Non-geodesic Spherical Funk Transforms with One and Two Centers

TL;DR

This paper investigates Funk-type transforms on spheres involving planes through one or two fixed points, establishing injectivity, inversion formulas, and demonstrating unique reconstruction with two centers.

Contribution

It introduces new injectivity conditions and inversion formulas for non-geodesic Funk transforms with multiple centers, expanding understanding of spherical integral geometry.

Findings

Injectivity conditions for transforms with one and two centers

Explicit inversion formulas derived for these transforms

Unique reconstruction possible with two distinct centers

Abstract

We study non-geodesic Funk-type transforms associated with cross-sections of the n-sphere by k-dimensional planes passing through an arbitrary fixed point inside the sphere. The main results include injectivity conditions for these transforms, inversion formulas, and connection with geodesic Funk transforms. We also show that, unlike the case of planes through a single common center, the integrals over spherical sections by planes through two distinct centers provide the corresponding reconstruction problem a unique solution.

Equations

$$(F_{a}f)(\tau )=\underset{S^n\cap \tau }{\int }f(x)d\sigma (x),\tau \in {\mathrm{Gr}}_a(n+1,k),$$

$${\mathrm{P}}_{a}x=\frac{a\cdot x}{|a{|}^2}a.$$

$${\phi }_{a}x=\frac{a-{\mathrm{P}}_{a}x-s_a{\mathrm{Q}}_{a}x}{1-x\cdot a},s_a=\sqrt{1-|a{|}^2},$$

$${\phi }_a(o)=a,{\phi }_a(a)=o,{\phi }_a({\phi }_{a}x)=x,$$

$$1-|{\phi }_{a}x{|}^2=\frac{(1-|a{|}^2)(1-|x{|}^2)}{(1-x\cdot a{)}^2},x\cdot a\ne 1.$$

$$\frac{1-a\cdot {\phi }_{a}x}{1+a\cdot {\phi }_{a}x}=\frac{1-|a{|}^2}{|a-x{|}^2}.$$

$$\underset{S^n}{\int }f(x)dx=s_a^n\underset{S^n}{\int }\frac{(f\circ {\phi }_a)(y)}{(1-a\cdot y{)}^n}dy,s_a=\sqrt{1-|a{|}^2}.$$

$$x=\sqrt{1-u^2}\theta +u\tilde{a},\tilde{a}=\frac{a}{|a|},|u|\le 1,\theta \in S^n\cap a^{\perp },$$

$$\underset{S^n}{\int }f(x)dx=\underset{-1}{\overset{1}{\int }}(1-u^2{)}^{(n-2)/2}du\underset{S^n\cap a^{\perp }}{\int }f\left(\sqrt{1-u^2}\theta +u\tilde{a}\right)d\theta .$$

$${\phi }_{a}x=-\sqrt{1-v^2}\theta +v\tilde{a},v=\frac{|a|-u}{1-|a|u}.$$

$$u=\frac{|a|-v}{1-|a|v}$$

$$\frac{du}{dv}=\frac{|a{|}^2-1}{(1-|a|v{)}^2},1-u^2=\frac{(1-|a{|}^2)(1-v^2)}{(1-|a|v{)}^2},$$

$$\underset{S^n}{\int }f(x)dx$$

$${\tau }_{a}x=\frac{(|a{|}^2-1)x+2(1-x\cdot a)a}{|x-a{|}^2}.$$

$${\phi }_a{\tau }_a={\tau }_o{\phi }_a.$$

$$\underset{S^n}{\int }f({\tau }_{a}x)dx=\underset{S^n}{\int }f(x){\left(\frac{1-|a{|}^2}{|a-x{|}^2}\right)}^{n}dx,$$

$$\underset{S^n}{\int }f(x)dx=\underset{S^n}{\int }f({\tau }_{a}x){\left(\frac{1-|a{|}^2}{|a-x{|}^2}\right)}^{n}dx.$$

$$\underset{S^n}{\int }f({\tau }_{a}x)dx$$

$$(F_{a}f)(\tau )=\underset{S^n\cap \tau }{\int }f(x)d\sigma (x),\tau \in {\mathrm{Gr}}_a(n+1,k),$$

$$(M_{a}f)(y)={\left(\frac{s_a}{1-a\cdot y}\right)}^{k-1}(f\circ {\phi }_a)(y),(N_a\Phi )(\tau )=\Phi ({\phi }_a\tau ),$$

$$F_{a}f=N_a{F}_o{M}_{a}f.$$

$$(F_x^{\ast }\phi )(r)=\underset{\{\zeta \in {\mathrm{Gr}}_o(n+1,k):d(x,\zeta )=r\}}{\int }\phi (\zeta )dm(\zeta ),0<r<1,$$

$$f(x)$$

$$({\tilde{F}}_o^{-1}\phi )(x)=\underset{s\to 1}{\lim }\frac{1}{2{\pi }^{k/2}}{\left(\frac{1}{2s}\frac{\partial }{\partial s}\right)}^{k/2}[s^{k-1}(F_x^{\ast }\phi )(s)].$$

$$(M_a^{-1}f)(x)=(1-a\cdot {\phi }_{a}x{)}^{k-1}(f\circ {\phi }_a)(x),N_a^{-1}\Phi =\Phi \circ {\phi }_a.$$

$$f\equiv {\tilde{F}}_a^{-1}g=M_a^{-1}{\tilde{F}}_o^{-1}{N}_a^{-1}g,$$

$$(F_{a}f)(L_{a,x})=f(x)+f({\tau }_{a}x).$$

$$f(x)=\frac{1}{2}(F_{a}f)(L_{a,x}).$$

$${\rho }_a(x)={\left(\frac{1-|a{|}^2}{|a-x{|}^2}\right)}^{k-1},(W_{a}f)(x)={\rho }_a(x)f({\tau }_{a}x),$$

$$(W_a{W}_{a}f)(x)={\left[\frac{1-|a{|}^2}{|a-x{|}^2}\frac{1-|a{|}^2}{|a-{\tau }_{a}x{|}^2}\right]}^{k-1}f(x).$$

Full text

1. Introduction

Let ${{S}}^{n}$ be the unit sphere in ${{R}}^{n+1}$. Given a point $a$ inside ${{S}}^{n}$, we denote by ${\rm Gr}_{a}(n+1,k)$, $1\leq k\leq n$, the Grassmann manifold of $k$-dimensional affine planes in ${{R}}^{n+1}$ passing through $a$. The aim of the paper is to study injectivity of the generalized Funk transform

[TABLE]

and obtain inversion formulas for $F_{a}$ in suitable classes of functions.

The classical case $F_{a}=F_{o}$, when $a=o$ is the origin, goes back to the pioneering works by Funk [2, 3] ($n=2$), which were inspired by Minkowski [10]. A generalization of the Funk transform $F_{o}$ to arbitrary $1\leq k\leq n$ is due to Helgason [8]; see also [9, 18, 20] and references therein. Operators of this kind play an important role in convex geometry, spherical tomography, and various branches of Analysis [4, 6, 7, 20, 13, 14].

The case when $a$ differs from the origin is relatively new in modern literature, though Funk-type transforms on ${{S}}^{2}$ for noncentral plane sections were considered by Gindikin, Reeds, and Shepp [7] in the framework of the kappa-operator theory. One should also mention non-geodesic Funk-type transforms studied by Palamodov [14, Section 5.2]. Inversion formulas for these transforms were obtained in terms of delta functions and differential forms. Operators (1.1) with $a\neq o$ are non-geodesic too, however, they differ from those in [14]. In particular, they are non-injective. Non-geodesic Funk-type transforms over subspheres of fixed radius were studied by the second co-author in [17], where the results fall into the scope of number theory.

The case $a\neq o$ in (1.1) with $k=n$ was considered by Salman; see [23] for $n=2$ and [24] for any $n\geq 2$. To avoid non-uniqueness, he imposed restriction on the support of the functions that makes his operator different from ours. The stereographic projection method of [23, 24] makes it possible to reduce inversion of Salman’s operator to the similar problem for a certain Radon-like transform over spheres in ${{R}}^{n}$.

The next step was due to Quellmalz [15] for $n=2$, who expressed $F_{a}$ through the totally geodesic Funk transform $F_{o}$ and thus explicitly inverted this operator on a certain subclass of continuous functions. If $a=o$ this subclass consists of even functions on ${{S}}^{n}$. The results from [15] were generalized by Quellmalz [16] and Rubin [21] to any $n\geq 2$ with $k=n$. The paper [21] also contains an alternative inversion method for Salman’s operators.

Our aim in the present article is two-fold. First, we characterize the kernel (the null subspace) of $F_{a}$ and the subclass of all continuous functions on which $F_{a}$ is injective. We also obtain inversion formulas for $F_{a}$ on that subclass for any $1\leq k\leq n$ and thus generalize the corresponding results from [21].

Second, to achieve uniqueness in the reconstruction problem, we consider sections by planes through two distinct centers. To the best of our knowledge, this approach is new. We shall prove that for any pair of distinct points $a$ and $b$ inside the sphere, the kernels of the corresponding transforms $F_{a}$ and $F_{b}$ have trivial intersection. The latter means that, unlike the case of a single center, the collection of data from two distinct centers provides the reconstruction problem a unique solution. We also develop an analytic procedure of the reconstruction, that reduces to a certain dynamical system on ${{S}}^{n}$.

The results of this paper extend to the case when the point $a$ lies outside ${{S}}^{n}$, and to arbitrary pairs of distinct centers $a,b$ in ${{R}}^{n+1}$. We plan to address these cases elsewhere.

Plan of the Paper. Section 2 contains notation and necessary preliminaries related to Möbius-type automorphisms of the sphere. In Section 3 we describe the kernel of the operator $F_{a}$ on continuous functions and characterize the subclass of functions on which $F_{a}$ is injective. We also obtain an explicit inversion formula for $F_{a}$ on that subclass. Section 4 deals with the system of two equations, $F_{a}f=g$, $F_{b}f=h$, corresponding to distinct centers $a$ and $b$ inside the sphere. Unlike the case of a single common center, such a system determines $f$ uniquely and the function $f$ can be reconstructed by a certain pointwise convergent series. Norm convergence of this series is studied in Section 5. It turns out that the series does not converge uniformly on the entire sphere ${{S}}^{n}$ (only on some compact subsets of ${{S}}^{n}$), however, it converges in the $L^{p}({{S}}^{n})$-norm for all $1\leq p\leq p_{0}$, $p_{0}=n/(k-1)$, and this bound is sharp. In Section 6 we prove Theorem 3.1, which was formulated without proof in Section 3. This theorem plays a key role in the paper. It states that the shifted transform $F_{a}$ is represented as $F_{a}=N_{a}F_{o}M_{a}$, where $N_{a}$ and $M_{a}$ are the suitable bijections and $F_{o}$ is the classical Funk transform corresponding to $a=o$.

The main results are contained in Theorems 3.4, 4.2, 5.2, and 5.4.

2. Preliminaries

2.1. Notation

In the following, ${{B}}^{n+1}=\{x\in{{R}}^{n+1}:|x|<1\}$ is the open unit ball in ${{R}}^{n+1}$, ${{S}}^{n}$ is its boundary, $x\cdot y$ is the usual dot product. The notation $C({{S}}^{n})$ and $L^{p}({{S}}^{n})$ for the corresponding spaces of continuous and $L^{p}$ functions on ${{S}}^{n}$ is standard. If $x$ is the variable of integration over ${{S}}^{n}$, then $dx$ stands for the $O(n+1)$-invariant surface area measure on ${{S}}^{n}$, so that $\int_{{{S}}^{n}}dx=2\pi^{(n+1)/2}/\Gamma((n+1)/2)$. We write $d\sigma(x)$ for the induced surface area measure on lower dimensional spherical sections. The letter $x$ can be replaced by another one, depending on the context.

We denote by \hbox{\frak M}_{n,m} the space of real matrices having $n$ rows and $m$ columns; $\mathrm{M}^{\prime}$ is 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}\} denotes the Stiefel manifold of orthonormal $m$-frames in ${{R}}^{n}$; ${\rm Gr}_{a}(n,m)$ is the Grassmann manifold of $m$-dimensional affine planes in ${{R}}^{n}$ passing through a fixed point $a$. We will be mainly dealing with the manifolds ${\rm St}(n\!+\!1,n\!+\!1\!-\!k)$, ${\rm Gr}_{a}(n+1,k)$, and ${\rm Gr}_{o}(n+1,k)$ (i.e. $a=o$), $1\leq k\leq n$. Given a frame $\xi\in{\rm St}(n\!+\!1,n\!+\!1\!-\!k)$, the notation $\xi^{\perp}$ stands for the $k$-dimensional linear subspace orthogonal to $\xi$; $\{\xi\}$ denotes an $(n+1-k)$-dimensional linear subspace spanned by $\xi$. All points in ${{R}}^{n+1}$ are identified with the corresponding column vectors.

2.2. Spherical Automorphisms

We recall some basic facts; see, e.g., Rudin [22, Section 2.2.1)], Stoll [25, Section 2.1]. Given a point $a\in{{B}}^{n+1}\setminus\{o\}$, we denote by $\mathrm{P}_{a}$ and $\mathrm{Q}_{a}=\mathrm{I}_{n+1}-\mathrm{P}_{a}$ the orthogonal projections of ${{R}}^{n+1}$ onto the direction of $a$ and the subspace $a^{\perp}$, respectively. If $x\in{{R}}^{n+1}$, then

[TABLE]

Let

[TABLE]

which is a one-to-one Möbius transformation satisfying

[TABLE]

[TABLE]

If $x\in{{S}}^{n}$, then

[TABLE]

Properties (2.2)-(2.3) can be checked by straightforward computation. By (2.3), $\varphi_{a}$ maps the ball ${{B}}^{n+1}$ onto itself and preserves ${{S}}^{n}$.

Remark 2.1*.*

It is known that the ball ${{B}}^{n+1}$ with the relevant metric can be considered as the Poincaré model of the real $(n+1)$-dimensional hyperbolic space ${{H}}^{n+1}$. There is an intimate connection between the Möbius transformations of ${{B}}^{n+1}$ and the group $O(1,n+1)$ in the hyperboloid model of ${{H}}^{n+1}$. In the present article we do not exploit this connection. An interested reader may be referred, e.g., to Beardon [1, Section 3.7], Gehring, Martin, Palka [5, Section 3.7], Mostow [12, Theorem 1.1].

Lemma 2.2**.**

For any $f\in L^{1}({{S}}^{n})$,

[TABLE]

Proof.

We write $x$ in spherical coordinates

[TABLE]

to obtain

[TABLE]

By (2.1),

[TABLE]

Note that the map $u\to v$ is an involution. Changing variable

[TABLE]

and taking into account that

[TABLE]

we have

[TABLE]

∎

We also define the reflection $\tau_{a}:\,{{S}}^{n}\to{{S}}^{n}$ about the point $a\in{{B}}^{n+1}$:

[TABLE]

It assigns to $x\in{{S}}^{n}$ the antipodal point $\tau_{a}x\in{{S}}^{n}$ that lies on the line passing through $x$ and $a$. A similar reflection map about the origin $o$ is denoted by $\tau_{o}$, so that $\tau_{o}x=-x$.

The map $\varphi_{a}$ intertwines reflections $\tau_{a}$ and $\tau_{o}$, that is,

[TABLE]

Indeed, $\varphi_{a}$ maps chords of the ball onto chords. Hence, for any $x\in{{S}}^{n}$, the segment $[x,\tau_{a}x]$ is mapped onto the segment $[\varphi_{a}x,\varphi_{a}\tau_{a}x]$. Since the first segment contains $a$, the second one contains $\varphi_{a}(a)=o$. The latter means that the points $\varphi_{a}x$ and $\varphi_{a}\tau_{a}x$ are symmetric with respect to the origin, that is, $\varphi_{a}\tau_{a}x=\tau_{o}\varphi_{a}x$.

Lemma 2.3**.**

If $f\in L^{1}({{S}}^{n})$ and $a\in{{B}}^{n+1}$, then

[TABLE]

[TABLE]

Proof.

By (2.9) and (2.5),

[TABLE]

It remains to apply (2.4). The second equality follows from the first one: just replace $f(x)$ by $f(\tau_{a}x)$ and use $\tau_{a}\tau_{a}x=x$. ∎

3. The Shifted Funk Transform

3.1. Inversion Procedure

The following theorem establishes connection between the shifted Funk transform

[TABLE]

and the classical Funk transform $F_{o}=F_{a}|_{a=o}$ that takes functions on ${{S}}^{n}$ to functions on ${\rm Gr}_{o}(n+1,k)$. Given a function $f$ on ${{S}}^{n}$ and a function $\Phi$ on ${\rm Gr}_{o}(n+1,k)$, we denote

[TABLE]

where $s_{a}=\sqrt{1-|a|^{2}}$ and $\varphi_{a}$ is an automorphism (2.1).

Theorem 3.1**.**

Let $1\leq k\leq n$, $a\in{{B}}^{n+1}$. If $f\in C({{S}}^{n})$, then

[TABLE]

The proof of this theorem is given in Section 6.

The Funk transform $F_{o}$ is injective on the subspace $C^{+}({{S}}^{n})$ of even functions, whilst the subspace $C^{-}({{S}}^{n})$ of odd functions is the kernel of $F_{o}$ in $C({{S}}^{n})$; see, e.g., [9, 18, 19, 20]. We denote by $\tilde{F}_{o}$ the restriction of $F_{o}$ onto $C^{+}({{S}}^{n})$.

There exist several different approaches to inversion of $\tilde{F}_{o}$. We recall one of them. Given $\varphi=\tilde{F}_{o}f$, $f\in C^{+}({{S}}^{n})$, consider the mean value operator

[TABLE]

where integration is performed with respect to the relevant probability measure over the set of all planes $\zeta\in{\rm Gr}_{o}(n+1,k)$ at geodesic distance $d(x,\zeta)=\cos^{-1}r$ from $x$.

Theorem 3.2**.**

(cf. [19, Theorem 5.3])* A function $f\in C^{+}({{S}}^{n})$ can be reconstructed from $\varphi=\tilde{F}_{o}f$ by*

[TABLE]

In particular, for $k$ even,

[TABLE]

The limit in these formulas is understood in the $\sup$-norm.

Now we proceed to inversion of $F_{a}$, which, by Theorem 3.1, is factorized as $F_{a}=N_{a}F_{o}M_{a}$. Here the operators $M_{a}$ and $N_{a}$ are injective, so that

[TABLE]

The following definition is motivated by the factorization $F_{a}=N_{a}F_{o}M_{a}$ and nicely agrees with the case $a=o$.

Definition 3.3**.**

A function $f\in C({{S}}^{n})$ is called $a$-even (or $a$-odd) if $M_{a}f$ is even (or odd, resp.) in the usual sense. The subspaces of all $a$-even and $a$-odd continuous functions on ${{S}}^{n}$ will be denoted by $C_{a}^{+}({{S}}^{n})$ and $C_{a}^{-}({{S}}^{n})$, respectively. The restriction of $F_{a}$ onto $C_{a}^{+}({{S}}^{n})$ will be denoted by $\tilde{F}_{a}$.

Theorem 3.4**.**

Let $1<k\leq n$. Then ${\hbox{\rm ker}}\,(F_{a})=C_{a}^{-}({{S}}^{n})$ and the restricted operator $\tilde{F}_{a}$ is injective. A function $f\in C_{a}^{+}({{S}}^{n})$ can be uniquely reconstructed from $g=\tilde{F}_{a}f$ by

[TABLE]

where $M_{a}^{-1}$, $\tilde{F}_{o}^{-1}$, and $N_{a}^{-1}$ are defined by (3.7) and Theorem 3.2.

This statement is an immediate consequence of (3.3) and the corresponding results for $F_{o}$.

Remark 3.5*.*

In the case $k=1$, which is not included in Theorem 3.4, the plane $\tau$ is a line and the integral (1.1) is a sum of the values of $f$ at the points where this line intersects the sphere. If $x$ is one of such points and $L_{a,x}$ is the line through $a$ and $x$, then

[TABLE]

The $a$-odd functions, for which $f(x)=-f(\tau_{a}x)$, form the kernel of the operator (3.9). An $a$-even function $f$, satisfying $f(x)=f(\tau_{a}x)$, can be reconstructed from $(F_{a}f)(L_{a,x})$ by the formula

[TABLE]

3.2. Alternative description of the subspaces $C_{a}^{\pm}({{S}}^{n})$

We set

[TABLE]

where $\tau_{a}$ is the reflection (2.8).

Lemma 3.6**.**

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

Proof.

The statement is obvious for $a=o$, when $(W_{0}f)(x)=f(-x)$. It is also obvious for any $a\in{{B}}^{n}$ if $k=1$. In the general case, taking into account that $\tau_{a}\tau_{a}x=x$, we have

[TABLE]

By (2.4) and (2.9), the expression in square brackets can be written as

[TABLE]

This gives the result. ∎

Theorem 3.7**.**

A function $f\in C({{S}}^{n})$ is $a$-even (or $a$-odd) if and only if $f=W_{a}f$ (or $f=-W_{a}f$, respectively).

Proof.

By Definition 3.3, $f\in C({{S}}^{n})$ is $a$-even if and only if $(M_{a}f)(y)=(M_{a}f)(-y)$ for all $y\in{{S}}^{n}$. The latter is equivalent to

[TABLE]

or (set $y=\varphi_{a}x$ and use (2.4) and (2.9))

[TABLE]

The proof for the $a$-odd functions is similar. ∎

Corollary 3.8**.**

Every function $f\in C({{S}}^{n})$ can be represented as a sum of its $a$-even and $a$-odd parts. Specifically,

[TABLE]

Proof.

The first equality follows from the second one. Further, by Lemma 3.6,

[TABLE]

Hence, by Theorem 3.7, $f_{a}^{+}$ is $a$-even and $f_{a}^{-}$ is $a$-odd. ∎

4. Reconstruction from Two Centers

As we have seen in Section 3, a generic function $f\in C({{S}}^{n})$ cannot be reconstructed from $F_{a}f$. Because $F_{a}f=\tilde{F}_{a}f_{a}^{+}$, one can reconstruct only the $a$-even part $f_{a}^{+}$ of $f$, whilst the $a$-odd part is lost. Our aim is to show that complete reconstruction becomes possible if we consider two distinct centers instead of one. Specifically, let $a,b\in{{B}}^{n+1}$, $a\neq b$. Consider the system of two equations

[TABLE]

and suppose that a function $f\in C({{S}}^{n})$ satisfies this system. Then $\tilde{F}_{a}f_{a}^{+}=g$, $\tilde{F}_{b}f_{b}^{+}=h$, and therefore, by (3.12),

[TABLE]

where

[TABLE]

Setting

[TABLE]

we obtain a pair of functional equations

[TABLE]

Then we substitute $f$ from the second equation into the right-hand side of the first one to get

[TABLE]

Iterating (4.4), we obtain

[TABLE]

This equation generates a dynamical system on ${{S}}^{n}$.

Lemma 4.1**.**

Let $a^{*}$ and $b^{*}$ be the points on ${{S}}^{n}$ that lie on the straight line through $a$ and $b$. Suppose that $a$ is closer to $a^{*}$ than $b$. If $W=W_{a}W_{b}$, then $\lim\limits_{m\to\infty}(W^{m}f)(x)=0$ for all $x\in{{S}}^{n}\setminus\{a^{*}\}$ and $1<k\leq n$. If $k=1$ and $x\in{{S}}^{n}\setminus\{a^{*}\}$, then $\lim\limits_{m\to\infty}(W^{m}f)(x)=f(b^{*})$.

Proof.

We observe that

[TABLE]

Denote

[TABLE]

Then $(Wf)(x)=\rho(x)f(\mathrm{T}x)$ and, by iteration,

[TABLE]

For any $x\neq a^{*}$, the mapping $\mathrm{T}$ preserves the circle $C_{x,a,b}$ in the 2-plane spanned by $x$, $a$ and $b$, and leaves the points $a^{*}$ and $b^{*}$ fixed. A simple geometric consideration in the 2-plane shows that the distance from the points $\mathrm{T}^{j}x\in C_{x,a,b}$ to $b^{*}$ monotonically decreases, and therefore, the sequence $\mathrm{T}^{j}x$ has a limit. This limit must be a fixed point of the mapping $\mathrm{T}$, and hence $\mathrm{T}^{j}x\to b^{*}$ as $j\to\infty$. Because $\rho$ is continuous, it follows that

[TABLE]

Using this fact, let us show that if $k>1$, then

[TABLE]

Once (4.10) has been proved, the statement of the lemma for $k>1$ will follow because the factor $f(\mathrm{T}^{m+1}x)$ has finite limit $f(b^{*})$.

To prove (4.10), it suffices to show that

[TABLE]

where, by (4.7),

[TABLE]

Let

[TABLE]

Taking into account that $|a^{*}|=|b^{*}|=1$ and using (4.13), we obtain

[TABLE]

[TABLE]

Hence

[TABLE]

The last inequality is an immediate consequence of the assumption $0<t<s<1$.

The case $k=1$ is simpler. In this case $\rho(x)=1$, and therefore, $(W^{m}f)(x)=f(\mathrm{T}^{m+1}x)\to f(b^{*})$ as $m\to\infty$, $x\in{{S}}^{n}\setminus\{a^{*}\}.$ ∎

The above reasoning yields the following preliminary conclusion. If $a\neq b$, then, by Theorem 3.4, the kernel of the map $f\to(F_{a}f,F_{b}f)$, $f\in C({{S}}^{n})$, is $C_{a}^{-}({{S}}^{n})\cap C_{b}^{-}({{S}}^{n})$. But if $f$ is odd with respect to both $a$ and $b$, then, by Theorem 3.7, $Wf=f$. By Lemma 4.1 it follows that $f(x)=0$ for all $x\in{{S}}^{n}\setminus\{a^{*}\}$. However, since $f$ is continuous, we must have $f=0$ everywhere on ${{S}}^{n}$. In particular, it follows that $f$ can be reconstructed from the knowledge of $F_{a}f$ and $F_{b}f$, or, what is the same, from $f^{+}_{a}$ and $f^{+}_{b}$.

More precisely, we have the following result.

Theorem 4.2**.**

Let $W_{a}$ and $W_{b}$ be involutions (4.2), $1<k\leq n$. If the system of equations $F_{a}f=g$ and $F_{b}f=h$ has a solution $f\in C({{S}}^{n})$, then this solution is unique and can defined by the pointwise convergent series

[TABLE]

where $W=W_{a}W_{b}$, $q=2\,[\tilde{F}_{a}^{-1}g-W_{a}\tilde{F}_{b}^{-1}h]$, $\tilde{F}_{a}^{-1}$ and $\tilde{F}_{b}^{-1}$ being defined as in (3.8). Alternatively,

[TABLE]

where $\tilde{W}=W_{b}W_{a}$ and $r=2\,[\tilde{F}_{b}^{-1}h-W_{b}\tilde{F}_{a}^{-1}g]$.

Proof.

To prove (4.16), it suffices to pass to the limit in (4.5), taking into account that, by Lemma 4.1, the remainder $(W^{m}f)(x)$ of the series (4.16) converges to zero for every $x\neq a^{*}$. An alternative formula (4.17) then follows if we interchange $a$ and $b$, $g$ and $h$. ∎

Remark 4.3*.*

In the case $k=1$, a function $f\in C({{S}}^{n})$ can be reconstructed from the system $F_{a}f=g$, $F_{b}f=h$ as follows. By Lemma 4.1, $(W^{m}f)(x)\to f(b^{*})$ as $m\to\infty$. Hence

[TABLE]

where $q(x)=2\,[(\tilde{F}_{a}^{-1}g)(x)-(\tilde{F}_{b}^{-1}h)(\tau_{a}x)]$. By (3.10),

[TABLE]

where the line $L_{a,x}$ passes through $a$ and $x$ and $L_{b,\tau_{a}x}$ passes through $b$ and $\tau_{a}x$. It follows that

[TABLE]

Similarly, $(\tilde{W}^{m}f)(x)\to f(a^{*})$, and we have

[TABLE]

[TABLE]

The series (4.18) and (4.20) reconstruct $f$ up to unknown additive constants $f(a^{*})$ or $f(b^{*})$, where $a^{*}$ and $b^{*}$ are the endpoints of the chord through $a$ and $b$. However, complete reconstruction is still possible, if we apply symmetrization, by summing (4.18) and (4.20). This gives the following result.

Theorem 4.4**.**

Let $k=1$. Then

[TABLE]

where $q$ and $r$ are defined by (4.19) and (4.21), respectively, $L_{a,b}$ is the line through $a$ and $b$, and $F_{a}(L_{a,b})=f(a^{*})+f(b^{*})\,(=F_{b}(L_{a,b}))$ is known. The values of $f$ at the points $a^{*}$ and $b^{*}$ can be reconstructed by continuity.

5. Norm Convergence of the Reconstructing Series

Reconstruction of $f$ by the pointwise convergent series (4.16) and (4.17) gives a little possibility to control the accuracy of the result because the rate of the pointwise convergence depends on the point. Therefore, it is natural to look at the convergence in certain normed spaces. Below we explore such convergence in the spaces $C({{S}}^{n})$ and $L^{p}({{S}}^{n})$. As above, we keep the notation $a^{*}$ and $b^{*}$ for the endpoints of the chord through $a$ and $b$.

Consider the most interesting case $k>1$. By (4.5), the convergence of the series (4.16) to $f$ is equivalent to convergence of its remainder $W^{m}f$ to [math] as $m\to\infty$. Thus, it suffices to confine to $W^{m}f$.

We first note that the series (4.16) may diverge at the point $a^{*}$. Indeed, because $(W^{m}f)(a^{*})=f(\mathrm{T}^{m+1}a^{*})\prod\limits_{j=0}^{m}\rho(\mathrm{T}^{j}a^{*})$ and $a^{*}$ is a fixed point of the mapping $\mathrm{T}$, we have

[TABLE]

Suppose that $a$ and $b$ are symmetric with respect to the origin and $|a|=|b|=1/2$. Then

[TABLE]

and therefore $(W^{m}f)(a^{*})=9^{(k-1)(m+1)}f(a^{*})\to\infty$ as $m\to\infty$ whenever $f(a^{*})\neq 0$. The latter means that if $f(a^{*})\neq 0$, then the series (4.16) diverges at $a^{*}$ and its uniform convergence on the entire sphere fails. Below it will be shown that the uniform convergence of this series fails for any $a,b\in{{B}}^{n+1}$.

To understand the type of convergence, we need a deeper insight in the dynamics of involved reflections.

5.1. Dynamics of the Double Reflection Mapping $\mathrm{T}=\tau_{b}\tau_{a}$

We know that the trajectory $\{\mathrm{T}^{m}x:\,m=0,1,2,\ldots\}$ of any point $x\in{{S}}^{n}\setminus\{a^{*}\}$ converges to the point $b^{*},$ which is the endpoint of the chord containing $a$ and $b$. Let us specify the character of this convergence.

Lemma 5.1**.**

The mapping $\mathrm{T}=\tau_{b}\tau_{a}$ maps the punctured sphere ${{S}}^{n}_{a}={{S}}^{n}\setminus\{a^{*}\}$ onto itself. The point $b^{*}$ is the attracting point of the dynamical system $T^{m}:\,{{S}}^{n}_{a}\to{{S}}^{n}_{a}$ uniformly on compact subsets, that is, for any open neighborhood $U\subset{{S}}^{n}_{a}$ of $b^{*}$ and any compact set $K\subset{{S}}^{n}_{a}$ there exists $\overline{m}$ such that $T^{m}K\subset U$ for all $m\geq\overline{m}$.

Proof.

The first statement is obvious, because $\mathrm{T}a^{*}=a^{*}$ and $\mathrm{T}^{-1}a^{*}=\tau_{a}\tau_{b}a^{*}=a^{*}$. The second statement follows by a standard argument for monotone pointwise convergence on compacts. In fact, it suffices to prove this statement for the sets $U$ and $K$ having the form

[TABLE]

where $B(a^{*},\varepsilon)$ and $B(b^{*},\delta)$ are geodesic balls in ${{S}}^{n}$ of sufficiently small radii.

The pointwise convergence yields that for any fixed $x_{0}\in K_{\delta}$ there exists a number $m_{0}$ such that $\mathrm{T}^{m_{0}}x_{0}\in U_{\varepsilon}$. By continuity, the same is true for every $x$ in some neighborhood $V_{x_{0}}$ of $x_{0}$. Thus, the compact $K_{\delta}$ is covered by open sets $V_{x}$, $x\in K_{\delta}$, and therefore we can cover $K_{\delta}$ by a finite family $\{V_{x_{1}},\ldots V_{x_{M}}\}$. For each $x_{i}$, there is a number $m_{i}$ such that $\mathrm{T}^{m_{i}}x_{i}\in U_{\varepsilon}$. Setting $\overline{m}=\max\{m_{1},...,m_{M}\}$, we have

[TABLE]

A simple geometric consideration shows that the sequence $\mathrm{T}^{m+1}K_{\delta}$ monotonically decreases, i.e., $\mathrm{T}^{m+1}K_{\delta}\subset\mathrm{T}^{m}K_{\delta}$. Hence $\mathrm{T}^{m}K_{\delta}\subset U_{\varepsilon}$ for all $m\geq\overline{m}$. ∎

5.2. Uniform Convergence on Compact Subsets of the Punctured Sphere

Theorem 5.2**.**

If $f\in C({{S}}^{n})$, then the series (4.16) converges to $f$ uniformly on compact subsets of the punctured sphere ${{S}}^{n}\setminus\{a^{*}\}$.

Proof.

Consider the remainder $(W^{m}f)(x)$ of the series (4.16). By (4.8),

[TABLE]

Because $\rho(b^{*})<1$ (see (4.15)), for a fixed $\gamma$ satisfying $\rho(b^{*})<\gamma<1$, there is an open neighborhood $U\subset{{S}}^{n}\setminus\{a^{*}\}$ of the point $b^{*}$ such that $0<\rho(y)<\gamma$ for all $y\in U$. On the other hand, Lemma 5.1 says that there exists $\overline{m}$ such that $\mathrm{T}^{m}K_{\delta}\subset U$ for $m\geq\overline{m}$ and hence $0<\rho(\mathrm{T}^{m}x)<\gamma$ for all $x\in K_{\delta}$ and $m\geq\overline{m}$. Thus

[TABLE]

for all $m\geq\overline{m}$ and all $x\in K_{\delta}$. It follows that $\omega_{m}(x)\to 0$ as $m\to\infty$ uniformly on $K_{\delta}$. Since $|f(T^{m}x)|\leq\|f\|_{C(S^{n})}$, we conclude that $W^{m}f\to 0$ uniformly on $K_{\delta}$. This gives the result. ∎

5.3. $L^{p}$-Convergence

Lemma 5.3**.**

The operators $W_{a}$, $W_{b}$, $W=W_{a}W_{b}$, and $\tilde{W}=W_{b}W_{a}$ are isometries of the space $L^{p_{0}}({{S}}^{n})$ with $p_{0}=n/(k-1)$.

Proof.

The statement about $W_{a}$ follows from (2.11), which reads

[TABLE]

The equality holds for any $f\in L^{1}({{S}}^{n})$ and therefore, if $f\in L^{p_{0}}({{S}}^{n})$, then, using $|f(x)|^{p_{0}}$ instead of $f$, we obtain $\|W_{a}f\|_{p_{0}}=\|f\|_{p_{0}}$. The statement for $W_{b}$ follows analogously. The operators $W$ and $\tilde{W}$ are also isometries, as the products of two isometries. ∎

Theorem 5.4**.**

Let $f\in C(S^{n})$, $p_{0}=n/(k-1)$. The series (4.16) and (4.17) converge to $f$ in the norm of $L^{p}({{S}}^{n})$ for any $1\leq p<p_{0}$. The convergence to $f$ fails in any space $L^{p}({{S}}^{n})$ with $p_{0}\leq p\leq\infty.$

Proof.

It is clear that $f$ belongs to $L^{p}({{S}}^{n})$ for any $1\leq p\leq\infty$. Fix $\delta>0$ and consider the function $W^{m}f=(W_{a}W_{b})^{m}f$. Suppose that $p<p_{0}$ and set $r=p_{0}/p>1$. We write

[TABLE]

where, as above, $K_{\delta}={{S}}^{n}\setminus B(a^{*},\delta)$. By Hölder’s inequality,

[TABLE]

Owing to Lemma 5.3, the operator $W^{m}$ preserves the $L^{p_{0}}$-norm, and therefore

[TABLE]

Hence

[TABLE]

where $A(\delta)$ is the $n$-dimensional surface area of the geodesic ball $B(a^{*},\delta)$. For the second integral in (5.1) we have

[TABLE]

where $\sigma_{n}$ is the area of the unit sphere ${{S}}^{n}$.

Now we fix sufficiently small $\varepsilon>0$. Using (5.2), let us choose $\delta>0$ so that $I_{1}(m,\delta)<\varepsilon/2$ for all $m\geq 0$. By Theorem 5.2, the inequality (5.3) implies that there exists $\tilde{m}=\tilde{m}(\delta)$ such that $I_{2}(m,\delta)<\varepsilon/2$ for all $m\geq\tilde{m}$. Hence, by (5.1), $\|W^{m}f\|^{p}_{p}<\varepsilon$ for $m\geq\tilde{m}$, and therefore $W^{m}f$ tends to [math] as $m\to\infty$ in the $L^{p}$-norm. The latter gives the desired convergence of the series (4.16).

On the other hand, if $p>p_{0}$, then, by Hölder’s inequality, $\|f\|_{p_{0}}=\|W^{m}f\|_{p_{0}}\leq c\,\|W^{m}f\|_{p}$, $c={\hbox{\rm const}}>0$. It follows that the $L^{p}$-norm of the remainder $W^{m}f$ of the series does not tend to [math] as $m\to\infty$, unless $f=0$.

The proof for $\tilde{W}f$ is similar. ∎

Remark 5.5*.*

As we can see, the iterative method in terms of the series (4.16) and (4.17) does not provide uniformly convergent reconstruction of continuous functions. The reconstruction is guaranteed only in the $L^{p}$-norm with $1\leq p<p_{0}=n/(k-1)$. For instance, in the case of the hyperplane sections, when $k=n$ and $p_{0}=1+1/(n-1)$, the $L^{2}$-convergence fails because $p_{0}$ does not exceed $2$. The less the dimension $k$ is, the greater exponent $p$ can be chosen. The case $p=1$ works for all $1<k\leq n$.

6. Proof of Theorem 3.1

We recall that $a\in{{B}}^{n+1}$, $s_{a}=\sqrt{1-|a|^{2}}$, and $\varphi_{a}$ is an automorphism (2.1). The following lemma allows us to exploit the language of Stiefel manifolds when dealing with affine planes.

Lemma 6.1**.**

Let $1\leq k\leq n$. The map $\varphi_{a}$ extends as a bijection from ${\rm Gr}_{a}(n+1,k)$ onto ${\rm Gr}_{o}(n+1,k)$. Specifically, if $\tau\in{\rm Gr}_{a}(n+1,k)$ is defined by

[TABLE]

then $\zeta\equiv\varphi_{a}\tau\in{\rm Gr}_{o}(n+1,k)$ has the form

[TABLE]

where

[TABLE]

Conversely, if $\zeta\in{\rm Gr}_{o}(n+1,k)$ is defined by (6.2), then $\tau\equiv\varphi_{a}\zeta$ has the form (6.1) with

[TABLE]

Proof.

Let $\tau\in{\rm Gr}_{a}(n+1,k)$ be defined by (6.1). Then

[TABLE]

By (2.1),

[TABLE]

Hence $\zeta=\{y\in{{R}}^{n+1}:\,(\mathrm{A}\xi)^{\prime}y=0\}$. Now (6.2) follows if we represent the $(n+1)\times(n+1-k)$ matrix $\mathrm{A}\xi$ in the polar form

[TABLE]

see, e.g., [11, pp. 66, 591].

Conversely, let $\zeta\in{\rm Gr}_{o}(n+1,k)$ be defined by (6.2). Then

[TABLE]

By (2.1), the equality $\eta^{\prime}\varphi_{a}x=0$ is equivalent to

[TABLE]

We write $\mathrm{A}_{1}\eta$ in the form $\mathrm{A}_{1}\eta=\xi\,\beta^{1/2}$ with $\,\beta=(\mathrm{A}_{1}\eta)^{\prime}(\mathrm{A}_{1}\eta)$ and $\xi=(\mathrm{A}_{1}\eta)\,\beta^{-1/2}\in{\rm St}(n\!+\!1,n\!+\!1\!-\!k)$. Then (6.7) yields $\xi^{\prime}x=\,\beta^{-1/2}\eta^{\prime}a$. To complete the proof, it remains to note that $\,\beta^{-1/2}\eta^{\prime}a=\xi^{\prime}a$. Indeed,

[TABLE]

∎

Proof of the Theorem. The case $k=1$ is almost obvious; cf. Remark 4.2. Assuming $1<k\leq n$, let $\tau\in{\rm Gr}_{a}(n+1,k)$ have the form (6.1) and write

[TABLE]

We make use of the standard approximation machinery. Given a sufficiently small $\varepsilon>0$, let

[TABLE]

where $\omega_{\varepsilon}$ is a smooth bump function supported on the ball in ${{R}}^{n+1-k}$ of radius $\varepsilon$ with center at the origin, so that $\lim\limits_{\varepsilon\to 0}\int_{|t|<\varepsilon}\omega_{\varepsilon}(t)\,g(t)\,dx=g(o)$ for any function $g$ which is continuous in a neighborhood of the origin.

STEP I. Let us show that

[TABLE]

We pass to bispherical coordinates (see, e.g., [20, p. 31])

[TABLE]

[TABLE]

and set $s=\cos\,\theta$. This gives

[TABLE]

where

[TABLE]

if $|y|\leq 1$ and $H(y)=0$, otherwise. Passing to the limit, we obtain

[TABLE]

where

[TABLE]

If the argument of $f$ is denoted by $x$, then $x-a$ lies in the subspace perpendicular to $\xi$. Further, the integration in (6.15) is performed over the $(k-1)$-dimensional sphere of radius $\sqrt{1\!-\!|\xi^{\prime}a|^{2}}$. Switching to the surface area measure, we can write (6.15) as

[TABLE]

as desired.

STEP II. Let us obtain an alternative expression for the limit (6.9), now in terms of the automorphism $\varphi_{a}$. By Lemma 2.2,

[TABLE]

where

[TABLE]

see (6.5). Denote

[TABLE]

Then

[TABLE]

As in (6.6), the polar decomposition yields

[TABLE]

Then we pass to bispherical coordinates (cf. (6.10))

[TABLE]

[TABLE]

and set $s=\cos\,\theta$. This gives

[TABLE]

or (set $z=s\psi\in\{\eta\}\sim{{R}}^{n+1-k}$, $|z|<1$)

[TABLE]

We set

[TABLE]

[TABLE]

so that $t=o$ if and only if $z=o$, where $o$ is the origin in the corresponding space. Further, we write (6.23) as

[TABLE]

and denote $m=n+1-k$. Because the $m\times m$ matrix $(\partial\Phi_{i}/\partial z_{j})(o,o)=\Lambda$ is invertible, there exists an inverse function $z=z(t)$, which is well-defined and differentiable in a small neighborhood of $t=0$. Hence, for sufficiently small $\varepsilon>0$,

[TABLE]

where

[TABLE]

Passing to the limit, we obtain

[TABLE]

where

[TABLE]

This gives

[TABLE]

Note that

[TABLE]

and therefore ${\hbox{\rm det}}(\alpha)={\hbox{\rm det}}(I_{n+1-k}-\xi^{\prime}aa^{\prime}\xi)$. The last expression can be transformed by making use of the known fact from Algebra (see, e.g., [11, Theorem A3.5]). Specifically, if $\mathrm{U}$ and $\mathrm{V}$ are $m\times n$ and $n\times m$ matrices, respectively, then

[TABLE]

By this formula, ${\hbox{\rm det}}(\alpha)=1-(a^{\prime}\xi)(\xi^{\prime}a)=1-|\xi^{\prime}a|^{2}$. Thus, changing notation, as in (3.2), we have

[TABLE]

where $\eta=(\mathrm{A}\xi)\,\alpha^{-1/2}$; cf. (6.16).

STEP III. Comparing (6.9) with (6.28) and switching backward to the Grassmannian language (use Lemma 6.1), we obtain the statement of the theorem.