereby advancing understanding of their mathematical properties and potential applications.

Contribution

It introduces new relationships between Funk-type transforms with different centers and dimensions, and provides inversion formulas and injectivity results.

Findings

Funk-type transforms with exterior centers relate to parallel slice transforms.

Parallel slice transforms are connected to Radon-John d-plane transforms.

The paper establishes invertibility and injectivity conditions for these transforms.

Abstract

We consider two families of Funk-type transforms that assign to a function on the unit sphere the integrals of that function over spherical sections by planes of fixed dimension. Transforms of the first kind are generated by planes passing through a fixed center outside the sphere. Similar transforms with interior center and with center on the sphere itself we studied in previous publications. Transforms of the second kind, or the parallel slice transforms, correspond to planes that are parallel to a fixed direction. We show that the Funk-type transforms with exterior center express through the parallel slice transforms and the latter are intimately related to the Radon-John d-plane transforms on the Euclidean ball. These results allow us to investigate injectivity of our transforms and obtain inversion formulas for them. We also establish connection between the Funk-type transforms of different dimensions with arbitrary center.

Equations

$$(F_{a}f)(\tau )=\underset{S^n\cap \tau }{\int }f(x)d\sigma (x),({\Pi }_{a}f)(\zeta )=\underset{S^n\cap \zeta }{\int }f(x)d\sigma (x).$$

$$F_{a}f=({\Pi }_a{M}_{a^{\ast }}f)\circ {\phi }_{a^{\ast }},$$

$$(M_{a^{\ast }}f)(y)={\left(\frac{s_{a^{\ast }}}{1-a^{\ast }\cdot y}\right)}^{k-1}(f\circ {\phi }_{a^{\ast }})(y),s_{a^{\ast }}=\sqrt{1-|a^{\ast }{|}^2};$$

$${\mathrm{P}}_{a}x=\frac{a\cdot x}{|a{|}^2}a,{\mathrm{Q}}_a={\mathrm{I}}_{n+1}-{\mathrm{P}}_a,a\ne 0,$$

$$(R\phi )(\tau )\equiv (R\phi )({\tau }_0,u)=\underset{{\tau }_0}{\int }\phi (u+v)dv$$

$$(I_{-,2}^{\alpha }f)(t)=\frac{2}{\Gamma (\alpha )}\underset{t}{\overset{\infty }{\int }}\frac{f(s)sds}{(s^2-t^2{)}^{1-\alpha }},\alpha >0.$$

$${\mathcal{D}}_{-,2}^{d/2}f=t^{2-d+2m}(-D{)}^{m+1}{t}^{d}g,g=I_{-,2}^{1-d/2+m}{t}^{-2m-2}f,D=\frac{1}{2t}\frac{d}{dt},$$

$${\mathcal{D}}_{-,2}^{d/2}f=(-D{)}^{d/2}f.$$

$$(R_x^{\ast }\Phi )(t)=\underset{SO(n)}{\int }\Phi (\gamma {\tau }_0+x+t\gamma e_n)d\gamma ,t>0,$$

$$\phi (x)\equiv (R^{-1}\Phi )(x)=\underset{t\to 0}{\lim }{\pi }^{-d/2}({\mathcal{D}}_{-,2}^{d/2}{R}_x^{\ast }\Phi )(t),$$

$$\zeta \equiv \zeta ({\zeta }_0,u)={\zeta }_0+u,$$

$${\zeta }_0\in {\mathcal{Z}}_a^0(n+1,k),u\in {\zeta }_0^{\perp }\cap a^{\perp },|u|<1.$$

$$({\Pi }_{a}f)(\zeta )=\underset{S^n\cap \zeta }{\int }f(x)d\sigma (x),\zeta \in {\mathcal{Z}}_a(n+1,k),$$

$${\mathrm{R}}_{a}x=x-2\frac{x\cdot a}{|a{|}^2}a$$

$$C_a^{\pm }(S^n)=\left\{f\in C(S^n):f(x)=\pm f({\mathrm{R}}_{a}x)\right\},$$

$$f_a^{+}=\frac{f+f\circ {\mathrm{R}}_a}{2},f_a^{-}=\frac{f-f\circ {\mathrm{R}}_a}{2}.$$

$$({\Pi }_{a}f)(\zeta )=(1-|u{|}^2{)}^{1/2}(R\phi )(a^{\perp }\cap \zeta ),$$

$$\phi (y)=2(1-|y{|}^2{)}^{-1/2}f(y+\sqrt{1-|y{|}^2}\tilde{a}),\tilde{a}=\frac{a}{|a|},$$

$$R^k=R{e}_1\oplus \cdots \oplus R{e}_{k-1}\oplus R{e}_{n+1}=R^{k-1}\oplus R{e}_{n+1}$$

$$\gamma e_{n+1}=\tilde{a},\gamma R^k={\zeta }_o,\gamma e_k=u/|u|.$$

$$t=|u|,x=\gamma (\xi +t{e}_k),\xi \in S^n\cap R^k.$$

$$({\Pi }_{a}f)(\zeta )=\underset{S_{\ast }^{k-1}}{\int }f(\gamma (\xi +t{e}_k))d\sigma (\xi ).$$

$$({\Pi }_{a}f)(\zeta )=2\underset{|\eta |<\sqrt{1-t^2}}{\int }f(\gamma (\eta +t{e}_k+\sqrt{1-t^2-|\eta {|}^2}{e}_{n+1}))$$

$$\times \sqrt{1+{\left(\frac{\partial {\eta }_{n+1}}{\partial {\eta }_1}\right)}^2+\dots +{\left(\frac{\partial {\eta }_{n+1}}{\partial {\eta }_{k-1}}\right)}^2}d\eta$$

$$=2\sqrt{1-t^2}\underset{|\eta |<\sqrt{1-t^2}}{\int }f(\gamma (\eta +t{e}_k+\sqrt{1-t^2-|\eta {|}^2}{e}_{n+1}))\frac{d\eta }{\sqrt{1-t^2-|\eta {|}^2}}.$$

$$({\Pi }_{a}f)(\zeta )\equiv ({\Pi }_{a}f)({\zeta }_0,u)=2\sqrt{1-|u{|}^2}\underset{\tau }{\int }\frac{f(y+\sqrt{1-|y{|}^2}\tilde{a})}{\sqrt{1-|y{|}^2}}dy,$$

$$\ker ({\Pi }_a)=C_a^{-}(S^n)=\{f\in C(S^n):f_a^{+}=0\}.$$

$${\Pi }_a^{-1}g=f_a^{+},g={\Pi }_{a}f\in {\Pi }_a(C(S^n)).$$

$$0$$

$${\Pi }_a({\Pi }_a^{-1}g)={\Pi }_a{f}_a^{+}={\Pi }_{a}f=g,$$

$$f(x)\equiv ({\Pi }_a^{-1}g)(x)=\frac{1}{2}|x\cdot \tilde{a}|(R^{-1}\Phi )({\mathrm{Q}}_{a}x),\tilde{a}=\frac{a}{|a|},$$

Full text

1. Introduction

The present investigation deals with a pair of integral operators on the unit sphere ${{S}}^{n}$ in ${{R}}^{n+1}$ that arise in spherical tomography [7] and resemble the classical Funk transform. These operators have the form

[TABLE]

In the first integral, a function $f$ is integrated over the spherical section ${{S}}^{n}\cap\,\tau$ by the $k$-dimensional plane $\tau$ passing through a fixed point $a\in{{R}}^{n+1}$, $1<k\leq n$. In the second integral, the spherical section ${{S}}^{n}\cap\,\zeta$ is determined by the $k$-plane $\zeta$, which is parallel to the vector $a\neq 0$. We call these operators the shifted Funk transform and the spherical slice transform, respectively. Our main concern is injectivity of the operators (1.1) and inversion formulas.

The case when $a=0$ is the origin of ${{R}}^{n+1}$ corresponds to the totally geodesic transform $F_{0}$, which goes back to Minkowski [13], Funk [3, 4], and Helgason [8]; see also [6, 9, 19, 22] and references therein. This operator annihilates odd functions and can be explicitly inverted on even functions.

The case $|a|=1$, when $F_{a}$ is injective, was considered by Abouelaz and Daher [1] (on zonal functions) and Helgason [9, p. 145] (on arbitrary functions) for $n=2$. The general case of all $n\geq 2$ for hyperplane sections was studied by the second co-author [22, Section 7.2]. The method of this work extends to all $1<k\leq n$ and inversion formulas can be obtained by making use of the stereographic projection and the inversion results for the Radon-John $d$-plane transforms.

Operators $F_{a}$ with $|a|<1$ were studied by Salman [26, 27] (in a different setting, caused by support restrictions) and later by Quellmalz [17, 18] and Rubin [23] for $k=n$. The case of all $1<k\leq n$ was considered by Agranovsky and Rubin [2]. The kernel of such operators consists of functions $f$ that are odd in a certain sense, whereas the corresponding even components of $f$ can be explicitly reconstructed from $F_{a}f$.

We give credit to related works by Gardner [5], Gindikin, Reeds, and Shepp [7], Kazantsev [11], Palamodov [15, 16], and many others, containing information on diverse operators of integral geometry on the unit sphere and extensive references on this subject.

The present article is focused on the unknown case $|a|>1$ of the exterior center. We shall prove that, unlike the case $|a|<1$, when $F_{a}$ is equivalent, in a certain sense, to the Funk transform $F_{0}$, now it is similarly related to the parallel slice transform $\Pi_{a}$. When $a$ belongs to the last coordinate axis, the operator $\Pi_{a}$ is known as the vertical slice transform. It was studied by Hielscher and Quellmalz [10] (for $n=2$) and Rubin [24] (for $k=n\geq 2$). In the present article we consider $\Pi_{a}$ for all $k\leq n$ and all $a\neq 0$.

Plan of the Paper and Main Results. Section 2 contains preliminaries, including notation and auxiliary facts about the Radon-John $d$-plane transforms.

Section 3 deals with the parallel slice transform $\Pi_{a}$. We show that $\Pi_{a}$ reduces to the Radon-John transform of functions on the unit ball in the subspace $a^{\perp}$ perpendicular to $a$. Hence $\Pi_{a}$ can be explicitly inverted on the subspace of continuous functions that are even with respect to reflection in $a^{\perp}$.

Section 4 contains auxiliary facts about Möbius automorphisms $\varphi_{a}$ and $\varphi_{a^{*}}$, $a^{*}=a/|a|^{2}$ (see (4.1)) that play an important role in the sequel. The definition of these automorphisms is borrowed from the book by Rudin [25, Section 2.2.1].

Section 5 contains main results of the paper related to injectivity and inversion of the shifted Funk transform $F_{a}$, $|a|>1$, on continuous functions. The consideration relies on the equality

[TABLE]

which links together $F_{a}$ and $\Pi_{a}$ by means of $\varphi_{a^{*}}$ and the bijection $M_{a^{*}}$ having the form

[TABLE]

see Theorem 5.1. This equality is the core of the paper. It paves the way to the study of injectivity and inversion of $F_{a}$ (see Theorems 5.3, 5.4). In the case $|a|<1$, a similar formula, but with the geodesic Funk transform $F_{0}$ in place of $\Pi_{a}$, was obtained by Quellmalz [17] for $n=2$ and generalized in [2, 18, 23] to higher dimensions. In order not to overload Section 5 with technicalities, we move the proof of (1.2) to Section 6.

Section 7 can be considered as an addendum to the previous sections and can be read independently. Here the main result is Theorem 7.1 that establishes connection between the shifted Funk transforms $F_{a}$ associated with planes of different dimensions. The point $a$ in this theorem is arbitrary. Theorem 7.1 reduces inversion of the lower dimensional transforms to those with $k=n$. Although the case $k=n$ is less technical, the payment for this simplicity is that the inversion procedure contains an additional integration step and the degree of the differential operator in the corresponding Radon-John inversion component becomes bigger, specifically, $n-1$ versus $k-1$. We think that exposition of both methods is, perhaps, instructive.

In our previous work [2], we also studied the paired shifted Funk transform $f\to\{F_{a}f,F_{b}f\}$ with two distinct centers $a$ and $b$ inside the sphere. This approach makes it possible to circumvent non-injectvity of the single-centered transform $F_{a}$. Under certain conditions, the results of [2] extend to arbitrary two centers in the space, e.g., one center inside ${{S}}^{n}$ and another outside, or both are outside. We plan to write a separate paper on this subject.

2. Preliminaries

2.1. Notation

In the following, ${{R}}^{n+1}$ is the real $(n+1)$-dimensional space of points $x=(x_{1},\ldots,x_{n+1})$; $e_{1},e_{2},\ldots,e_{n+1}$ are the coordinate unit vectors; $x\cdot y$ denotes the usual dot product. All points in ${{R}}^{n+1}$ are identified with the corresponding column vectors. For the sake of simplicity, we use the same notation “$\,0\,$” for the zero vector or a number and for the origin of the corresponding coordinate space. Precise meaning of this notation is clear from the context.

We write ${{B}}^{n+1}=\{x\in{{R}}^{n+1}:|x|<1\}$ for the open unit ball in ${{R}}^{n+1}$; ${{S}}^{n}$ is the boundary of ${{B}}^{n+1}$. If $x\in{{S}}^{n}$, then $dx$ stands for the Riemannian measure on ${{S}}^{n}$. The notation $d\sigma(x)$ is used for the lower-dimensional surface area measure on subspheres of ${{S}}^{n}$. Given a point $a\in{{R}}^{n+1}\setminus\{0\}$, let $a^{*}=a/|a|^{2}$ be the inversion of $a$ with respect to ${{S}}^{n}$, $\tilde{a}=a/|a|\in{{S}}^{n}$, and $a^{\perp}$ be the linear subspace orthogonal to the vector $a$.

We set \hbox{\frak M}_{n,m} to be the space of all real matrices having $n$ rows and $m$ columns; $\mathrm{M}^{\prime}$ denotes the transpose of the matrix $\mathrm{M}$; $\mathrm{I}_{m}$ is the identity $m\times m$ matrix. For $n\geq m$, {\rm St}(n,m)=\{\mathrm{M}\in\hbox{\frak M}_{n,m}:\mathrm{M}^{\prime}\mathrm{M}=\mathrm{I}_{m}\} is the Stiefel manifold of orthonormal $m$-frames in ${{R}}^{n}$. Given $a\in{{R}}^{n+1}\setminus\{0\}$, we write ${\rm St}(a^{\perp},m)$ for the subset of ${\rm St}(n+1,m)$ formed by $m$-frames in $a^{\perp}$.

We denote by ${\mathcal{T}}_{a}(n+1,k)$ a family of all $k$-dimensional affine planes $\tau$ in ${{R}}^{n+1}$ passing through the point $a$ and intersecting the ball ${{B}}^{n+1}$. If $a\neq 0$, the notation $\mathcal{Z}_{a}(n+1,k)$ is used for the set of all $k$-dimensional affine planes $\zeta$ that are parallel to the vector $a$ and meet ${{B}}^{n+1}$. We also set

[TABLE]

for the corresponding projection maps.

The standard notation $C({{S}}^{n})$ and $L^{p}({{S}}^{n})$ is used for the corresponding spaces of continuous and $L^{p}$ functions on ${{S}}^{n}$.

2.2. The Radon-John Transform

Let $d$ and $n$ be positive integers, $d<n$. We denote by ${\rm Gr}(n,d)$ the set of all $d$-dimensional affine planes in ${{R}}^{n}$; ${\rm Gr}_{0}(n,d)$ is the Grassmann manifold of $d$-dimensional linear subspaces of ${{R}}^{n}$ . Each $d$-plane $\tau$ is parameterized as $\tau=\tau(\tau_{0},u)$, where $\tau_{0}\in{\rm Gr}_{0}(n,d)$ and $u\in\tau_{0}^{\perp}$, the orthogonal complement of $\tau_{0}$ in ${{R}}^{n}$. The Radon-John $d$-plane transform of a function $\varphi$ on ${{R}}^{n}$ is defined by

[TABLE]

provided this integral is meaningful; see, e.g., [6, 9, 12, 20].

A great deal of inversion formulas for this transform is known. We recall one of them, following [21]. Consider the Erdélyi-Kober type fractional integral

[TABLE]

The relevant fractional derivative of order $d/2$ can be defined by

[TABLE]

where $m=[d/2]$ is the integer part of $d/2$ and the powers of $t$ stand for the corresponding multiplication operators. If $d$ is even, this formula can be replaced by the simple one:

[TABLE]

Given a function $\Phi$ on ${\rm Gr}(n,d)$ and a point $x\in{{R}}^{n}$, we set

[TABLE]

which is the mean value of $\Phi$ over all $d$-planes at distance $t$ from $x$. Here $\tau_{0}$ is an arbirary $d$-plane passing through the origin.

Theorem 2.1**.**

(cf. [21, Theorem 3.5])* A function $\varphi\in L^{p}({{R}}^{n})$, $1\leq p<n/d$, can be recovered from $\Phi=R\varphi$ by the formula*

[TABLE]

where the limit is understood in the $L^{p}$-norm.

More inversion formulas in different classes of functions can be found in [9, 20].

3. Parallel Slice Transforms

We recall that given a point $a\in{{R}}^{n+1}$, $|a|>1$, and an integer $k$, $1<k\leq n$, the notation $\mathcal{Z}_{a}(n+1,k)$ stands for the set of all $k$-dimensional affine planes in ${{R}}^{n+1}$ that are parallel to the vector $a$ and meet the open ball ${{B}}^{n+1}$. Let $\mathcal{Z}_{a}^{0}(n+1,k)$ be the subset of $\mathcal{Z}_{a}(n+1,k)$ that consists of all planes containing both $a$ and the origin. Every plane $\zeta\in\mathcal{Z}_{a}(n+1,k)$ has the form

[TABLE]

where

[TABLE]

Consider the parallel slice transform

[TABLE]

assuming $f\in C({{S}}^{n})$. This operator annihilates all functions which are odd with respect to the subspace $a^{\perp}$. Specifically, let

[TABLE]

be the reflection (or inversion) mapping in the hyperplane $a^{\perp}$. We set

[TABLE]

[TABLE]

Then for $f\in C_{a}^{-}({{S}}^{n})$ we have $\Pi_{a}f=0$. In other words, $\Pi_{a}f=0$ whenever $f^{+}_{a}=0$. As we shall see below, the last equality characterizes the kernel of the operator $\Pi_{a}$. To prove this fact, we establish connection between $\Pi_{a}$ and the Radon-John transform $R$. This connection also yields a series of inversion formulas for $\Pi_{a}$.

Lemma 3.1**.**

Let $\zeta\equiv\zeta(\zeta_{0},u)\in\mathcal{Z}_{a}(n+1,k)$. Then

[TABLE]

where $\varphi$ is a function on the ball $\{y\in a^{\perp}:|y|<1\}$ defined by

[TABLE]

and $R$ is the Radon-John transform over $(k-1)$-planes in $a^{\perp}$.

Proof.

To simplify the geometry of our consideration, let

[TABLE]

and choose $\gamma\in O(n+1)$ so that

[TABLE]

We set

[TABLE]

The new variable $\xi$ ranges on the sphere ${{S}}^{k-1}_{*}$ of radius $\sqrt{1-t^{2}}$, centered at the origin, and lying in the coordinate plane ${{R}}^{k}$. Thus we have

[TABLE]

This integral can be computed in a standard way, using projection $\eta$ of $\xi$ onto the coordinate plane ${{R}}^{k-1}$. This gives

[TABLE]

Denoting the argument of $f$ by $y$ and returning to the original notation, we obtain

[TABLE]

where $\tau=\{y\in a^{\perp}\cap\zeta:\,|y|<1,{\rm P}_{u}y=u\}$, ${\rm P}_{u}y$ being the orthogonal projection of $y$ onto the direction of $u$. Now the result follows. ∎

Theorem 3.2**.**

The operator $\Pi_{a}$ is injective on the subspace $C^{+}_{a}({{S}}^{n})$. Moreover,

[TABLE]

Proof.

The embedding $C^{-}_{a}({{S}}^{n})\subset{\hbox{\rm ker}}(\Pi_{a})$ is obvious. Conversely, if $\Pi_{a}f=0$ for some $f\in C({{S}}^{n})$, then $\Pi_{a}f^{+}_{a}=\Pi_{a}(f-f^{-}_{a})=0-0=0$, and therefore, by Lemma 3.1, the corresponding function $\varphi$ in (3.7) is zero. The latter means that $f^{+}_{a}=0$. It remains to note that the injectivity of $\Pi_{a}$ on $C^{+}_{a}({{S}}^{n})$ follows from Theorem 2.1 for the Radon-John transform when the latter acts on $(k-1)$-planes in $a^{\perp}\sim{{R}}^{n}$. ∎

The inversion formula for $\Pi_{a}f$ on functions $f\in C^{+}_{a}({{S}}^{n})$ follows from the corresponding result for the Radon-John transform. We define the operator $\Pi_{a}^{-1}:\Pi_{a}(C({{S}}^{n}))\to C_{a}^{+}({{S}}^{n})$ by setting

[TABLE]

This definition does not depend on the choice of the representative $f$ in $g=\Pi_{a}f$. Indeed, if $g=\Pi_{a}f_{1}=\Pi_{a}f_{2}$, then $f_{1}-f_{2}\in{\hbox{\rm ker}}(\Pi_{a})$, and therefore

[TABLE]

Because the restriction of $\Pi_{a}$ onto $C_{a}^{+}({{S}}^{n})$ is injective, it follows that $(f_{1})_{a}^{+}=(f_{2})_{a}^{+}$.

If $g\in\Pi_{a}(C({{S}}^{n}))$, that is, $g=\Pi_{a}f$ for some $f\in C({{S}}^{n})$, then

[TABLE]

so that $\Pi_{a}^{-1}$ is the right inverse of the operator $\Pi_{a}$, when the latter acts on $C({{S}}^{n})$. However, for functions $f\in C_{a}^{+}({{S}}^{n})$ we have $\Pi_{a}^{-1}\Pi_{a}f=f$, which means that $\Pi_{a}^{-1}$ can also be considered as the left inverse of the restriction of $\Pi_{a}$ onto $C_{a}^{+}({{S}}^{n})$. In the last setting, an explicit expression for $\Pi_{a}^{-1}$ can be obtained by making use of the following theorem.

Theorem 3.3**.**

Let $1\!<k\leq n$, $a\!\in\!{{B}}^{n+1}\!\setminus\{0\}$. A function $f\in C^{+}_{a}({{S}}^{n})$ can be reconstructed from $g(\zeta)=(\Pi_{a}f)(\zeta)$ by the formula

[TABLE]

where ${\rm Q}_{a}x$ is the orthogonal projection of $x$ onto $a^{\perp}$,

[TABLE]

[TABLE]

$R^{-1}\Phi$* is the inverse Radon-John transform of $\Phi$ on $a^{\perp}$ (cf. (2.3)).*

Proof.

Let ${{S}}^{n}_{a+}=\{x\in{{S}}^{n}:x\cdot a\geq 0\}$. Because $f\in C^{+}_{a}({{S}}^{n})$, it suffices to prove (3.10) for $x\in{{S}}^{n}_{a+}$. By Lemma 3.1, the function $\Phi$ in (3.11) is represented by the Radon-John transform $R\varphi$ with

[TABLE]

The latter belongs to $L^{1}$ on the ball $\{y\in a^{\perp}:|y|<1\}$ and is continuous inside this ball. Now, because $(1-|y|^{2})^{1/2}=x\cdot\tilde{a}$, the result follows. ∎

4. Basic Automorphisms of ${{S}}^{n}$

Given a point $a\in{{R}}^{n+1}\setminus\{0\}$, we define the projection maps

[TABLE]

Let first $|a|<1$. Consider the map

[TABLE]

which is a one-to-one Möbius transformation satisfying

[TABLE]

[TABLE]

see, e.g., Rudin [25, Section 2.2.1], Stoll [28, Section 2.1]. By (4.3), $\varphi_{a}$ maps the ball ${{B}}^{n+1}$ onto itself and preserves ${{S}}^{n}$.

In the case $|a|>1$ we replace $a$ by its Kelvin transformation $a^{*}=a/|a|^{2}$, so that $|a^{*}|<1$. Then

[TABLE]

and $\varphi_{a^{*}}(a)=\infty$ in the suitable sense. The latter means that $\varphi_{a^{*}}$ maps planes through $a$ to planes parallel to the vector $a$, and vice versa.

To give this statement precise meaning, let ${\mathcal{T}}_{a}(n+1,k)$ (or $\mathcal{Z}_{a}(n+1,k)$) be the set of all $k$-planes in ${{R}}^{n+1}$ intersecting the open unit ball ${{B}}^{n+1}$ and passing through $a$ (or parallel to the vector $a$, resp.). It is convenient to use the language of Stiefel manifolds. Specifically, every plane $\tau\in{\mathcal{T}}_{a}(n+1,k)$ has the form

[TABLE]

[TABLE]

Similarly, every plane $\zeta\in\mathcal{Z}_{a}(n+1,k)$ is given by

[TABLE]

[TABLE]

Lemma 4.1**.**

Let $|a|>1$, $1\leq k\leq n$. The involutive map $\varphi_{a^{*}}$ extends as a bijection from ${\mathcal{T}}_{a}(n+1,k)$ onto $\mathcal{Z}_{a}(n+1,k)$. Specifically, if $\tau$ has the form (4.5), then $\zeta=\varphi_{a^{*}}\tau$ has the form (4.6) with

[TABLE]

[TABLE]

Conversely, if $\zeta$ has the form (4.6), then $\tau=\varphi_{a^{*}}\zeta$ has the form (4.5) with

[TABLE]

Proof.

Let $\tau\in{\mathcal{T}}_{a}(n+1,k)$ be defined by (4.5). Because $|\xi^{\prime}a|$ is the Euclidean distance from $\tau$ to the origin, the condition $|\xi^{\prime}a|<1$ is equivalent to $\tau\cap{{B}}^{n+1}\neq\emptyset$. The $\varphi_{a^{*}}$-image of $\tau$ is

[TABLE]

By (4.1),

[TABLE]

and therefore

[TABLE]

We represent the $(n+1)\times(n+1-k)$ matrix $\mathrm{Q}_{a^{*}}\xi$ in the polar form

[TABLE]

see, e.g., [14, pp. 66, 591]. Here $\rho=(\mathrm{Q}_{a^{*}}\xi)^{\prime}(\mathrm{Q}_{a^{*}}\xi)$ is a positive definite $(n+1-k)\times(n+1-k)$ matrix and

[TABLE]

Now (4.11) can be written as

[TABLE]

Because $\eta$ is orthogonal to the vector $a$, the $k$-plane (4.11) is parallel to $a$. Further, we observe that

[TABLE]

because the $\varphi_{a^{*}}$-image of any line interval in $\tau\cap{{B}}^{n+1}$ is the line interval in $\varphi_{a^{*}}\tau\cap{{B}}^{n+1}$. It follows that the $k$-plane $\varphi_{a^{*}}\tau$ has non-empty intersection with ${{B}}^{n+1}$, and therefore, $\varphi_{a^{*}}\tau\in\mathcal{Z}_{a}(n+1,k)$.

Conversely, let $\zeta\in\mathcal{Z}_{a}(n+1,k)$ be defined by (4.6). Then $\varphi_{a^{*}}\zeta=\{x\in{{R}}^{n+1}:\,\eta^{\prime}\varphi_{a^{*}}x=t\}$. Using (4.1) and keeping in mind that $\eta$ is orthogonal to the vector $a$, we obtain

[TABLE]

Hence the equality $\eta^{\prime}\varphi_{a^{*}}x=t$ is equivalent to $s_{a^{*}}\eta^{\prime}x=t(1-x\cdot a^{*})$ or

[TABLE]

Setting $\tilde{\xi}=s_{a^{*}}\eta+a^{*}t^{\prime}$, we can write (4.16) in the required form as $\tilde{\xi}^{\prime}x=\tilde{\xi}^{\prime}a$. To complete the proof, it remains to normalize $\tilde{\xi}$ by writing it in the polar form

[TABLE]

∎

Now we introduce another automorphism that plays an important role in our paper. Given two distinct points $a$ and $x$, let $L_{a,x}$ be the straight line through these points. Assuming $|a|>1$ and $x\in{{S}}^{n}$, we define the reflection map

[TABLE]

where $\tilde{x}$ is the second point at which $L_{a,x}$ meets ${{S}}^{n}$. If $L_{a,x}$ is tangent to ${{S}}^{n}$, we simply set $\tilde{x}=x$. A straightforward computation gives

[TABLE]

Lemma 4.2**.**

Let $\mathrm{R}_{a}=\mathrm{I}_{n+1}-2\mathrm{P}_{a}$ be the reflection in $a^{\perp}$, as in (3.3). Then

[TABLE]

Proof.

By Lemma 4.1, the map $\varphi_{a^{*}}$ transforms the segment $[x,\tau_{a}x]$ into the segment $[\varphi_{a^{*}}x,\varphi_{a^{*}}\tau_{a}x]$ on the line $\varphi_{a^{*}}L_{a,x}$, which is parallel to the vector $a$. Hence, by the definition of the reflection $\mathrm{R}_{a}$, we have $\mathrm{R}_{a}\varphi_{a^{*}}x=\varphi_{a^{*}}\tau_{a}x.$ Now (4.18) follows because $\varphi_{a^{*}}$ is an involution. ∎

5. Injectivity and Inversion of $F_{a}$

Our treatment of the shifted Funk transform

[TABLE]

relies on the following theorem establishing connection between $F_{a}$ and the parallel slice transform $\Pi_{a}$. For $y\in{{S}}^{n}$ we denote

[TABLE]

where $\varphi_{a^{*}}$ is the spherical automorphism, which is defined by (4.1) with $a$ replaced by $a^{*}=a/|a|^{2}$.

Theorem 5.1**.**

If $f\in C({{S}}^{n})$, then

[TABLE]

We prove this theorem in the next section and now continue our investigation, taking into account that $M_{a^{*}}$ is one-to-one and its inverse has the form

[TABLE]

Injectivity of $\Pi_{a}$ was discussed in Theorem 3.2. It follows that $F_{a}$ is injective on functions $f\in C({{S}}^{n})$ if and only if

[TABLE]

or

[TABLE]

Further, the kernel of $F_{a}$ is formed by functions $f$, for which $M_{a^{*}}f$ belongs to the kernel of $\Pi_{a}$, that is $(M_{a^{*}}f)(y)=-(M_{a^{*}}f)(\mathrm{R}_{a}y)$. Setting $y=\varphi_{a^{*}}x$,

[TABLE]

and noting that $\varphi_{a^{*}}\mathrm{R}_{a}\varphi_{a^{*}}x=\tau_{a}x$ (see Lemma 4.2), we can write (5.4) in the equivalent form

[TABLE]

The weight function $\rho_{a^{*}}(x)$ can be represented in a simpler form

[TABLE]

(we leave straightforward computation to the reader).

Lemma 5.2**.**

The operator $W_{a}$ is an involution, i.e., $W_{a}W_{a}f=f$.

Proof.

Taking into account that $\tau_{a}\tau_{a}x=x$, we have $(W_{a}W_{a}f)(x)=\rho_{a^{*}}(x)\rho_{a^{*}}(\tau_{a}x)\,f(x)$. By (5.6), $\rho_{a^{*}}(x)\rho_{a^{*}}(\tau_{a}x)=1$. This gives the result. ∎

The above reasoning shows that $W_{a}$ serves as a reflection map in our consideration and the following statements hold.

Theorem 5.3**.**

Let $1<k\leq n$.

1. (a)

The operator $F_{a}$ is injective on functions $f\in C({{S}}^{n})$ satisfying the functional equation

[TABLE] 2. (b)

The kernel of $F_{a}$ has the form

[TABLE]

Theorem 5.4**.**

Let $a\in{{R}}^{n+1}$, $|a|>1$. A function $f\in C({{S}}^{n})$ satisfying (5.9) can be uniquely reconstructed from $g=F_{a}f$ by the formula

[TABLE]

where $M_{a^{*}}^{-1}$ and $\Pi_{a}^{-1}$ are defined by (5.3) and (3.10), respectively.

6. Proof of Theorem 5.1

We proceed as in our previous work [2], however, with several changes caused by the essence of the matter. Let $\tau\in{\mathcal{T}}_{a}(n+1,k)$ have the form (4.5) and write

[TABLE]

By Lemma 4.1, it suffices to show that

[TABLE]

where $\tau_{\xi}$ and $\zeta_{\eta,t}=\varphi_{a^{*}}\tau_{\xi}$ are related by (4.7)-(4.8). We set

[TABLE]

where $\omega_{\varepsilon}$ is a radial bump function supported on the ball in ${{R}}^{n+1-k}$ of radius $\varepsilon$ with center at the origin, so that for any function $g$, which is continuous in a neighborhood of the origin, we have

[TABLE]

One can show that

[TABLE]

The proof of this equality is a verbatim copy of Step I in the similar proof in [2, Section 6], and we skip it.

Let us obtain an alternative expression for the limit (6.3), now in terms of the automorphism $\varphi_{a^{*}}$. We make use of the equality

[TABLE]

which was proved in [2, Lemma 2.1]. By this equality,

[TABLE]

where (cf. (4.10))

[TABLE]

Denote

[TABLE]

and write $\mathrm{Q}_{a^{*}}\xi$ in the polar form

[TABLE]

This gives

[TABLE]

The frame $\eta\in{\rm St}(n\!+\!1,n\!+\!1\!-\!k)$ spans an $(n+1-k)$-dimensional subspace that will be denoted by $\{\eta\}$.

Let us pass to bispherical coordinates (see, e.g., [22, p. 31])

[TABLE]

[TABLE]

and set $s=\cos\,\theta$. We obtain

[TABLE]

or (set $z=s\psi\in\{\eta\}$, $|z|<1$)

[TABLE]

We set

[TABLE]

[TABLE]

to get

[TABLE]

Denote $t=\lambda^{-1}\delta$ and change variable

[TABLE]

so that $r=0$ if and only if $z=t$. One can write (6.12) as

[TABLE]

Because the matrix $(\partial\Phi/\partial z)(0,0)=\lambda$ is invertible, there exists an inverse function $z=z_{\varphi}(r)$, which satisfies $z_{\varphi}(0)=t$ and is differentiable in a small neighborhood of $r=0$. Hence, for sufficiently small $\varepsilon>0$,

[TABLE]

Passing to the limit and noting that

[TABLE]

we obtain

[TABLE]

or

[TABLE]

Using the notation in (5.1) and scaling the measure $d\varphi$, we can write the last formula as

[TABLE]

where $h=(1-|t|^{2})\,{\hbox{\rm det}}(\rho)$, $\;t=-s_{a^{*}}\rho^{-1/2}\xi^{\prime}a$, $\;\rho=(\mathrm{Q}_{a^{*}}\xi)^{\prime}(\mathrm{Q}_{a^{*}}\xi)$.

Let us simplify the expression for $h$. We have

[TABLE]

This expression can be transformed by making use of the known fact from Algebra (see, e.g., [14, Theorem A3.5]). Specifically, if $A$ and $B$ are $p\times q$ and $q\times p$ matrices, respectively, then

[TABLE]

By this formula,

[TABLE]

and therefore

[TABLE]

where $\mathrm{P}_{a}$ stands for the orthogonal projection map onto the direction of the vector $a$. Thus,

[TABLE]

Because $\rho=(\mathrm{Q}_{a^{*}}\xi)^{\prime}(\mathrm{Q}_{a^{*}}\xi)=\xi^{\prime}\mathrm{Q}_{a}\xi$, we continue:

[TABLE]

and (6.16) gives

[TABLE]

It follows that

[TABLE]

Comparing this formula with (6.3), we obtain

[TABLE]

which coincides with (6.1).

7. The Link Between Shifted Funk Transforms over Spherical Sections of Different Dimensions

The link between integral-geometrical objects of different dimensions is a part of the theory of these objects that can be useful in many occurrences. It is intuitively clear that to integrate a function $f$ over an $\ell$-dimensional section, say $\alpha$, it suffices first to integrate this function over each section $\beta$ of lower dimension $k<\ell$ and then inegrate the result over all $\beta$ in $\alpha$. Specifically,

[TABLE]

where integration is performed with respect to the corresponding measures.

This procedure is well known for diverse Radon-like transforms, including $F_{a}$ with $a=0$. Below we establish such connection for transforms $F_{a}f$ with arbitrary center $a$ in ${{R}}^{n+1}$.

We slightly change our notation and write $F_{a}f$ as

[TABLE]

As before, it is assumed that $|\xi^{\prime}a|<1$, because otherwise, the integral is identically zero.

Some more notations are in order. In the sequel ${{R}}^{n+1}={{R}}^{k}\times{{R}}^{n-k+1}$,

[TABLE]

${{S}}^{k-1}$ is the unit sphere in ${{R}}^{k}$,

[TABLE]

Given $\xi\in{\rm St}(n\!+\!1,n\!+\!1\!-\!k)$, we denote by $r_{\xi}$ an arbitrary rotation satisfying $r_{\xi}\xi_{0}=\xi$. Then

[TABLE]

The corresponding normalized transform has the form

[TABLE]

where $d_{*}\varphi$ stands for the probability measure on ${{S}}^{k-1}$.

Fix any integer $\ell$, so that $k<\ell\leq n$. Every frame $\eta\in{\rm St}(n+1,n+1-\ell)$ extends to a “bigger” frame $\xi\in{\rm St}(n\!+\!1,n\!+\!1\!-\!k)$, so that $\xi=[\tilde{\eta},\eta]$, $\tilde{\eta}\in{\rm St}(\eta^{\perp},\ell-k)$, where ${\rm St}(\eta^{\perp},\ell-k)$ is the Stiefel manifold of all $(\ell-k)$-frames in the $\ell$-dimensional subspace $\eta^{\perp}$. Here the notation $[\tilde{\eta},\eta]$ is used for the $(n+1)\times(n+1-k)$ matrix composed by the columns of $\tilde{\eta}$ and $\eta$.

Theorem 7.1**.**

Let $f\in L^{1}({{S}}^{n})$, $a\in{{R}}^{n+1}$, $1<k<\ell\leq n$. Then

[TABLE]

Proof.

We denote the left-hand side of (7.5) by $(Af)(\eta,a)$ and the right-hand side by $(Bf)(\eta,a)$. Setting $f_{\gamma}=f\circ\gamma$, $\gamma\in O(n+1)$, we have

[TABLE]

Hence it suffices to prove (7.5) for the case when $\eta$ is the coordinate frame of the subspace

[TABLE]

that is, $\eta=\eta_{0}$, where

[TABLE]

In this case we can write $A\equiv(Af)(\eta_{0},a)$ as

[TABLE]

Here $e_{1}$ is interpreted as the first coordinate unit vector in

[TABLE]

Similarly, for $B\equiv(Bf)(\eta_{0},a)$ we have

[TABLE]

where ${\rm St}(\ell,\ell-k)$ is the Stiefel manifold of orthonormal $(\ell-k)$-frames in ${{R}}^{\ell}$,

[TABLE]

Let us transform $(\dot{F}_{k}f)([\tilde{\alpha}\zeta_{0},\eta_{0}];a)$ using (7.4) with $\xi=[\tilde{\alpha}\zeta_{0},\eta_{0}]$. In this case, $r_{\xi}$ takes $\xi_{0}$ to $[\tilde{\alpha}\zeta_{0},\eta_{0}]=\tilde{\alpha}[\zeta_{0},\eta_{0}]=\tilde{\alpha}\xi_{0}$. Hence $\xi^{\prime}a=(\tilde{\alpha}\xi_{0})^{\prime}a$, and we can write

[TABLE]

Setting

[TABLE]

and noting that $\xi_{0}=\tilde{\omega}\xi_{0}$, we obtain

[TABLE]

and therefore

[TABLE]

Now we change the order of integration and set

[TABLE]

This gives

[TABLE]

Let

[TABLE]

where $b_{1}$ and $b_{2}$ are the $(\ell-k)$-vector and the $(n+1-\ell)$-vector, respectively. Then the unit vector $v_{0}$ can be represented in the form

[TABLE]

with some $\theta\in{{S}}^{\ell-1}$ and $\beta\in O(\ell)$.

To find a simple expression for $b_{2}$, we set \tilde{\eta}_{0}=\left[\begin{array}[]{c}\tilde{0}\\ \mathrm{I}_{n+1-\ell}\end{array}\right], where $\tilde{0}$ is the zero matrix having $\ell-k$ rows and $n+1-\ell$ columns. Then, by (7.11),

[TABLE]

Thus,

[TABLE]

Substituting this expression in (7.10) and changing variable $\gamma\beta=\delta$, we obtain

[TABLE]

The latter coincides with (7.7) up to notation. ∎

Remark 7.2*.*

An interested reader might be advised to obtain an analogue of (7.1) for the parallel slice transform $\Pi_{a}$.