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\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





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\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 $0\le n<m$ the orthogonality relations

$$

\int\limits_{-1}^1C_n(t)C_m(t)(1-t^2)^{(d-3)/2}dt=0\,.

$$

There holds $C_n(1)\ne 0$, [Seeley, 1966], (3). For $n$ odd (even) $C_n$ 

is an odd

(even)

function [Erd\'elyi et al., 1953], \S 10.9, (16). References to Gegenbauer 

polynomials are

[Erd\'elyi et al., 1953] and [Tricomi, 1955].



For suitable measures or distributions on $[-1,1]$ a formula similar to the

Funk--Hecke theorem holds, cf.\ for example Lemma 3.2.

\vskip2truecm



%%%%%%%%%%PARAGRAPH 3%%%%%%%%%%%



\heading 3. The operators $R^{(m)}_{\psi }$ and $R^P_{\psi }$\endheading

\vskip0.4truecm



Now we generalize the definition of the operator $R^{(1)}_{\psi }$ from

[Makai--Martini--\'Odor].

Let $d\ge 2$, and let $m\ge 1$ be an integer. We let

$Lip^{m-1}(S^{d-1}):=\{f\colon S^{d-1}\to\Bbb R\,\,\vert\,\, f$

has in some $C^\infty$--atlas of $S^{d-1}$ all partial derivatives of order

at most $m-1$, and all these are Lipschitz functions$\}$. We write

$Lip^0(S^{d-1})=Lip(S^{d-1})$. Let $-\pi/2\le\psi\le\pi/2$,

$\,f\in Lip^{m-1}(S^{d-1})$ and $\xi\in S^{d-1}$. Using polar coordinates

with

pole $\xi$, we define the integro--differential transform

$R_\psi^{(m)}f$

of $f$ by

$$(R_\psi^{(m)}f)(\xi)=

\int\limits_{S^{d-1}\cap\xi^\bot}

\frac{\partial^mf}{\partial\psi^m}(\eta,\psi)d\eta\,,$$

provided the

right--hand side exists. For 

$\psi=0$ we

drop the lower index.

The case $m=1$ of the following lemma is Lemma 3.6 of [Makai--Martini--\'Odor].



\proclaim{Lemma 3.1} Let $d\ge 2,\,\,m\ge 1$, $-\pi/2\le\psi\le\pi/2$

and

$f,g\in Lip^{m-1}(S^{d-1})$. Then, for almost all $\xi\in S^{d-1}$ we have

for almost all $\eta\in

S^{d-1}\cap\xi^\bot$ that $

({\partial^mf}/{\partial\psi^m})(\eta,\psi)$ exists, and,  

for almost all $\xi\in S^{d-1}$, the

integral defining $(R_\psi^{(m)}f)(\xi)$ exists. We have

$R_\psi^{(m)}f\in L^\infty(S^{d-1})$, and $R_\psi^{(m)}$ is

symmetric, i.e., $\int\limits_{S^{d-1}}

(R_\psi^{(m)}f)(\xi)g(\xi)d\xi=

\int\limits_{S^{d-1}}f(\xi)(R_\psi^{(m)}g)(\xi)d\xi$.

\endproclaim



\demo{Proof} The claimed existence of $\partial^mf/\partial\psi^m$ for

$\psi=\pm\pi/2$ follows from [Whitney, 1957], Ch.IX, Theorem 11A, by which

Lipschitz functions $S^{d-1} \to \Bbb R$ are almost everywhere 

differentiable, and 

otherwise

follows like

in Lemma 3.5 of [Makai--Martini--\'Odor], using the manifold 

$M_{\psi }=\{(\xi,\omega)\,\,\vert\,\,\xi,\omega\in S^{d-1}$,

$\langle\xi,\omega\rangle=\sin\psi\}$. 

Namely, $M_{\psi }$ is an $S^{d-2}$--bundle over $S^{d-1}$, via either 

projection $\pi _1:(\xi ,\omega ) \mapsto \xi, \,\,\pi _2 :(\xi ,\omega ) 

\mapsto \omega $. Applying locally the Fubini theorem to both $\pi _2 $ and 

$\pi _1$, we gain in turn, letting $\omega =(\eta , \psi )$, that $V_{2d-3}

(\{(\xi ,\omega )\in M_{\psi } \mid f(\omega )$ is not $m$ times 

differentiable at $\omega =(\eta , \psi )\})=0$, and $V_{d-1}(\{\xi \in 

S^{d-1} \mid \{ \omega \in S^{d-1} \mid \langle \xi ,\omega \rangle = \sin 

\psi ,\,\,f (\omega )$ is not $m$ times differentiable at $\omega =(\eta ,

\psi )\}$ does not have $(d-2)$--Lebesgue measure $0\})=0$ ($V_i$ denoting 

invariant Lebesgue type measure in the respective spaces). In other words: 

for almost all $\xi \in S^{d-1}$ we have for almost all $\omega \in 

\{\omega ' \in S^{d-1} \mid \langle \xi , \omega ' \rangle =\sin \psi \}$ 

that $f(\omega )$ is $m$ times differentiable at $\omega =(\eta ,\psi )$.

For almost all $\xi\in S^{d-1}$

we

have that the function $\eta\mapsto ({\partial^mf}/{\partial\psi^m})       

(\eta,\psi)$ is the almost everywhere limit of the continuous

functions

$$

\eta\mapsto\left(\frac{\partial^{m-1}f}{\partial\psi^{m-1}}

(\eta,\psi+\varepsilon)-

\frac{\partial^{m-1}f}{\partial\psi^{m-1}}

(\eta,\psi)\right)\biggm/\varepsilon

$$

with absolute value below a constant depending of $f$. Therefore the

function

$\eta\mapsto ({\partial^mf}/$ 

\newline ${\partial\psi^m})(\eta,\psi)$ is

integrable and $R_\psi^{(m)}f$ is bounded, for almost all $\xi$. Also

$R_\psi^{(m)}f$ is measurable, since by Lebesgue's dominated 

convergence theorem it is the almost everywhere limit

of

the (uniformly bounded set of) continuous functions

$$

\xi\mapsto\int\limits_{S^{d-1}\cap\xi^\bot}

\left(\frac{\partial^{m-1}f}{\partial\psi^{m-1}}

(\eta,\psi+\varepsilon)-

\frac{\partial^{m-1}f}{\partial\psi^{m-1}}(\eta,\psi)

\right)\varepsilon^{-1}d\eta\,.

$$

Hence $R_\psi^{(m)}f\in L^\infty(S^{d-1})$.



Further we have

$$

\int\limits_{S^{d-1}}

(R_\psi^{(m)}f)(\xi)g(\xi)d\xi=$$

$$\lim_{\varepsilon\to 0}

\int\limits_{S^{d-1}}\left[

\int\limits_{S^{d-1}\cap\xi^\bot}\left(

\frac{\partial^{m-1}f}{\partial\psi^{m-1}}

(\eta,\psi+\varepsilon)-

\frac{\partial^{m-1}f}{\partial\psi^{m-1}}

(\eta,\psi)\right)\varepsilon^{-1}

d\eta\right]g(\xi)d\xi\,.

$$

An induction on $m$ shows that this equals

$\int\limits_{S^{d-1}}f(\xi)(R_\psi^{(m)}g)(\xi)d\xi$. The induction

basis is $m=1$, which follows from Lemma 3.6 of [Makai--Martini--\'Odor].

\qued

\enddemo



Defining $R_\psi^{(0)}$ in the analogous way, i.e., by

$$(R_\psi^{(0)}f)(\xi)=

\int\limits_{S^{d-1}\cap\xi^\bot}

f(\eta,\psi)d\eta\,,$$

we have

the following lemma, that generalizes Lemma 3.7 of [Makai--Martini--\'Odor].

Its statement in the case $m=0$ is due to [Radon, 1917] (for $d=3$), and to 

[Schneider, 1969], formula (5) (an alternative proof cf. in 

[Makai--Martini--\'Odor], Lemma 3.7), while in the case $m=1$ it is due to

[Makai--Martini--\'Odor], Lemma 3.7.



\proclaim{Lemma 3.2} Let $d\ge 2$, $m\ge 0$, $-\pi/2\le\psi\le\pi/2$, 

and

let $Y_n\colon S^{d-1}\to\Bbb R$ be a spherical harmonic of degree $n$. Then 

$Y_n$

is an eigenfunction of $R_\psi^{(m)}$, i.e\.,

$R_\psi^{(m)}Y_n=\lambda_nY_n$, with

$$

\lambda_n=V_{d-2}(S^{d-2})C_n^{(d-2)/2}(1)^{-1}

\left(\frac{d}{d\psi}\right)^mC_n^{(d-2)/2}(\sin\psi)\,\,.

$$

\endproclaim



\demo{Proof} By the case $m=0$ of our statement (referred to above) we have

$$

\int \limits _ {S ^{d-1} \cap \xi ^ \bot}

Y_n(\eta, \psi) d \eta =

V_{d-2}(S^{d-2})C_n(1)^{-1}C_n(\sin\psi)

\cdot Y_n(\xi)\,.

$$

From this case subsequent differentiations with respect to $\psi$

prove the

statement for any

$m\ge 1$. 

\qued

\enddemo



The following theorem is a generalization of Theorem 3.8 of 

[Makai--Martini--\'Odor].

The statement corresponding to the case $m=0$, $\psi=0$ in 

this theorem is the Funk integral

theorem (for $f\in 

C(S^{d-1})$),

cf.\ [Funk, 1913], Kap. 2, [Lifshitz--Pogorelov, 1954], 

[Schneider, 1969], [Helgason, 1980], Ch. 3, \S 1.B and [Helgason, 1984], while 

the case $m=1$ is contained in [Makai--Martini--\'Odor], Theorem 3.8.



\proclaim{Theorem 3.3} Let $d\ge 2$, $m\ge 1$ and

$-\pi/2\le\psi\le\pi/2$.

Then the null--space of the operator $R_\psi^{(m)}\colon

Lip^{m-1}(S^{d-1})\to

L^\infty(S^{d-1})$ equals $\{f\in Lip^{m-1}(S^{d-1})\,\,\vert$ the Fourier

expansion $\sum\limits_{n=0}^\infty Y_n(f)$ of $f$ satisfies that

$\left(d/d \psi \right)^mC_n^{(d-2)/2}(\sin\psi)\ne

0$

implies $Y_n(f)=0\}$. In particular, for $\psi=0$ and $m$ odd (even)

the

null--space of $R^{(m)}=R^{(m)}_0$ 

contains $\{f\in Lip^{m-1}(S^{d-1})\,\,\vert

\,\,f$

is even ($f$ is the sum of an odd function and a constant)$\}$.

\endproclaim



\demo{Proof} For the first statement we proceed analogously to 

[Alexandroff, 1937], [Petty, 1961], [Schneider, 1969], [Schneider, 1970], 

[Falconer, 1983].

Let $f\in Lip^{m-1}(S^{d-1})$. Then, by 3.1,

$R_\psi^{(m)}f\in L^\infty(S^{d-1})\subset L^2(S^{d-1})$, and, by

completeness of spherical harmonics, $R_\psi^{(m)}f=0$ holds a.e\. if

and

only if for each $n\ge 0$, and each spherical harmonic $Y_n$ of degree $n$

we

have $0=\langle R_\psi^{(m)}f,Y_n\rangle$, where $\langle\,\,,\,\,

\rangle$ now denotes 

scalar product in $L^2(S^{d-1})$. Letting $\sum\limits_{n=0}^\infty Y_n(f)$

be the

Fourier expansion of $f$, we have by 3.1 and 3.2 that $$\langle

R_\psi^{(m)}f,

Y_n\rangle=\langle f,R_\psi^{(m)}Y_n\rangle=\lambda_n\langle

f,Y_n\rangle=$$

$$V_{d-2}(S^{d-2})C_n(1)^{-1}

\left(\frac{d}{d\psi}\right)^mC_n(\sin\psi)\cdot\langle Y_n(f),

Y_n\rangle .$$ For fixed $n$ and $Y_n$ arbitrary this is $0$ if and only if

$\left(d/d \psi \right)^mC_n(\sin\psi)\cdot Y_n(f)=0$. This

implies the first statement.



For the second statement observe that for $f$ constant we have

$R_\psi^{(m)}f=0$. Furthermore, for $\psi=0$, $m$ odd (even) and $f$

even

(odd) by 3.1 we have for almost all $\xi\in S^{d-1}$ that for almost all

$\eta\in S^{d-1}\cap\xi^\bot$ both

$({\partial^mf}/{\partial\psi^m})(\eta,0)$ and

$({\partial^mf}/{\partial\psi^m})(-\eta,0)$ exist and then have sum

0,

thus $(R^{(m)}f)(\xi)=0$ a.e\.

\qued

\enddemo



More generally, let $d\ge 2$, $m\ge 1$ an integer, $P$ a polynomial of

degree $m$, and $-\pi/2\le\psi\le\pi/2$. Then we define for $f\in

Lip^{m-1}(S^{d-1})$ and $\xi\in S^{d-1}$

$$

(R_\psi^{P}f)(\xi)=\int\limits_{S^{d-1}\cap\xi^\bot}

\left(P\left(\frac{\partial}{\partial\psi}\right)\right)

f(\eta,\psi)d\eta\,,$$

provided that the right--hand side exists. For $\psi =0$ we drop the lower 

index. Then, analogously like above, we have



\proclaim{Theorem 3.4} Let $d\ge 2$ be a fixed integer. Let $m\ge 1$ be an

integer, $P$ a real polynomial of degree $m$, and $-

\pi/2\le\psi\le\pi/2$. Furthermore, let 

$f,g\in Lip^{m-1}(S^{d-1})$. Then, for almost

all

$\xi\in S^{d-1}$ we have for almost all $\eta\in

S^{d-1}\cap\xi^\bot$ that

$\left(P\left({\partial}/{\partial\psi}\right)\right)f(

\eta,\psi)$ exists, and, for almost

all

$\xi\in S^{d-1}$, the integral defining 

$(R_\psi^{P}f)(\xi)$

exists. We have $R_\psi^{P}f\in L^\infty(S^{d-1})$, and $R_{\psi }^P$ is 

symmetric, i.e., $\int\limits_{S^{d-1}}

(R_\psi^Pf)(\xi)g(\xi)d\xi=

\int\limits_{S^{d-1}}f(\xi)(R_\psi^Pg)(\xi)d\xi$. The null--space

of the operator $R_\psi^{P}\colon Lip^{m-1}(S^{d-1})\to

L^\infty(S^{d-1})$ equals $\{ f\in Lip^{m-1}(S^{d-1})\,\,\vert\,\,$the

Fourier expansion

$\sum\limits_{n=0}^\infty Y_n(f)$ of $f$ satisfies that $\left(P

\left(d/d \psi \right)\right)C_n^{(d-2)/2}(\sin\psi)

\ne

0$ implies $Y_n(f)=0\}$.

\endproclaim 



\demo{Proof}

Let $P(t)=\sum_{r=0}^m c_r t^r$.

The existence statements, as well as 

the statement $R_{\psi}^{P}f \in L^{\infty}(S^{d-1})$

and symmetry of $R_{\psi}^{P}$, follow from 3.1. Then 

3.2 implies its

analogue for $R^{P}_{\psi}$, with eigenvalue 

$$ 

\lambda _n= V_{d-2} (S^{d-2}) C_n^{(d-2)/2} (1)^{-1} 

\(P \(

\frac {d} {d \psi }

\)\)

C_n^{(d-2)/2}(\sin \psi )\,. $$

Then, like in 3.3, we obtain that we have $(R_{\psi}^

{P} f)(\xi)=0$ a.e.\

if and only if

$$

\(P \(\frac {d} {d \psi }\)\)

C_n^{(d-2)/2}(\sin \psi) \ne  0

$$

implies $Y_n(f)=0$, where $\sum_{n=0}^{\infty}Y_n(f)$ is the

Fourier expansion of $f$.

\qued

\enddemo



{\bf {Remark.}} By Lemma 3.1 and Theorem 3.4 $R^{(m)}_{\psi }$ and 

$R^P_{\psi }$ are symmetric operators. Actually it is easily seen that their 

closures are self--adjoint operators in $L^2(S^{d-1})$, with domain

$\{\sum Y_n \mid Y_n$ is a spherical harmonic of degree $n,\,\,\sum (1+

\lambda _n ^2) \|Y_n\|^2 < \infty \}$, and are given by $\sum Y_n \mapsto \sum 

\lambda _nY_n$, where the $\lambda _n$ are the corresponding eigenvalues.



\vskip2truecm



%%%%%%%%%%%%%%%PARAGRAPH 4%%%%%%%%%%%%%%%%%%%%



\heading 4.\ The null--spaces of the operators $R^{(m)}$ and $R^P$\endheading

\vskip0.4truecm



Now, for $d\ge 2$, we turn to the case of general $m\ge 1$, and investigate

the

null--space of $R^{(m)}\colon Lip^{m-1}(S^{d-1})\to L^\infty(S^{d-1})$.



For $d=2$ the complete answer is given by



\proclaim{Proposition 4.1} Let $d=2$ and $m\ge 1$. Then the null--space of the

operator

$R^{(m)}\colon $ 

\newline

$Lip^{m-1}

(S^{d-1}) \to L^\infty(S^{d-1})$ equals for $m$ odd

(even) $\{f\in Lip^{m-1}(S^{d-1})\,\,\vert \,\,f$ is even ($f$ is the sum

of an odd

function and a constant)$\}$.

\endproclaim



 

\demo{Proof}

By 3.3, we have that $f\in \Lip^{m-1}(S^{d-1})$ satisfies $R^{(m)}f=0$ a.e.\

if and only if

$

\left. \(

d/{d \psi }

\)^m 

C^0_n(\sin \psi )\right|_{\psi=0}\allowbreak\ne 0

$

implies $Y_n(f)=0$. However, by [M\"uller, 1966], p. 33,

we have for some $a_n\ne 0$ that 

$

 C^0_n(\sin \psi)=C^0_n[\cos(\pi /2-\psi)]=a_n\cos [n(\pi /2-

\psi)].

$

A straightforward calculation gives then that

$$

\left. \(

\frac {d} {d \psi }

\)^m 

\cos [n(\pi /2-\psi )] \right|_{\psi=0}

$$

equals $0$ if $n=0$ or if $n-m$ is odd, and equals

$

 (-1)^{\lfloor n/2 \rfloor}(-1)^{\lfloor m/2 \rfloor} n^m\ne 0$

if $n-m$ is even and $n\ge 1$.

Then 3.3 implies our statement.

\qued



\proclaim{Lemma 4.2} 

Let $d \ge 3$ and $ m\ge 1$. Then there exists a polynomial 

$\varGamma _{d,m,n}$ of $n$, of degree $m$, such that for 

$n-m$ even we have  

$$

\left. \(\frac {d} {d \psi }\)^m 

C^{(d-2)/2}_n(\sin \psi)\right|_{\psi=0}=0

$$

if and only if $\varGamma _{d,m,n}=0$. We have $\varGamma_{d,0,n}=1$,

$\varGamma_{d,1,n}=n+d-3$, $\varGamma_{d,2,n}=n(n+d-3)$ and 

$\varGamma_{d,3,n}=(n+d-3)(n^2 +(d-2)n-(d-2))$.

\endproclaim



\demo{Proof} By [Tricomi, 1955], p. 182, for $d\ge 3$ we have

$$

C^{(d-2)/2}_n(\sin \psi)=C^{(d-2)/2}_n[\cos (\pi /2-

\psi)]=$$

$$(-1)^n \sum_{i=0}^n

\binom{-d/2+1}{i}

\binom{-d/2+1}{n-i}

\cos [(n-2i)(\pi /2-\psi)]\,.

$$

Using 

$$

\(\frac {d} {d \psi }\)^m \cos \psi =

\cos (\psi+m\pi /2)\,,

$$

we have for $n-m$ even that

$$

\gamma_{d,m,n}:=

\left. 

\(\frac {d} {d \psi }\)^m 

C^{(d-2)/2}_n(\sin \psi)

\right|_{\psi=0}=

$$

$$

(-1)^n \sum_{i=0}^n

\binom{-d/2+1}{i}

\binom{-d/2+1}{n-i}[-(n-2i)]^m

\cos [(n-2i)\pi /2+m\pi /2)]=

$$

$$

(-1)^n(-1)^{(n+m)/2}

\sum_{i=0}^n (-1)^i

\binom{-d/2+1}{i} \binom{-d/2+1}{n-i}(2i-n)^m.

$$

Here $(2i-n)^m=\sum\limits_{p=0}^m(-1)^p\binom{m}{p}2^{m-p}n^pi^{m-p}$,

and 

$$

i^{m-p}=\sum_{q=0}^{m-p}c_{m-p, q}\binom{i}{q}q!\,,

$$

for some constants $c_{m-p, q}$, where $c_{m-p, m-p}=1$. Hence

$$

(2i-n)^m=\sum_{q=0}^m f_{m, q}(n)\binom{i}{q}q!\,,

$$

where 

$$

   f_{m,q}(n)=\sum_{p=0}^{m-q}(-1)^p\binom{m}{p} 2^{m-p} c_{m-p, q} n^p.

$$

Therefore 

$$

\gamma _{d,m,n}=(-1)^n (-1)^{(n+m)/2}\sum_{q=0}^m	f_{m,q}(n) \delta 

_{d,n,q}\,,

$$

where

$$

\delta _{d,n,q}=

\sum_{i=0}^n (-1)^i

\binom{-d/2+1}{i} \binom{-d/2+1}{n-i}

\binom{i}{q}q!\,.

$$



Here $\delta _{d,n,q}$ is the  coefficient of $x^n$ in the 

power series expansion of

$$

x^q\left[

\(\frac{

d} {dx

}\)^q (1-x)^{(2-d)/2}

\right] (1+x)^{(2-d)/2}=

$$

$$

(-1)^q \(-\frac{d}{2}+1\)\(-\frac{d}{2}\)\dots 

\(-\frac{d}{2}-q+2\)x^q

(1-x^2)^{-{d/2}-q+1}(1+x)^q =

$$

$$

(-1)^q \(-\frac{d}{2}+1\)\(-\frac{d}{2}\)\dots 

\(-\frac{d}{2}-q+2\)x^q

\sum_{k=0}^\infty (-1)^k \binom{-d/2-q+1}{k}

x^{2k}\sum_{l=0}^q \binom{q}{l}x^l,

$$

i.e., 

$$

\delta _{d,n,q}=

(-1)^q \(-\frac{d}{2}+1\)\(-\frac{d}{2}\)\dots 

\(-\frac{d}{2}-q+2\)

\times

$$

$$

\sum_{k\ge 0,\, 0\le l\le q,\, 2k+l=n-q} (-1)^k

\binom{-d/2-q+1}{k}\binom{q}{l}\,.

$$

Here in the last summation we have $2k=n-q-l\in [n-2q, n-q]$,

thus 

$\lceil n/2 \rceil-q\le k\le \lfloor (n-q)/2 \rfloor\le 

\lfloor n/2 \rfloor$.

Therefore we introduce the notation $k=\lfloor n/2 \rfloor-j$;

then $0\le \lfloor n/2 \rfloor-\lfloor (n-q)/2 \rfloor=

q/2+\delta \le j\le q-\varepsilon $. Here $\delta =0$ for $q$ 

even, $\delta =1/2$ or

$-1/2$ for $q$ odd and $n$ even or odd,

respectively, and

$\varepsilon =0$ for $n$ even, $\varepsilon =1$ for $n$ odd. Furthermore,

$l=n-2k-q=\varepsilon +2j-q$.



We have 

$$

\delta _{d,n,q}\binom{-d/2+1}{\lfloor n/2 \rfloor}^{-1}=

\sum_{q/2+\delta \le j\le q-\varepsilon ,\,\, j\le 

\lfloor n/2 \rfloor}

\delta _{d, n, q, j}\,,

$$

where 

$$

\delta _{d, n, q, j}=

(-1)^q 

\(-\frac{d}{2}+1\)\(-\frac{d}{2}\)\dots 

\(-\frac{d}{2}-q+2\)\times

$$

$$

(-1)^{\lfloor {n/2} \rfloor-j}

\binom{-d/2-q+1}{\lfloor n/2 \rfloor-j}

\binom{-d/2+1}{\lfloor n/2 \rfloor}^{-1}

\binom{q}{\varepsilon +2j-q}=

$$

$$

(-1)^{\lfloor {n/2} \rfloor}(-1)^{q-j}

\(-\frac{d}{2}+1\)\(-\frac{d}{2}\)\dots\(-\frac{d}{2}-q+2\)

\times

$$

$$

\(-\frac{d}{2}-q+1\)\dots

\(-\frac{d}{2}-q-\left\lfloor\frac{n}{2}\right\rfloor+j+2\)

\times

$$

$$

\(\left\lfloor\frac{n}{2}\right\rfloor-j\)!^{-1}

\(-\frac{d}{2}+1\)^{-1}\dots

\(-\frac{d}{2}-\left\lfloor\frac{n}{2}\right\rfloor+2\)^{-1}

\left\lfloor\frac{n}{2}\right\rfloor!

\binom{q}{\varepsilon +2j-q}=

$$

$$

(-1)^{\lfloor n/2 \rfloor}(-1)^{q-j}

\(-\frac{d}{2}-\left\lfloor\frac{n}{2}\right\rfloor+1\)\dots

\(-\frac{d}{2}-\left\lfloor\frac{n}{2}\right\rfloor-q+j+2\)

\times

$$

$$

\left\lfloor\frac{n}{2}\right\rfloor\dots

\(\left\lfloor\frac{n}{2}\right\rfloor-j+1\)

\binom{q}{\varepsilon +2j-q}=

(-1)^{\lfloor n/2 \rfloor} g_{d , q, j}(n)\,,

$$

where $g_{d , q, j}(n)$ equals, for $n$ having the parity of $m$,

a polynomial of $n$, of degree 

$(q-j)+j=q$, and of leading coefficient $2^{-q}\binom{q}{\varepsilon +2j-q}$.

However, for $0\le \lfloor n/2 \rfloor<j$ we have

$g_{d,q,j}(n)=0$, thus

$$

\delta _{d,n,q}\binom{-d/2+1}{\lfloor n/2 \rfloor}^{-1}=

(-1)^{\lfloor n/2 \rfloor}\sum_{j=q/2+\delta }^{q-\varepsilon }

g_{d, q, j}(n)\,.

$$

Collecting these results, we have 

$$

(-1)^n(-1)^{(n+m)/2}(-1)^{\lfloor n/2 \rfloor}

\gamma _{d, m, n}\binom{-d/2+1}{\lfloor n/2 \rfloor}^{-1}=$$

$$(-1)^{\lfloor n/2 \rfloor}

\sum_{q=0}^m f_{m, q}(n)\delta _{d, n, q}

\binom{-d/2+1}{\lfloor n/2 \rfloor}^{-1}=

$$

$$

\sum_{q=0}^m f_{m, q}(n)

\sum_{j=q/2+\delta }^{q-\varepsilon } g_{d, q, j}(n):=

\varGamma _{d, m, n}\,.

$$

Here each $f_{m, q}(n)$ or $g_{d, q, j}(n)$ equals, for

$n$ having the parity of $m$, a polynomial of $n$, of degree $m-q$

or $q$, respectively.

Moreover, $\delta $ and 

$\varepsilon $ in the bounds of the inner summation only

depend on the parities of $n$ and $q$.

Therefore $\varGamma _{d,m,n}$ equals, for $n$ having the parity of $m$,

a polynomial 

of $n$, of degree $\le m$.

Furthermore, we have $\gamma _{d,m,n}=0$ if and only if 

$\varGamma _{d, m, n}=0$.



Now we show that $\varGamma _{d, m, n}$ equals, for $n$ having the parity

of $m$, a 

polynomial of $n$, of degree exactly $m$. More exactly, we show

that its

leading coefficient is $1$.

We have 

$$

\varGamma _{d, m, n}=\sum_{q=0}^m f_{m, q}(n)\sum_{j=q/2+\delta }^

{q-\varepsilon }

g_{d, q, j}(n)\,.

$$

Here, by $c_{q, q}=1$, the leading term of $f_{m, q}(n)$ is

$(-1)^{m-q}\binom{m}{q}2^q n^{m-q}$.

Similarly, the leading term of $g_{d, q, j}(n)$ is

$2^{-q}\binom{q}{\varepsilon +2j-q}n^q$. Hence the coefficient of $n^m$

in $\varGamma _{d, m, n}$ is

$$

\sum_{q=0}^m(-1)^{m-q}\binom{m}{q}

\sum_{j=q/2+\delta }^{q-\varepsilon }\binom{q}{\varepsilon +2j-q}\,.

$$

A small discussion, taking into consideration the parities of $m$ and $q$,

shows that this equals

$$

\sum_{q=0}^m(-1)^{m-q}\binom{m}{q}

\sum_{0\le \varepsilon +2j-q\le q}\binom{q}{\varepsilon +2j-q}=

$$

$$

\sum_{q=1}^m(-1)^{m-q}\binom{m}{q}2^{q-1}+\frac{1}{2}+(-1)^m \frac{1}{2}=

$$

$$

(2-1)^m\frac{1}{2}- 

(-1)^m\frac{1}{2}+\frac{1}{2}+(-1)^m\frac{1}{2}=1.

$$

The above considerations show that, for any given $m$, 

we can evaluate $\gamma _{d,m, n}$ for all $n$. Performing the above

calculations for $m \le 3$, we obtain the formulas for 

$\varGamma _{d,m,n}$, given in the lemma. 

\qued

\enddemo



Using the above considerations we prove the following theorem, that is a 

generalization of the last statement of Theorem 3.8 of 

[Makai--Martini--\'Odor].



\proclaim{Theorem 4.3}

Let $d\ge 3$  be a fixed integer. Then, for any integer  $m\ge 1$,

there exists a set $A_m$ of non--negative integers of the same

parity as $m$, with cardinality $|A_m|\le m$, such that the following holds.

The null--space of the operator $R^{(m)} \colon Lip^{m-1}(S^{d-1})

\to L^\infty (S^{d-1})$ equals $\{ f \in Lip^{m-1}(S^{d-1})\,\,|

\,\,f$ is of the form $f=g+\sum\limits_{n\in A_m}Y_n$,

where $g$ is even (odd) for $m$ odd (even), and $Y_n$

is a spherical harmonic of degree $n\}$. In particular, for

$m=2$, $3$ we have $A_2=\{0\}$, $A_3=\emptyset$.

\endproclaim



\demo{Proof}

By 3.3, we have that $f \in Lip^{m-1} (S^{d-1})$ satisfies 

$R^{(m)}f=0$ a.e.\

if and only if, for the Fourier expansion   

$\sum_{n=0}^\infty Y_n(f) $ of $f$, we have that

$\gamma _{d, m, n}:=\left.\(

d/{d \psi }

\)^m C_n^{(d-2)/2}(\sin 

\psi )

\right|_{\psi =0}\ne 0$

implies $Y_n(f)=0$. Since $C_n$ is odd (even) for $n$ odd

(even), therefore

for $n-m$ odd we have

$\gamma _{d, m, n}=0$. So we only need to consider the case $n-m$ even.

Then by 4.2 we have $\gamma _{d, m, n}=0$ if and only if 

$\varGamma _{d, m, n}=0$,

and $\varGamma _{d, m, n}$ equals, for $n$ having the parity

of $m$, a polynomial of $n$, of degree $m$.



We let $A_m=\{n\,\,|\,\, \text{$n\ge 0$ is an integer, $n-m$ is even,

$\varGamma _{d,m,n}=0$}\}$. Then $\{ f \in Lip^{m-1} $

$(S^{d-1})\,\,|\,\,

R^{(m)}f=0$ a.e\.$\}=\bigl \{ f \in Lip^{m-1}(S^{d-1})\,\,|\,\,f= \sum

\{Y_n(f)\,\,|\,\, n \ge 0$ is an integer, and either $2 \nmid 

(n-m),$ or $(2 \mid (n-m)$ and $n \in A_m)\}\bigr \} $.



For the cases $m=2$, $3$, we consider the equation

$\varGamma _{d,m,n}=0$ from 4.2. For $m=2$ its only non--negative  even root

is $n=0$. Now let $m=3$ and $n\ge 0$ odd. Then we have $n+d-3\ge 1$. 

Furthermore, 

the discriminant of $n^2+(d-2)n-(d-2)$ is $d^2-4$, that is not

a perfect square for $d\ge 3$, thus the roots of this polynomial 

are irrational.

\qued

\enddemo



\proclaim{Theorem 4.4}

Let $d \ge 2$ be a fixed integer. Let $m \ge 1$ be an integer, $P$ a 

polynomial of degree $m$, and $\psi =0$. If 

$P$ is odd

(even) for $m$ odd (even), then there exists a set $A_P$ of

non--negative

integers of the same parity as $m$, with cardinality $\vert A_P\vert\le m$,

such that the following holds. The null--space of the operator 

$R^P=R_0^{P}$ equals $\{f\in

Lip^{m-1}(S^{d-1})\,\,\vert \,\,f$ is of the form $f=g+

\sum\limits_{n\in A_P}Y_n$, where $g$ is even (odd)

for $m$ odd (even), and $Y_n$ is a spherical harmonic of

degree $n\}$.

\endproclaim



\demo{Proof} Suppose that $P$ is such as given in the theorem. Then we

 have 

$

%\left .

\(P \(

d / {d \psi }

\)\) C_n^{(d-2)/2}$

\newline $\left . (\sin \psi )

\right|_{\psi =0}=0

$

for $n-m$ odd. 

Now let us suppose that $n-m$ is even.

First we suppose that $d\ge 3$. We have, like at 4.3 and 4.2,

$$

\left .

\(P \(\frac {d} {d \psi }\)\)

C_n^{(d-2)/2}(\sin \psi )

\right|_{\psi =0}=

\sum_{r=0}^m c_r \gamma _{d, r, n}=

$$

$$

(-1)^n (-1)^{\lfloor {n/2} \rfloor}

\binom{-d/2+1}{\lfloor n/2 \rfloor}

\sum_{r=0}^m c_r (-1)^{(n+r)/2} \varGamma _{d, r, n}

$$

(observe that $n \equiv r\,\, ({\text {mod} }\,\, 2) $).

This expression equals $0$ if and only if the last sum equals $0$.

Furthermore, this last sum equals an $m$--th degree  polynomial of $n$,

if $n-m$ is even.

Now let $A_{P}$ be the set of those non--negative integer

roots $n$

of this polynomial, for which $n-m$ is even.



Now let $d=2$. Then we have from the proof of 4.1, with the same $a_n\ne 0$,

$$

\left. \(P \(\frac {d} {d \psi }\)\)

C_n^0(\sin \psi )\right|_{\psi =0}=

a_n (-1)^{\lfloor n/2 \rfloor}

\sum_{r=0}^m c_r (-1)^{\lfloor r/2 \rfloor} n^r .

$$

Then we define $A_{P}$ using this last sum, like above.

\qued

\enddemo



\vskip1truecm



{\bf Acknowledgment.} The authors express their gratitude to R. Gardner 

for his several valuable remarks concerning this paper. 





\Refs

\define\mr{{}}

\define\MR{\rm \bf MR}



\ref \key Adams, 1975

\by R. A. Adams

\book Sobolev Spaces

\publaddr Academic Press Inc., New York--London

\yr 1975,

\mr \MR56\#9247

\endref \vskip .1cm



\ref\key Alexandroff, 1937

\by A. D. Alexandroff

\paper Zur Theorie der gemischten Volumina von konvexen K\"orpern II.,

Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen

\jour Mat. Sb.

\vol 2

\yr 1937

\pages 1205--1238

(Russian with German summary), Zbl. {\bf 18},276

\endref \vskip .1cm



\ref \key Blaschke, 1949

\by W. Blaschke

\book Kreis und Kugel

\yr 1949

\publaddr Chelsea Publ. Co., New York, reprint

\yr 1949,

\mr \MR17--887

\endref \vskip .1cm



%\ref \key Bonnesen, 1926

%\by T. Bonnesen

%\paper On Minkowski's inequalities for convex bodies II (in Danish)

%\jour Mat. Tidsskr. B

%\yr 1926

%\pages 74--80,

%Jahrbuch Fortschr. Math. {\bf 53},710--711

%\endref \vskip .1cm



%\ref \key Croft et al., 1991

%\by H. T. Croft--K. J. Falconer--R. K. Guy

%\book Unsolved Problems in Geometry

%\publaddr Springer--Verlag, New York

%\yr 1991,

%\mr \MR92c:52001

%\endref \vskip .1cm



\ref\key Erd\'elyi et al., 1953

\by  A. Erd\'elyi--W. Magnus--F. Oberhettinger--F. G. Tricomi

\book Higher transcendental functions II.

\publaddr  McGraw--Hill, New York--Toronto--London

\yr 1953,

\mr \MR15--419

\endref \vskip .1cm



\ref \key Falconer, 1983

\by K. J. Falconer

\paper Applications of a result on spherical integration to the theory

of convex sets

\jour Amer. Math. Monthly

\vol 90

\yr 1983

\pages 690--693,

\mr \MR85f:52012

\endref \vskip .1cm



\ref \key Funk, 1913

\by P. Funk

\paper \"Uber Fl\"achen mit lauter geschlossenen geod\"atischen Linien

\jour Math. Ann.

\vol 74

\yr 1913

\pages 278--300,

Jahrbuch Fortschr. Math. {\bf 44},692

\endref \vskip .1cm



%\ref \key Funk, 1916

%\by P. Funk

%\paper \"Uber eine geometrische Anwendung der Abelschen Integralgleichung

%\jour Math. Ann.

%\vol 77

%\yr 1916

%\pages 129--135,

%Jahrbuch Fortschr. Math. {\bf 45},533

%\endref \vskip .1cm



\ref \key Gardner, 1995

\by R. J. Gardner

\book Geometric Tomography, Encyclopedia of Math. and its Appls. {\bf 58}

\publaddr Cambridge Univ. Press, Cambridge

\yr 1995,

\mr \MR96j:52006

\endref \vskip .1cm



\ref \key Groemer, 1993

\by H. Groemer

\paper Fourier series and spherical harmonics in convexity

\jour  in:

Handbook of Convex Geometry (eds. P. M. Gruber, J. M. Wills), 

North Holland, Amsterdam, 1993

\pages 1259--1295,

\mr \MR94j:52001

\endref \vskip .1cm



\ref \key Groemer, 1996

\by H. Groemer

\book Geometric Applications of Fourier Series and Spherical Harmonics,

Encyclopedia of Math. and its Appls. {\bf 61}

\publaddr Cambridge Univ. Press, Cambridge

\yr 1996,

\mr \MR97j:52001

\endref \vskip .1cm



\ref\key Hammer, 1954

\by P. C. Hammer

\paper Diameters of convex bodies

\jour  Proc. Amer. Math. Soc.

\vol 5

\yr 1954

\pages  304--306,

\mr \MR15--819

\endref \vskip .1cm



\ref \key Helgason, 1980

\by S. Helgason

\book The Radon Transform

\publaddr Birkh\"auser, Boston, Mass.

\yr 1980,

\mr \MR83f:43012

\endref \vskip .1cm



\ref \key Helgason, 1984

\by S. Helgason

\book Groups and Geometric Analysis. Integral Geometry,

Invariant Differential Operators, and Spherical Functions

\publaddr Academic Press, Inc., Orlando, Fla.

\yr 1984,

\mr \MR86c:22017

\endref \vskip .1cm



%\ref \key Klee, 1969

%\by V. Klee

%\paper Is a body spherical if its $HA$ measurements are constant?

%\jour Amer. Math. Monthly, Research Problems

%\vol 76

%\yr 1969

%\pages 539--542

%\endref \vskip .1cm



\ref \key Lifshitz--Pogorelov, 1954

\by I. M. Lifshitz--A. V. Pogorelov

\paper On the determination of Fermi surfaces and electron velocities

in metals by  the oscillation  of magnetic susceptibility (in Russian)

\jour Dokl. Akad. Nauk SSSR

\vol 96

\yr 1954

\pages 1143--1145,

Zbl. {\bf 57}.448

\endref \vskip .1cm



%\ref\key Lutwak, 1988

%\by E. Lutwak

%\paper Intersection bodies and dual mixed volumes

%\jour Adv. in Math.

%\vol 71

%\yr 1988

%\pages 232--261,

%\mr \MR90a:52023

%\endref \vskip .1cm



%\ref \key Mackintosh, 1963

%\by A. R. Mackintosh

%\paper The Fermi surface of metals

%\jour Scientific American

%\vol 209

%\yr 1963, July

%\pages 110--120

%\endref \vskip .1cm



%\ref\key Makai--Martini, 1996

%\by E. Makai, Jr.--H. Martini

%\paper On bodies associated with a given convex body

%\jour Canad. Math. Bull.

%\vol 39

%\yr 1996

%\pages 448--459,

%\mr \MR97m:52012

%\endref \vskip .1cm



\ref\key Makai--Martini--\'Odor

\by E. Makai, Jr.--H. Martini--T. \'Odor

\paper Maximal sections and centrally symmetric bodies

\jour Mathematika, accepted for publication

\vol 

\yr 

\pages 

\mr \MR

\endref \vskip .1cm



%\ref\key Martini, 1992

%\by H. Martini

%\paper Extremal equalities for cross--sectional measures of convex bodies

%\jour Proc. 3rd Geometry Congress, Aristoteles Univ. Thessaloniki, 

%Thessaloniki 1992

%\pages 285--296,

%\mr \MR93f:52014

%\endref \vskip .1cm



%\ref \key Martini, 1994

%\by H. Martini

%\paper Cross--sectional measures

%\jour Intuitive Geom. (Szeged 1991), Coll. Math. Soc. J. Bolyai 63, 

%North Holland, Amsterdam--Oxford--New York, 1994

%\pages 269--310,

%\mr \MR97b:52008

%\endref \vskip .1cm



\ref\key M\"uller, 1966

\by C. M\"uller

\book  Spherical Harmonics

\publaddr  Lecture Notes in Math. 17, Springer, Berlin

\yr   1966,

\mr \MR33\# 7593

\endref \vskip .1cm



%\ref \key Petty, 1952

%\by C. M. Petty

%\book On Minkowski geometries, Ph.D. dissertation

%\publaddr Univ. South California, Los Angeles

%\yr 1952

%\endref \vskip .1cm



\ref \key Petty, 1961

\by C. M. Petty

\paper Centroid surfaces

\jour Pacific J. Math. 

\vol 11 

\yr 1961

\pages 1535--1547,

\mr \MR24\#A3558

\endref \vskip .1cm



\ref \key Radon, 1917

\by J. Radon

\paper \"Uber die Bestimmung von Funktionen durch ihre Integralwerte l\"angs

gewisser Mannigfaltigkeiten

\jour Ber. Verh. S\"achs. Akad. Wiss. Leipzig, Math. Nat. Kl.

\vol 69

\yr 1917

\pages 262--277,

Jahrbuch Fortschr. Math. {\bf 46},436

\endref \vskip .1cm



\ref \key Sansone, 1959

\by G. Sansone

\book Orthogonal Functions

\publaddr Rev. ed., Interscience, London--New York

\yr 1959,

\mr \MR21 \#2140

\endref \vskip .1cm



%\ref\key Schneider, 1967

%\by R. Schneider

%\paper Zu einem Problem von Shephard \"uber 

%die Projektionen konvexer K\"orper

%\jour Math. Z.

%\vol 101

%\yr 1967

%\pages 71--82,

%\mr \MR36\#2059

%\endref \vskip .1cm



\ref\key Schneider, 1969

\by R. Schneider

\paper Functions on a sphere with vanishing integrals over certain

subspheres

\jour J. of Math. Anal. and Appl.

\vol 26

\yr 1969

\pages 381--384,

\mr \MR38\#6004

\endref \vskip .1cm



\ref\key Schneider, 1970

\by R. Schneider

\paper \"Uber eine Integralgleichung in der Theorie der konvexen K\"orper

\jour Math. Nachr.

\vol 44

\yr 1970

\pages 55--75,

\mr \MR43\#1043

\endref \vskip .1cm



\ref\key Schneider, 1993

\by R. Schneider

\book Convex Bodies: the Brunn--Minkowski Theory,

Encyclopedia of Math. and its Appls. {\bf 44}

\publaddr Cambridge Univ. Press, Cambridge

\yr 1993,

\mr \MR94d:52007

\endref \vskip .1cm



\ref\key Seeley, 1966

\by  R. T. Seeley

\paper 	Spherical harmonics

\jour  Amer. Math. Monthly

\vol 73

\yr 1966

\pages 	115--121,

\mr \MR34\#1577

\endref \vskip .1cm



%\ref \key Shoenberg, 1960

%\by D. Shoenberg

%\paper The de Haas--van Alphen effect

%\jour in: The Fermi Surface, W. A. Harrison--M. B. Webb (eds.),

%Wiley, New York, 1960

%\pages 74--83

%\endref \vskip .1cm



\ref \key Tricomi, 1955

\by F. G. Tricomi

\book Vorlesungen \"uber Orthogonalreihen, Grundlehren Math.\ Wiss.\

{\bf 76}

\publaddr Springer-- Verlag, Berlin--G\"ottingen--Heidelberg

\yr 1955,

\mr \MR17--30

\endref \vskip .1cm



\ref \key Tricomi, 1957

\by F. G. Tricomi

\book Integral Equations

\publaddr Interscience, New York--London

\yr 1957,

\mr \MR20\#1177

\endref \vskip .1cm



%\ref \key Ungar, 1954

%\by P. Ungar

%\paper Freak theorem about functions on a sphere

%\jour J. London Math. Soc.

%\vol 29

%\yr 1954

%\pages 100--103,

%\mr \MR15--299

%\endref \vskip .1cm



\ref\key Whitney, 1957

\by  H. Whitney

\book Geometric Integration Theory

\publaddr Princeton Univ. Press, Princeton, N.J.

\yr 1957,

\mr \MR 19--309

\endref \vskip .1cm



%\ref \key Zaks, 1971

%\by J. Zaks

%\paper Nonspherical bodies with constant $HA$--measurements exist

%\jour Amer. Math. Monthly

%\vol 78

%\yr 1971

%\pages 513--516,

%\mr Zbl. {\bf 216},429

%\endref \vskip .1cm



\ref \key Ziemer, 1989

\by W. P. Ziemer

\book Weakly Differentiable Functions. Sobolev Spaces and Functions of

Bounded Variation

\publ Springer Verlag

\publaddr   New York

\yr 1989,

\mr \MR91e:46046

\endref \vskip .1cm

\endRefs



\enddocument 



