% \input c:/tibor/defs/def.tex \parindent=12pt \font\bigrm=cmbx10 scaled\magstep3 \font\normalrm=cmbx10 scaled\magstep2 \topmatter \title ON AN INTEGRO-DIFFERENTIAL TRANSFORM ON THE SPHERE\endtitle \author E. Makai, Jr.$^*$, H\. Martini, T\. {\'O}dor$^{**}$ \endauthor \thanks * Partially supported by Hungarian National Foundation for Scientific Research, Grant No\. T-031931 and FKFP Grant No. 0391/1997 \newline ** Partially supported by Hungarian National Foundation for Scientific Research, Grant number 4427 \endthanks \rightheadtext\nofrills{\sevenrm On an integro-differential transform on the sphere} \leftheadtext\nofrills{\sevenrm E. Makai, Jr.$^*$, H\. Martini, T\. {\'O}dor$^{**}$} \address $^*$, $^{**}$: Alfr\'ed R\'enyi Mathematical Institute of the Hungarian Academy of Sciences, P.O.B.\ 127, H-1364 Budapest, Hungary; \newline Technische Universit\"at Chemnitz, Fakult\"at f\"ur Mathematik, D-09107 Chemnitz, Germany \endaddress \email makai\@renyi.hu, martini\@mathematik.tu-chemnitz.de, odor\@renyi.hu \endemail %\date %\today %\enddate \subjclass 1991 {\it Mathematics Subject Classification.} Primary: 44A12; Secondary: 33C55 \endsubjclass \keywords integro--differential operator, spherical harmonics, Funk--Hecke theorem \endkeywords \magnification=\magstep1 \documentstyle{amsppt} \NoBlackBoxes %\pagewidth{6.6truein} %\pageheight{9.2truein} %\vcorrection{-0.2truecm} %\hcorrection{0.4truecm} \define\defeq {\overset{\operatorname{def}} \to =} \define\submitted {\text{$\(\text{submitted}\)$ }} \define\preprint {\text{$\(\text{preprint}\)$ }} \define\manuscript {\text{$\(\text{manuscript}\)$ }} \define\inpreparation {\text{$\(\text{in preparation}\)$ }} \redefine\toappear {\text{$\(\text{to appear}\)$ }} \def \today {\ifcase \month \or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi\ \number \day, \number \year} \redefine \sgn{\operatorname{sgn\,}} \define \pr{\operatorname{pr\,}} \redefine \conv{\operatorname{conv}} \define \hull{\operatorname{hull\,}} \define \const{\operatorname{const\,}} \redefine \dim{\operatorname{dim\,}} \define \constant{\operatorname{constant\,}} \redefine \Im{\operatorname{Im\,}} \define \Ker{\operatorname{Ker}} \define \cl{\operatorname{cl\,}} \define \card{\operatorname{card\,}} \define \aff{\operatorname{aff\,}} \define \Lin{\operatorname{Lin\,}} \define \lin{\operatorname{lin\,}} \define \relint{\operatorname{relint\,}} \define \Int{\operatorname{int\,}} \define \Rng{\operatorname{Rng\,}} \define \interior{\operatorname{int\,}} \define \supp {\operatorname{supp\,}} \redefine \Cup{\bigcup} \redefine \<{\left\langle} \redefine \>{\right\rangle} \redefine \ang{\sphericalangle} \define \({\left(} \define \){\right)} \define \[{\left[} \define \]{\right]} \redefine \Cap{\bigcap} \define \all{\forall} %\redefine \Bbb{{}} \define \Vol{\operatorname{Vol}} \define \id{\operatorname{id\,}} \define\Laplace{\varDelta} \def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt \hbox {\vrule width.#2pt height#1pt \kern#1pt \vrule width.#2pt} \hrule height.#2pt}}}} \def\square{\mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3} \redefine \qed{$\square$} \define \qued{\text{\,\,\qed\,\,\,}} %\baselineskip=16.5pt %\tolerance=5000 %\parskip=10pt plus 2pt %\pagewidth{15truecm} %\pageheight{23truecm} %\hoffset=+1truecm %\NoBlackBoxes \redefine\d{\operatorname{d}} \define\Lip{\operatorname{Lip}} \define\ndivide{{\not |}} \def\br{\Bbb R} \baselineskip=16.5pt \abstract In a recent paper the authors have proved that a convex body $K \subset \Bbb R ^d$, containing the origin $0$ in its interior, is symmetric with respect to $0$ if and only if $V_{d-1}(K \cap H') \ge V_{d-1}(K \cap H)$ for all hyperplanes $H,H'$ such that $H$ and $H'$ are parallel and $H' \ni 0$ ($V_{d-1}$ is $(d-1)$--measure). For the proof the authors employed a new type of integro--differential transform, that let to correspond to a sufficently nice function $f$ on $S^{d-1}$ the function $R^{(1)}f$, where $(R^{(1)}f)(\xi )= \int\limits _{S^{d-1} \cap \xi ^\bot } ({\partial f}/ {\partial \psi }) d \eta $ --- where $\xi \in S^{d-1}$ and $\psi $ is geographic latitude --- and determined the null--space of the operator $R^ {(1)}$. In this paper we extend the definition to any integer $m \ge 1$, defining $(R^{(m)}f)(\xi )$ analogously as for $m=1$, but using $ {\partial ^m f}/{\partial \psi ^m}$ rather than $ {\partial f}/{\partial \psi }$. We investigate the null--space of the operator $R^{(m)}$, determining it nearly. \endabstract \endtopmatter \document \vskip .4cm %%%%%%%%%%%%%%INTRODUCTION%%%%%%%%%%%%%%%%%%%%%% \heading 1.\ Introduction \endheading \vskip .4cm In [Makai--Martini--\'Odor] we have proved that a convex body $K \subset \Bbb R ^d$, containing the origin $0$ in its interior, has the property that for every hyperplane $H$ the hyperplane $H'$ parallel to $H$ and passing through $0$ satisfies $V_{d-1} (K \cap H') \ge V_{d-1} (K \cap H) $ ($V_{d-1}$ is $(d-1)$--volume), if and only if this convex body $K$ is symmetric with respect to $0$. The case $d=2$ was proved by [Hammer, 1954]. (In fact he proved the analogous statement for $1$--dimensional sections for $K \subset \Bbb R ^d$, and we have proved the analogous statement for $k$--dimensional sections, $1 \le k \le d-1$.) We have proved our result by using a new Radon--type transformation, which can be considered as a common generalization of partial differential operators and Radon--type transformations. Moreover we have proved our theorem by using spherical harmonics and the Funk--Hecke theorem. Our integro--differential transformation let to correspond to a sufficiently nice function $f\colon S^{d-1}\to \Bbb R$ the function $R^{(1)}f\colon S^{d-1}\to \Bbb R$, where $(R^{(1)}f)(\omega)$ is the integral over the large $S^{d-2}$ of $S^{d-1}$, with pole $\omega$, of the derivative of $f$ in the direction $\omega$. To prove our theorem we have established that the null--space of this operator $R$ consists of the even functions in the domain of $R$. We investigate generalizations of this transformation, where we integrate over the same large $S^{d-2}$ the $m$'th derivative of $f$ with respect to the geographic latitude ($\omega$ considered as the north pole). The case $m=0$ is the spherical Radon transformation (Funk transformation), the case $m=1$ is the above transformation $R^{(1)}$. We obtain that for $m \ge 1$ the null--space of this operator is the direct sum of the even functions (for $m$ odd) or odd functions (for $m$ even) in the domain of this operator, respectively, and a subspace of finite dimension. For general analytical background we refer to the books [Tricomi, 1957], [Adams, 1975] and [Ziemer, 1989]. \vskip2truecm %%%%%%%%PARAGRAPH 2%%%%%%%%%%%%%% \heading 2. Preliminaries \endheading \vskip .4cm As usual, $\Bbb R^d$ denotes the $d$--dimensional Euclidean space which is endowed with the standard inner product and norm $|\,\cdot \,|$ structure. We will suppose $d \ge 2$. The origin is denoted by $0$ and $V_{d-1}$ is the $(d-1)$--dimensional volume on the hyperplanes. Let $S^{d-1}$ denote the unit sphere with centre $0$; its variable point will be denoted by $\omega, \,\, \xi$ or $\eta$. For $\omega \in S^{d-1}$ and $t \in \Bbb R$ let $H(\omega, t)$ be the hyperplane given by the equation $ \langle x, \omega \rangle = t$. We write $\omega ^\bot$ for $H(\omega, 0)$. Often we will use a polar coordinate system on $S^{d-1}$, with pole some $\xi \in S^{d-1}$. That is, any $\omega \in S^{d-1}$ can be written as $\xi \sin \psi + \eta \cos \psi$, where $\eta \in S^{d-1} \cap \omega ^\bot $ and $- \pi /2 \le \psi \le \pi /2 $ (thus $\psi $ is the {\it geographic latitude}, which will be more convenient for us than the costumarily used $ \phi = \pi /2 - \psi $); then we write $\omega = (\eta, \psi)$. A real function defined on $S^{d-1}$ is called {\it {even (odd)}}, if for all $ \omega \in S^{d-1}$ we have $ f(- \omega )=f(\omega)\,\,\,\,(f(-\omega) =-f(\omega))$. We turn to spherical harmonics, which are higher--dimensional generalizations of the trigonometric functions $\cos (nx)$, $\sin(nx)$ (these are obtained for $d=2$). Standard references are [M{\"u}ller, 1966], [Seeley, 1966], [Erd\'elyi et al., 1953] and, for $d=3$ in more detail, [Sansone, 1959]; further references, with some geometrical applications, are e.g\. [Blaschke, 1949], [Gardner, 1995], Appendix C, and also the survey paper [Groemer, 1993] and the books [Schneider, 1993], pp\. 428--432, as well as [Groemer, 1996], which contain ample further bibliography. A polynomial $f\colon\Bbb R ^d\to\Bbb R$ is {\it harmonic\/}, if $\sum\limits_{i=1}^d\left({\partial}/{\partial x_i}\right)^2f=0$. (This is invariant under the choice of an orthonormal base.) For an integer $n\ge 0$ a {\it spherical harmonic (of degree $n$)\/} in $d$ dimensions is the restriction of a homogeneous harmonic polynomial $f\colon\Bbb R ^d \to\Bbb R$ (of degree $n$) to $S^{d-1}$. (Since $d$ will be fixed, later we will not refer to the dimension.) The spherical harmonics of degree $n$ form a finite dimensional vector space. Choosing from this subspace an orthonormal base $\{Y_{ni}\,\,\vert\,\,1\le i\le N(d,n)\}$ (orthonormality meant in the space $L^2(S^{d-1})$, for Lebesgue measure on $S^{d-1}$), their union for each $n\ge 0$ is a complete orthonormal system in $L^2(S^{d-1})$. Thus each $f\in L^2(S^{d-1})$ has a Fourier expansion $\sum\limits_{n=0}^\infty\left( \sum\limits_{i=1}^{N(d,n)} c_{ni}Y_{ni}\right)$. We will write $\sum\limits_{i=1}^{N(d,n)}c_{ni}Y_{ni}=Y_n(f)$, thus the Fourier expansion of $f$ is $\sum\limits_{n=0}^\infty Y_n(f)$. The spherical harmonics are the eigenfunctions of many linear operators commuting with rotations. For example, the Funk--Hecke theorem ([Seeley, 1966], Theorem 3) says that if $F$ is measurable on $[-1,1]$, with $\int\limits_{-1}^1\vert F(t)\vert(1-t^2)^{(d-3)/2}dt<\infty$, then any spherical harmonic $Y_n$ of $n$--th degree is an eigenfunction of the integral operator $f\mapsto g=g(\xi)=\int\limits_{S^{d-1}}F(\langle\xi,\eta\rangle)f(\eta)d\eta$, that is $$\int\limits_{S^{d-1}} F(\langle\xi,\eta\rangle)Y_n(\eta)d\eta=\lambda_nY_n(\xi)\,,$$ where the eigenvalue $\lambda_n$ equals $$ \lambda_n=V_{d-2}(S^{d-2})C_n(1)^{-1}\int\limits_{-1}^1F(t)C_n(t) (1-t^2)^{(d-3)/2}dt\,. $$ Here $V_{d-2}$ means $(d-2)$--dimensional volume, and $C_n(t)= C_n^{(d-2)/2}(t)$ is the $n$'th Gegenbauer polynomial, of order $(d-2)/2$, that is a non--zero polynomial of degree $n$, satisfying for $0\le n