Intersection Bodies and Generalized Cosine Transforms

TL;DR

This paper explores the relationships between generalized cosine transforms and intersection bodies, introducing the class of $\\lambda$-intersection bodies and providing new characterizations and proofs related to their geometric properties.

Contribution

It establishes the interrelation of different generalized cosine transforms and applies these to characterize and analyze $\\lambda$-intersection bodies, including new proofs and examples.

Findings

Restrictions of Radon and cosine transforms preserve geometric structure.

New characterizations of $\\lambda$-intersection bodies are provided.

Examples illustrate the properties of these bodies.

Abstract

Intersection bodies represent a remarkable class of geometric objects associated with sections of star bodies and invoking Radon transforms, generalized cosine transforms, and the relevant Fourier analysis. The main focus of this article is interrelation between generalized cosine transforms of different kinds in the context of their application to investigation of a certain family of intersection bodies, which we call $\lam$-intersection bodies. The latter include $k$-intersection bodies (in the sense of A. Koldobsky) and unit balls of finite-dimensional subspaces of $L_p$-spaces. In particular, we show that restrictions onto lower dimensional subspaces of the spherical Radon transforms and the generalized cosine transforms preserve their integral-geometric structure. We apply this result to the study of sections of $\lam$-intersection bodies. New characterizations of this class of bodies are obtained and examples are given. We also review some known facts and give them new proofs.

Equations

$$(Mf)(\theta )={\int }_{S^{n-1}\cap {\theta }^{\perp }}f(u)d_{\theta }u,f(u)={\rho }_L^{n-1}(u),$$

$${\mathrm{vol}}_k(K\cap \xi )={\mathrm{vol}}_{n-k}(L\cap {\xi }^{\perp })\forall \xi \in G_{n,k}.$$

$$\int_{{\mathbb{R}}^{n}}||x||_{K}^{-k}\phi(x)\,dx=\int_{S^{n-1}}\Big{[}\int_{0}^{\infty}t^{k-1}\hat{\phi}(t\theta)\,dt\Big{]}\,d\mu(\theta),$$

$$\int_{{\mathbb{R}}^{n}}||x||_{K}^{-p}\phi(x)\,dx=\int_{S^{n-1}}\Big{[}\int_{0}^{\infty}t^{p-1}\hat{\phi}(t\theta)\,dt\Big{]}\,d\mu(\theta),$$

$$B_{q,\ell }^n=\{x=(x^{\prime },x^{\prime \prime }):|x^{\prime }{|}^q+|x^{\prime \prime }{|}^q\le 1;x^{\prime }\in {\mathbb{R}}^{n-\ell },x^{\prime \prime }\in {\mathbb{R}}^{\ell }\},q>0.$$

$$({\Pi }_{t}f)(\theta )=(1-t^2){\int }_{S^{n-1}}f(u)|\theta -tu{|}^{-n}du,0<t<1,$$

$$(R_{i}f)(\xi )={\int }_{\theta \in S^{n-1}\cap \xi }f(\theta )d_{\xi }\theta ,(R_i^{\ast }\phi )(\theta )={\int }_{\xi \ni \theta }\phi (\xi )d_{\theta }\xi ,$$

$$(R_i^{\ast }\phi )(\theta )={\int }_{SO(n-1)}\phi (r_{\theta }\gamma p_0)d\gamma ,\theta \in S^{n-1},$$

$${\int }_{G_{n,i}}(R_{i}f)(\xi )\phi (\xi )d\xi ={\int }_{S^{n-1}}f(\theta )(R_i^{\ast }\phi )(\theta )d\theta .$$

$$(R_i\mu ,\phi )={\int }_{S^{n-1}}(R_i^{\ast }\phi )(\theta )d\mu (\theta ),(R_i^{\ast }m,f)={\int }_{G_{n,i}}(R_{i}f)(\xi )dm(\xi ),$$

$$(R_i^{\alpha }f)(\xi )={\gamma }_{n,i}(\alpha ){\int }_{S^{n-1}}|{\text{Pr}}_{{\xi }^{\perp }}\theta {|}^{\alpha +i-n}f(\theta )d\theta ,$$

$$({\stackrel{\ast }{R}}_i{}^{\alpha }\phi )(\theta )={\gamma }_{n,i}(\alpha ){\int }_{G_{n,i}}|{\text{Pr}}_{{\xi }^{\perp }}\theta {|}^{\alpha +i-n}\phi (\xi )d\xi ,$$

$${\gamma }_{n,i}(\alpha )=\frac{{\sigma }_{n-1}\Gamma ((n-\alpha -i)/2)}{2{\pi }^{(n-1)/2}\Gamma (\alpha /2)},Re\alpha >0,\alpha +i-n\ne 0,2,4,\dots .$$

$$(Mf)(u)={\int }_{\{\theta :\theta \cdot u=0\}}f(\theta )d_u\theta =(R_{n-1}f)(u^{\perp }),u\in S^{n-1},$$

$$(M^{\alpha }f)(u)=(R_{n-1}^{\alpha }f)(u^{\perp })={\gamma }_n(\alpha ){\int }_{S^{n-1}}f(\theta )|\theta \cdot u{|}^{\alpha -1}d\theta ,$$

$$\gamma_{n}(\alpha)\!=\!{\sigma_{n-1}\,\Gamma\big{(}(1\!-\!\alpha)/2\big{)}\over 2\pi^{(n-1)/2}\Gamma(\alpha/2)},\qquad Re\,\alpha\!>\!0,\quad\alpha\!\neq\!1,3,5,\ldots.$$

$$({\tilde{M}}^{\alpha }f)(u)={\int }_{S^{n-1}}f(\theta )|\theta \cdot u{|}^{\alpha -1}d\theta .$$

$$\underset{\alpha \to 0}{\lim }{R}_i^{\alpha }f$$

$$\underset{\alpha \to 0}{\lim }{\stackrel{\ast }{R}}_i{}^{\alpha }\phi$$

$$\underset{\alpha \to 0}{\lim }{M}^{\alpha }f$$

$$u\in S^{n-1}\cap \xi \sim S^{i-1},v\in S^{n-1}\cap {\xi }^{\perp }\sim S^{n-i-1},0\le \psi \le \pi /2.$$

$$d\theta =c{\sin }^{i-1}\psi {\cos }^{n-i-1}\psi d\psi dudv,c={\sigma }_{i-1}{\sigma }_{n-i-1}/{\sigma }_{n-1}.$$

$$(R_i^{\alpha }f)(\xi )$$

$$c_i(\alpha )=\frac{c{\gamma }_{n,i}(\alpha )\Gamma (\alpha /2)}{2}=\frac{{\sigma }_{i-1}{\sigma }_{n-i-1}}{2}\frac{\Gamma ((n-\alpha -i)/2)}{2{\pi }^{(n-1)/2}}\to \frac{{\sigma }_{i-1}}{2{\pi }^{(i-1)/2}}$$

$$F(t)=(1-t^2{)}^{i/2-1}{\int }_{S^{n-1}\cap {\xi }^{\perp }}dv{\int }_{S^{n-1}\cap \xi }f(u\sqrt{1-t^2}+vt)du.$$

$$\underset{\alpha \to 0}{\lim }\frac{1}{\Gamma (\alpha /2)}{\int }_0^1{t}^{\alpha /2-1}F(t)dt=F(0)={\int }_{S^{n-1}\cap \xi }f(u)du=(R_{i}f)(\xi ),$$

$$m_{j,\alpha}\!=\!\left\{\begin{array}[]{cl}\!(-1)^{j/2}\,\displaystyle{\frac{\Gamma(j/2+(1-\alpha)/2)}{\Gamma(j/2+(n-1+\alpha)/2)}}&\mbox{\rm if $j$ is even},\\ 0&\mbox{\rm if $j$ is odd};\end{array}\right.$$

$$(M^{\alpha }f,\omega )=(f,M^{\alpha }\omega )=\sum _{j,k}{m}_{j,\alpha }{f}_{j,k}{\omega }_{j,k},\omega \in \mathcal{D}(S^{n-1});\alpha \ne 1,3,5,\dots .$$

$$M^{\alpha }{M}^{\beta }f=f.$$

$$(M{)}^{-1}=c_{n-1}{M}^{2-n},c_{n-1}=\frac{{\sigma }_{n-2}}{2{\pi }^{(n-2)/2}}.$$

$$a_{\alpha ,\beta }(j)=\frac{\Gamma (j/2+(1-\alpha )/2)}{\Gamma (j/2+(n-1+\alpha )/2)}\frac{\Gamma (j/2+(n-1+\beta )/2)}{\Gamma (j/2+(1-\beta )/2)},$$

Full text

The research was supported in part by the NSF grant DMS-0556157 and the Louisiana EPSCoR program, sponsored by NSF and the Board of Regents Support Fund.

Contents

1. Introduction.

2. Preliminaries.

3. Analytic families of the generalized cosine transforms.

4. Positive definite homogeneous distributions.

5. $\lambda$-intersection bodies.

6. Examples of $\lambda$-intersection bodies.

7. $(q,\ell)$-balls.

8. The generalized cosine transforms and comparison of volumes.

9. Appendix.

1. Introduction

This is an updated and extended version of our previous preprint [R5].

Intersection bodies interact with Radon transforms and encompass diverse classes of geometric objects associated to sections of star bodies. The concept of intersection body was introduced in the remarkable paper by Lutwak [Lu] and led to a breakthrough in the solution of the long-standing Busemann-Petty problem; see [G], [K4], [Lu], [Z2] for references and historical notes.

We remind some known facts that will be needed in the following. An origin-symmetric (o.s.) star body in ${\mathbb{R}}^{n}$, $n\geq 2$, is a compact set $K$ with non-empty interior such that $tK\subset K\;\forall t\in[0,1]$, $K=-K$, and the radial function $\rho_{K}(\theta)=\sup\{\lambda\geq 0:\,\lambda\theta\in K\}$ is continuous on the unit sphere $S^{n-1}$. In the following, ${\mathcal{K}}^{n}$ denotes the set of all o.s. star bodies in ${\mathbb{R}}^{n}$, $G_{n,i}$ is the Grassmann manifold of $i$-dimensional linear subspaces of ${\mathbb{R}}^{n}$, and ${\hbox{\rm vol}}_{i}(\cdot)$ denotes the $i$-dimensional volume function. The Minkowski functional of a body $K\in{\mathcal{K}}^{n}$ is defined by $||x||_{K}={\hbox{\rm min}}\{a\geq 0\,:\,x\in aK\}$, so that $||\theta||_{K}=\rho_{K}^{-1}(\theta)$, $\theta\in S^{n-1}$.

Definition 1.1**.**

[Lu]** A body $K\in{\mathcal{K}}^{n}$ is an intersection body of a body $L\in{\mathcal{K}}^{n}$ if $\rho_{K}(\theta)={\hbox{\rm vol}}_{n-1}(L\cap\theta^{\perp})$ for every $\theta\in S^{n-1}$, where $\theta^{\perp}$ is the central hyperplane orthogonal to $\theta$.

By taking into account that ${\hbox{\rm vol}}_{n-1}(L\cap\theta^{\perp})$ in Definition 1.1 is a constant multiple of the Minkowski-Funk transform

[TABLE]

Goodey, Lutwak and Weil [GLW] generalized Definition 1.1 as follows.

Definition 1.2**.**

A body $K\in{\mathcal{K}}^{n}$ is an intersection body if $\rho_{K}=M\mu$ for some even non-negative finite Borel measure $\mu$ on $S^{n-1}$.

A sequence of bodies $K_{j}\in{\mathcal{K}}^{n}$ is said to be convergent to $K\in{\mathcal{K}}^{n}$ in the radial metric if $\lim\limits_{j\to\infty}||\rho_{K_{j}}-\rho_{K}||_{C(S^{n-1})}=0$.

Proposition 1.3**.**

The class of intersection bodies is the closure of the class of intersection bodies of star bodies in the radial metric.

Proposition 1.4**.**

If $K$ is an intersection body in ${\mathbb{R}}^{n}$, $n>2$, then for every $i=2,3,\ldots,n-1$ and every $\eta\in G_{n,i}$, $K\cap\eta$ is an intersection body in $\eta$.

Regarding these two important propositions see [FGW], [GW] and a nice historical survey in [G].

Different generalizations of the concept of intersection body associated to lower dimensional sections were suggested in the literature; see, e.g., [K4], [RZ], [Z1]. The following one, which plays an important role in the study of the lower dimensional Busemann-Petty problem, is due to Zhang [Z1].

Definition 1.5**.**

We say, that a body $K\in{\mathcal{K}}^{n}$ belongs to Zhang’s class $\mathcal{Z}^{n}_{i}$ if there is a non-negative finite Borel measure $m$ on the Grassmann manifold $G_{n,i}$ such that $\rho_{K}^{n-i}=R^{*}_{i}m$, where $R^{*}_{i}$ is the dual spherical Radon transform; see (2.2), (2.5).

Another generalization was suggested by Koldobsky [K2] and described in detail in [K4]. This class of bodies will be our main concern.

Definition 1.6**.**

[K4, p. 71]** A body $K\in{\mathcal{K}}^{n}$ is a $k$-intersection body of a body $L\in{\mathcal{K}}^{n}$ $($we write $K={\mathcal{I}}{\mathcal{B}}_{k}(L))$ if

[TABLE]

We denote by ${\mathcal{I}}{\mathcal{B}}_{k,n}$ the set of all bodies $K\in{\mathcal{K}}^{n}$ satisfying (1.1) for some $L\in{\mathcal{K}}^{n}$.

When $k=1$, this definition coincides with Definition 1.1 up to a constant multiple. An analog of Definition 1.2 was given in the Fourier analytic terms as follows.

Definition 1.7**.**

[K4, Definition 4.7]** A body $K\in{\mathcal{K}}^{n}$ is a $k$-intersection body if there is a non-negative finite Borel measure $\mu$ on $S^{n-1}$, so that for every Schwartz function $\phi$,

[TABLE]

where $\hat{\phi}$ denotes the Fourier transform of $\phi$.

The set of all $k$-intersection bodies in ${\mathbb{R}}^{n}$ will be denoted by ${\mathcal{I}}^{n}_{k}$.

Keeping in mind Proposition 1.3 for $k=1$, one can alternatively define the class ${\mathcal{I}}^{n}_{k}$ as a closure of ${\mathcal{I}}{\mathcal{B}}_{k,n}$ in the radial metric; cf. [Mi1, p. 532]. However, to apply results from [K4] to such class, equivalence of this definition to Definition 1.7 must be proved. We will do this in the more general situation in Section 5.2.

From Definitions 1.6 and 1.7 it is not clear, for which bodies $L\in{\mathcal{K}}^{n}$ the relevant $k$-intersection body $K={\mathcal{I}}{\mathcal{B}}_{k}(L)$ does exist. It is also not obvious which bodies actually constitute the class ${\mathcal{I}}^{n}_{k}$. The following important characterization is due to Koldobsky.

Theorem 1.8**.**

[K4, Theorem 4.8]** A body $K\in{\mathcal{K}}^{n}$ is a $k$-intersection body if and only if $||\cdot||_{K}^{-k}$ represents a positive definite tempered distribution on ${\mathbb{R}}^{n}$, that is, the Fourier transform $(||\cdot||_{K}^{-k})^{\wedge}$ is a positive tempered distribution on ${\mathbb{R}}^{n}$.

The concept of $k$-intersection body is related to another important development. For $K\in{\mathcal{K}}^{n}$, the quasi-normed space $({\mathbb{R}}^{n},||\cdot||_{K})$ is said to be isometrically embedded in $L_{p},\;p>0$, if there is a linear operator $T:{\mathbb{R}}^{n}\to L_{p}([0,1])$ so that $||x||_{K}=||Tx||_{L_{p}([0,1])}$.

Theorem 1.9**.**

[K4, Theorem 6.10]** The space $({\mathbb{R}}^{n},||\cdot||_{K})$ embeds isometrically in $L_{p},\;p>0,\;p\neq 2,4,\ldots\,$, if and only if $\Gamma(-p/2)(||\cdot||_{K}^{p})^{\wedge}$ is a positive distribution on ${\mathbb{R}}^{n}\setminus\{0\}$.

Following Theorems 1.9 and 1.8, one can formally say that $K\in{\mathcal{I}}^{n}_{k}$ if and only if $({\mathbb{R}}^{n},||\cdot||_{K})$ embeds isometrically in $L_{-k}$. This observation, combined with Definition 1.7, was used by A. Koldobsky to define the concept of “isometric embedding in $L_{p}$” for negative $p$.

Definition 1.10**.**

[K4, Definition 6.14]** Let $0<p<n$, $K\in{\mathcal{K}}^{n}$. The space $({\mathbb{R}}^{n},||\cdot||_{K})$ is said to be isometrically embedded in $L_{-p}$ if there is a non-negative finite Borel measure $\mu$ on $S^{n-1}$, so that for every Schwartz function $\phi$,

[TABLE]

where $\hat{\phi}$ denotes the Fourier transform of $\phi$.

Origin-symmetric bodies $K$ in this definition can be regarded as “unit balls of $n$-dimensional subspaces of $L_{-p}$”. Comparing Definitions 1.10 and 1.7, one might call these bodies “$p$-intersection bodies”. Since the meaning of the space $L_{-p}$ itself is not specified in Definition 1.10 and since our paper is mostly focused on geometric properties of bodies (rather than embeddings in $L_{p}$), in the following we prefer to adopt another name “$\lambda$-intersection body”, where $\lambda$ is a real number, that will be specified in due course. We denote the set of all $\lambda$-intersection bodies in ${\mathbb{R}}^{n}$ by ${\mathcal{I}}^{n}_{\lambda}$.

Contents of the paper. We will focus on intimate connection between intersection bodies, spherical Radon transforms, and generalized cosine transforms; see definitions in Section 2.2. This approach is motivated by the fact that the volume of a central cross section of a star body is expressed through the spherical Radon transform, and the latter is a member of the analytic family of the generalized cosine transforms. These transforms were introduced by Semyanistyi [Se] and arise (up to naming and normalization) in different contexts of analysis and geometry; see, e.g., [K4], [R1]-[RZ], [Sa2], [Sa3], [Str1], [Str2].

Sections 2-4 provide analytic background for geometric considerations in Sections 5-7. In Section 2 we establish our notation and define the generalized cosine transforms on the sphere and the relevant dual transforms on Grassmann manifolds. In Section 3 we present basic properties of these transforms, establish new relations between spherical Radon transforms and the generalized cosine transforms, and prove “restriction theorems”, which are akin to trace theorems in Sobolev spaces. Section 4 deals with positive definite homogeneous distributions, that can be characterized in terms of the generalized cosine transforms. This section serves as a preparation for the forthcoming definition of the concept of $\lambda$-intersection body. We investigate which $\lambda$’s are appropriate and why. In Section 5 we switch to geometry and define the class ${\mathcal{I}}^{n}_{\lambda}$ of $\lambda$-intersection bodies. The case $0<\lambda<n$ corresponds to the “unit balls of $L_{-p}$-spaces” in the spirit of Definition 1.10. The reader will find in this section new proofs of some known facts. We introduce the notion of $\lambda$-intersection body of a star body in ${\mathbb{R}}^{n}$, which extends Definition 1.6 to all $\lambda<n,\;\lambda\neq 0$. The class of all such bodies will be denoted by ${\mathcal{I}}{\mathcal{B}}^{n}_{\lambda}$. We will prove that for all $\lambda<n$, $\lambda\neq 0,-2,-4,\ldots\,$, the class ${\mathcal{I}}^{n}_{\lambda}$ is the closure of ${\mathcal{I}}{\mathcal{B}}^{n}_{\lambda}$ in the radial metric. The case $\lambda=1$ gives Proposition 1.3. It will be proved that all $m$-dimensional central sections of $\lambda$-intersection bodies are $\lambda$-intersection bodies in the corresponding $m$-planes provided $\lambda<m,\;\lambda\neq 0$.

The natural question arises: How to construct $\lambda$-intersection bodies? In Section 6 we give a series of examples; some of them are known and some are new. They can be obtained by utilizing auxiliary statements from Section 3. In particular, the famous embedding of Zhang’s class $\mathcal{Z}^{n}_{n-k}$ into ${\mathcal{I}}^{n}_{k}$, which was first established in [K3] and studied in [Mi1], [Mi2], will be generalized to the case, when $k$ is replaced by any $\lambda\in(0,n)$. Section 7 is devoted to the so called $(q,\ell)$-balls, defined by

[TABLE]

We show that if $0<q\leq 2$, then $B^{n}_{q,\ell}\in{\mathcal{I}}^{n}_{\lambda}$ for all $\lambda\in(0,n)$. If $q>2$ and $n-3\leq\lambda<n$, we still have $B^{n}_{q,\ell}\in{\mathcal{I}}^{n}_{\lambda}$. If $q>2$ and $0<\lambda<\lambda_{0}=\max(n-\ell,\ell)-2$, then $B^{n}_{q,\ell}\not\in{\mathcal{I}}^{n}_{\lambda}$. The case, when $q>2,\;\ell>1$, and $\lambda_{0}\leq\lambda<n-3$ represents an open problem.

In Section 8 we remind the generalized Busemann-Petty problem (GBP) for $i$-dimensional central sections of o.s. convex bodies in ${\mathbb{R}}^{n}$. This challenging problem is still open for $i=2$ and $i=3$ ($n\geq 5$). It actually inspires the whole investigation. Using properties of the generalized cosine transforms, we give a short direct proof of the fact that an affirmative answer to GBP implies that every smooth o.s. convex body in ${\mathbb{R}}^{n}$ with positive curvature is an $(n-i)$-intersection body. This fact was discovered by A. Koldobsky. The original proof in [K3] is based on the embedding ${\mathcal{I}}^{n}_{n-i}\subset\mathcal{Z}^{n}_{i}$ and Zhang’s result [Z1, Theorem 6]. The latter heavily relies on the Hahn-Banach separation theorem. Our proof is more constructive and almost self-contained. We conclude the paper by Appendix, which is added for convenience of the reader.

The list of references at the end of the paper is far from being complete. Further references can be found in cited books and papers.

Acknowledgement. I am grateful to Professor Alexander Koldobsky, who shared with me his knowledge of the subject. Special thanks go to Professors Erwin Lutwak, Deane Yang, and Gaoyong Zhang for useful discussions.

2. Preliminaries

2.1. Notation.

In the following, ${\mathbb{N}}=\{1,2,\ldots\,\}$ is the set of all natural numbers, $S^{n-1}$ is the unit sphere in ${\mathbb{R}}^{n}$ with the area $\,\sigma_{n-1}=2\pi^{n/2}/\Gamma(n/2)$; $C_{e}(S^{n-1})$ is the space of even continuous functions on $S^{n-1}$; $SO(n)$ is the special orthogonal group of ${\mathbb{R}}^{n}$; for $\theta\in S^{n-1}$ and $\gamma\in SO(n)$, $d\theta$ and $d\gamma$ denote the relevant invariant probability measures; ${\mathcal{D}}(S^{n-1})$ is the space of $C^{\infty}$-functions on $S^{n-1}$ equipped with the standard topology, and ${\mathcal{D}}^{\prime}(S^{n-1})$ stands for the corresponding dual space of distributions. The subspaces of even test functions (distributions) are denoted by ${\mathcal{D}}_{e}(S^{n-1})$ ( ${\mathcal{D}}_{e}^{\prime}(S^{n-1})$); $G_{n,i}$ denotes the Grassmann manifold of $i$-dimensional subspaces $\xi$ of $\mathbb{R}^{n}$ with the $SO(n)$-invariant probability measure $d\xi$; ${\mathcal{D}}(G_{n,i})$ is the space of infinitely differentiable functions on $G_{n,i}$.

We write ${\mathcal{M}}(S^{n-1})$ and ${\mathcal{M}}(G_{n,i})$ for the spaces of finite Borel measures on $S^{n-1}$ and $G_{n,i}$; ${\mathcal{M}}_{+}(S^{n-1})$ and ${\mathcal{M}}_{+}(G_{n,i})$ are the relevant spaces of non-negative measures; ${\mathcal{M}}_{e+}(S^{n-1})$ denotes the space of even measures $\mu\in{\mathcal{M}}_{+}(S^{n-1})$. Given a function $\varphi$ on $G_{n,i}$, we denote $\varphi^{\perp}(\eta)=\varphi(\eta^{\perp}),\quad\eta\in G_{n,n-i}$. Similarly, given a measure $\mu\in{\mathcal{M}}(G_{n,n-i})$, the corresponding “orthogonal measure” $\mu^{\perp}$ in ${\mathcal{M}}(G_{n,i})$ is defined by $(\mu^{\perp},\varphi)=(\mu,\varphi^{\perp})$, $\varphi\in C(G_{n,i})$.

Let $\{Y_{j,k}\}$ be an orthonormal basis of spherical harmonics on $S^{n-1}$. Here $j=0,1,2,\dots,$ and $k=1,2,\dots,d_{n}(j)$, where $d_{n}(j)$ is the dimension of the subspace of spherical harmonics of degree $j$. Each function $\omega\in{\mathcal{D}}(S^{n-1})$ admits a decomposition $\omega=\sum_{j,k}\omega_{j,k}Y_{j,k}$ with the Fourier-Laplace coefficients $\omega_{j,k}=\int_{S^{n-1}}\omega(\theta)Y_{j,k}(\theta)d\theta$, which decay rapidly as $j\to\infty$. Each distribution $f\in{\mathcal{D}}^{\prime}(S^{n-1})$ can be defined by $(f,\omega)=\sum_{j,k}f_{j,k}\omega_{j,k}$ where $f_{j,k}=(f,Y_{j,k})$ grow not faster than $j^{m}$ for some integer $m$. We will need the Poisson integral, which is defined for $f\in L^{1}(S^{n-1})$ by

[TABLE]

and has the Fourier-Laplace decomposition $\Pi_{t}f=\sum_{j,k}t^{j}f_{j,k}Y_{j,k}$ [SW]. For $f\in{\mathcal{D}}^{\prime}(S^{n-1})$, this decomposition serves as a definition of $\Pi_{t}f$. For harmonic analysis on the unit sphere, the reader is referred to [Le], [Mü], [Ne], [SW], and a survey article [Sa3].

2.2. Basic integral transforms

For integrable functions $f$ on $S^{n-1}$ and $\varphi$ on $G_{n,i}$, $1\leq i\leq n-1$, the spherical Radon transform $(R_{i}f)(\xi),\,\xi\in G_{n,i}$, and its dual $(R_{i}^{*}\varphi)(\theta),\,\theta\in S^{n-1}$, are defined by

[TABLE]

where $d_{\xi}\theta$ and $d_{\theta}\xi$ denote the probability measures on the manifolds $S^{n-1}\cap\xi$ and $\{\xi\in G_{n,i}:\xi\ni\theta\}$, respectively. The precise meaning of the second integral is

[TABLE]

where $p_{0}$ is an arbitrarily fixed coordinate $i$-plane containing the north pole $e_{n}$ and $r_{\theta}\in SO(n)$ is a rotation satisfying $r_{\theta}e_{n}=\theta$.

Operators $R_{i}$ and $R_{i}^{*}$ extend to finite Borel measures in a canonical way, using the duality

[TABLE]

Specifically, for $\mu\in{\mathcal{M}}(S^{n-1})$ and $m\in{\mathcal{M}}(G_{n,i})$, we define $R_{i}\mu\in{\mathcal{M}}(G_{n,i})$ and $R_{i}^{*}m\in{\mathcal{M}}(S^{n-1})$ by

[TABLE]

where $\varphi\in C(G_{n,i}),\;f\in C(S^{n-1})$.

The generalized cosine transforms are defined by

[TABLE]

[TABLE]

[TABLE]

Here $\text{\rm Pr}_{\xi^{\perp}}\theta$ stands for the orthogonal projection of $\theta$ onto $\xi^{\perp}$, the orthogonal complement of $\xi\in G_{n,i}$. If $f$ and $\varphi$ are smooth enough, then integrals (2.2) can be regarded (up to a constant multiple) as members of the relevant analytic families (2.6) and (2.7); cf. Lemma 3.1. The particular case $i=n-1$ in (2.2) corresponds to the Minkowski-Funk transform

[TABLE]

which integrates a function $f$ over great circles of codimension $1$. This transform is a member of the analytic family

[TABLE]

[TABLE]

The values $\alpha=1,3,5,\ldots$ are poles of the Gamma function $\Gamma((1\!-\!\alpha)/2)$. In some occasions we include these values into consideration and set

[TABLE]

Historical notes. Regarding spherical Radon transforms (2.2) and the Minkowski-Funk transform (2.8), see [GGG], [He], [R2], [R3]. The first detailed investigation of the analytic family $\{M^{\alpha}\}$ is due to Semyanistyi [Se], who showed that these operators naturally arise in the Fourier analysis of homogeneous functions. The case $\alpha=2$ in (2.11) was known before, thanks to W. Blaschke, A.D. Alexandrov, and P. Lévy. Integrals (2.9) (sometimes with different normalization) arise in diverse areas of analysis and geometry; see [K4], [R1] - [R3], [Sa3], [Str1], and references therein. In convex geometry and Banach space theory, operators (2.11) with $\alpha-1$ replaced by $p$ are known as the $p$-cosine transforms. More general analytic families (2.6) and (2.7) were introduced in [R2].

3. Analytic Families of the Generalized Cosine Transforms

3.1. Basic properties

Below we review basic properties of integrals (2.6), (2.7), (2.9); see [R2], [R3] for more details. For integrable functions $f$ and $\varphi$ and $Re\,\alpha>0$, integrals (2.6), (2.7) and (2.9) are absolutely convergent. When $f$ and $\varphi$ are infinitely differentiable, these integrals extend meromorphically to all $\alpha\in{\mathbb{C}}$.

Lemma 3.1**.**

If $f$ and $\varphi$ are continuous functions, then

[TABLE]

Hence, the Radon transform, its dual, and the Minkowski-Funk transform can be regarded (up to a constant multiple) as members of the corresponding analytic families $\{R_{i}^{\alpha}\}$, $\{\overset{*}{R}_{i}{}^{\alpha}\}$, $\{M^{\alpha}\}$.

Proof.

Formulas (3.2) and (3.3) follow from (3.1). To prove (3.1), we write (2.6) in bi-spherical coordinates $\theta=u\sin\,\psi+v{\hbox{\rm cos}}\,\psi$, where

[TABLE]

[TABLE]

This gives

[TABLE]

where

[TABLE]

as $\alpha\to 0$, and

[TABLE]

Since

[TABLE]

we are done. ∎

Analytic continuation of integrals (2.9) can be realized in spherical harmonics as $M^{\alpha}f\!=\!\sum\limits_{j,k}m_{j,\alpha}f_{j,k}Y_{j,k}$, where

[TABLE]

see [R1], [R3]. If $f\!\in\!{\mathcal{D}}^{\prime}(S^{n-1})$, then $M^{\alpha}f$ is a distribution defined by

[TABLE]

Lemma 3.2**.**

Let $\alpha,\beta\in{\mathbb{C}};\;\alpha,\beta\neq 1,3,5,\ldots\,$. If $\alpha+\beta=2-n$ and $f\in{\mathcal{D}}_{e}(S^{n-1})$ $($or $f\in{\mathcal{D}}^{\prime}_{e}(S^{n-1}))$, then

[TABLE]

If $\,\alpha,2-n-\alpha\neq 1,3,5,\ldots$, then $M^{\alpha}$ is an automorphism of the spaces ${\mathcal{D}}_{e}(S^{n-1})$ and ${\mathcal{D}}^{\prime}_{e}(S^{n-1})$.

Proof.

The equality (3.5) is equivalent to $m_{j,\alpha}m_{j,\beta}=1$, $\;\alpha+\beta=2-n$. The latter follows from (3.4). The second statement is a consequence of the standard theory of spherical harmonics [Ne], because the Fourier-Laplace multiplier $m_{j,\alpha}$ has a power behavior as $j\to\infty$. ∎

Corollary 3.3**.**

The Minkowski-Funk transform on the spaces ${\mathcal{D}}_{e}(S^{n-1})$ and ${\mathcal{D}}^{\prime}_{e}(S^{n-1})$ can be inverted by the formula

[TABLE]

Note that there is a wide variety of diverse inversion formulas for the Minkowski-Funk transform (see [GGG], [He], [R3] and references therein), but all of them are, in fact, different realizations of (3.6), depending on classes of functions.

3.2. Auxiliary statements

We establish some connections between operator families defined above.

Lemma 3.4**.**

Let $\alpha,\beta\in{\mathbb{C}};\;\alpha,\beta\neq 1,3,5,\ldots\,$. If $Re\,\alpha>Re\,\beta$, then $M^{\alpha}=M^{\beta}A_{\alpha,\beta}$, where $A_{\alpha,\beta}$ is a spherical convolution operator with the Fourier-Laplace multiplier

[TABLE]

so that $a_{\alpha,\beta}(j)\sim(j/2)^{\beta-\alpha}$ as $j\to\infty$. If $\alpha$ and $\beta$ are real numbers satisfying $\alpha>\beta>1-n,\;\alpha+\beta<2$, then $A_{\alpha,\beta}$ is an integral operator such that $A_{\alpha,\beta}f\geq 0$ for every non-negative $f\in L^{1}(S^{n-1})$.

Proof.

The first statement follows from (3.4). To prove the second one, we consider integral operators

[TABLE]

expressed through the Poisson integral (2.1). The Fourier-Laplace multipliers of $Q_{+}^{\mu,\lambda}$ and $Q_{-}^{\mu,\nu}$ are

[TABLE]

They can be easily computed by taking into account that $\Pi_{t}\sim t^{j}$ in the Fourier-Laplace terms. If $f\in L^{1}(S^{n-1})$ and $0<\mu<\nu<n$, then integrals (3.8) and (3.9) are absolutely convergent and obey $Q_{\pm}^{\mu,\nu}f\geq 0$ when $f\geq 0$. Comparing (3.10) and (3.7), we obtain a factorization $A_{\alpha,\beta}=Q_{+}^{\alpha-\beta,1-\beta}Q_{-}^{\alpha-\beta,1-\beta}$ (set $\mu=\alpha-\beta,\;\nu=1-\beta$), which implies the second statement of the lemma. ∎

It is convenient to introduce a special notation for the spherical Radon transform and the generalized cosine transform with orthogonal argument. Assuming $\xi\in G_{n,i}$, we denote

[TABLE]

Lemma 3.5**.**

Let $f\in L^{1}(S^{n-1}),\quad Re\,\alpha>0;\quad\alpha\neq 1,3,5,\ldots\,$. Then

[TABLE]

or (replace $i$ by $n-i$)

[TABLE]

If $f\in{\mathcal{D}}_{e}(S^{n-1})$, then (3.12) and (3.13) extend to $Re\,\alpha\leq 0$ by analytic continuation.

Proof.

For $Re\,\alpha>0$,

[TABLE]

Since $|\theta\cdot u|=|\text{\rm Pr}_{\xi}\theta||v_{\theta}\cdot u|$ for some $v_{\theta}\in S^{n-1}\cap\xi$, by changing the order of integration, we obtain

[TABLE]

The inner integral is independent on $v_{\theta}$ and can be easily evaluated:

[TABLE]

This implies (3.12). ∎

The following statement is dual to Lemma 3.5.

Lemma 3.6**.**

Let $\mu\in{\mathcal{M}}(G_{n,i}),\;\alpha\neq 1,3,5,\ldots\,$. Then

[TABLE]

in the ${\mathcal{D}}^{\prime}(S^{n-1})$-sense. If $Re\,\alpha>0$ and $\mu$ is absolutely continuous with density $\varphi\in L^{1}(G_{n,i})$, then

[TABLE]

almost everywhere on $S^{n-1}$. If $\varphi\in{\mathcal{D}}(G_{n,i})$, then (3.15) extends to all complex $\alpha\neq 1,3,5,\ldots$ by analytic continuation.

Proof.

Let $\omega\in{\mathcal{D}}_{e}(S^{n-1})$ (it suffices to consider only even test functions). By (2.4) and (3.12),

[TABLE]

This gives the result. ∎

The next statement contains explicit representations of the right inverse of the dual Radon transform $R_{i}^{*}$ (note that $R_{i}^{*}$ is non-injective on ${\mathcal{D}}(G_{n,i})$ when $1<i<n-1$).

Lemma 3.7**.**

Every function $f\!\in\!{\mathcal{D}}_{e}(S^{n-1})$ is represented as $f\!=\!R_{i}^{*}Af$, where $A:{\mathcal{D}}_{e}(S^{n-1})\to{\mathcal{D}}(G_{n,i})$,

[TABLE]

[TABLE]

Proof.

The coincidence of expressions in (3.16) follows from (3.13). To prove the first equality, we invoke spherical convolutions defined by analytic continuation of the integral

[TABLE]

$Re\,\alpha>0,\quad\alpha-n\neq 0,2,4,\ldots\,$, so that $Q^{0}f=f$ [R2]. By Theorem 1.1 from [R2], $R^{*}_{i}R_{i}^{\alpha}f=c_{1}^{-1}Q^{\alpha+i-1}f$, and therefore (set $\alpha=1-i$), $R^{*}_{i}R_{i}^{1-i}f=c_{1}^{-1}f$, as desired. ∎

The next statement provides an intriguing factorization of the Minkowski-Funk transform in terms of Radon transforms associated to mutually orthogonal subspaces. This factorization can be useful in different occurrences.

Theorem 3.8**.**

For $f\in L^{1}(S^{n-1})$ and $0<i<n$,

[TABLE]

Proof.

By (2.3),

[TABLE]

The inner integral is independent on $w\in S^{n-1}\cap{\mathbb{R}}^{n-i}$ and equals $(Mf)(\theta)$. This gives (3.18). ∎

3.3. Restriction theorems

Theorems of such type deal with traces of functions on lower dimensional subspaces and are well known, for instance, in the theory of function spaces. To the best of our knowledge, traces of functions represented by Radon transforms or, more generally, by the generalized cosine transforms , were not studied systematically and deserve particular attention, because they provide analytic background to a series of results related to sections of star bodies; cf. [R3, Sec. 3.5], [FGW]. Given a subspace $\eta\in G_{n,m}$