NEW ZEALAND JOURNAL OF MATHEMATICS Volume 29 (2000), 11-18

THE POMPEIU PROPERTY ON THE SPHERE

D.H. A r m it a g e

(Received December 1998)

Abstract. Let /j be a finite signed measure on the unit sphere S in Rn, where n > 3. It is shown that there exists a non-null continuous function f on S such that f s f o T dfi = 0 for every rotation T of Rn if and only if there is a non­negative integer m such that j s H d i — 0 for every homogeneous harmonic polynomial H of degree m on Rn. Some known results are recovered as special cases, and a recent theorem of Laquer and Ullrich about the Pompeiu property for sectors in the unit sphere of M3 is generalized to higher dimensions.

1. Introduction

Let a denote (n — l)-dimensional measure on the unit sphere 5'n_1 in Rn. Except where the contrary is stated, we assume that n > 3. As usual, C(Sn~1) denotes the space of real-valued continuous functions on 5n_1 and SO(n) denotes the group of rotations (orientation-preserving orthogonal transformations) of Kn. Let E be a compact subset of 5n_1. If there is no function / G C(S'n_1 )\{0} such that

[ f o T d a = 0 Je

for every T G SO (n), then we say that E has the Pompeiu property (PP). (For the origin of this terminology and a survey of the literature, see [11].) Several results are known about PP. Many years ago, Ungar [10] characterized spherical caps in S2 which have P P , and his result was generalized and extended to higher dimensions by Schneider [7]. More recently, Ullrich [9] improved a partial result of Laquer [5] to show that a sector (a set bounded by two great semicircles meeting at an angle less than tt) in S2 must have PP. The problem of identifying regular spherical polygons in S2 which have PP was discussed in [1].

It is convenient to introduce a more general notion. We shall say that a finite signed measure /i on Sn~1 has PP if there is no function / G C'(5'n_1 )\{0} such that

[ fo T d fi = 0 (1)

for every T G 50(n). For each m G N = {0 ,1 ,2 ,...} , we define Hm to be the vector space of all homogeneous harmonic polynomials of degree m on Rn, and we say that /i annihilates Hm if

j H dfi — 0Js" - 1

for all H G Hm-

1991 AMS Mathematics Subject Classification: 33D30, 31B05, 28A25.

12 D.H. ARMITAGE

Theorem 1.1. A finite signed measure n on Sn 1 fails to have PP if and only if // annihilates Hm for some m G N.

After proving this theorem, we shall show how several results follow from it.

2. Proof of Theorem 1.1

The elementary facts about harmonic polynomials that are used in the proof can be found in [3, Chapter 5].

The ‘if’ statement in the theorem is easy. The spaces Hm are invariant under rotations of IRn, so if (i annihilates Hm-, then (1 ) holds for all T G SO(n) and allf e n m-

To prove the ‘only if’ part, fix z G <S'n_1 and defineQ = {T G SO{n) : T(z) = z}.

Then Q is a compact subgroup of S'O(n); in fact Q is isomorphic to SO(n — 1). Let v denote Haar measure on Q, normalized so that v(G) = 1. For each / € C(Mn) (resp. C(Sn~1)) define / on Mn (resp. 5'n_1) by

f(x) = [ f oU{x) dv{U)Jg

and define Ff on SO(n) by

Ff (T) = / f 7 f ( x ) dfi(x).J 5 n_1

Recall that there is an element Im,z of 'Hm such that l m z o V — 7TO;2 for allV € Q and Im,z(z) = 1; moreover if H G H m and H o V = H for all V G Q, then H — H (z )ImjZ. For each x G Sn~1 let Tx be an element of SO(n) such that Tx(z) = x. The key to the ‘only if’ part of the theorem is the equation

[ Ff (Tx)H(x)da(x) — [ Im,zd[i [ fH da (H G Wm, m G N). (2)Jsn~1 JSn~x JSn~xTo prove (2), suppose first that / G Hk for some k. Since / o U is harmonic

for each U G Q, it follows that f is harmonic on Rn, and clearly / is homogeneous of degree k. Hence f E Hk- Also, by the invariance property of Haar measure, / o V = f for all V G Q and therefore / = f(z)Ik,z — f{z)Ik,z- If T G SO(n), then / o T EHk and the above argument shows that

f T f = fo T {z )I k,z.

Now suppose that / is a harmonic polynomial, no longer necessarily homoge­neous. Writing f = f 0 + fi + - - - + fj, where fk G Hk, we obtain that

jT ° t = £ / i » r ( 2) /M

k=0

for each T G SO(n). In particular,j

f °T X = £ /* ( * ) ! * , , k=0

THE POMPEIU PROPERTY ON THE SPHERE 13

L

and hence

Ff(Tx) = * > ( * ) [ Ik,z dfi (3)k=0

for each x G Sn~1. If H G Hm, then (3) and the orthogonality equation

GH da = 0 (G eH k ,H e H m,k ^ m)lSn-l

show that (2) holds.To establish (2) for an arbitrary function / G C(Sn~1), use the fact that f can

be uniformly approximated on 5n_1 by harmonic polynomials.To complete the proof of the theorem, suppose that there exists / G C'(S'n_1 )\{0}

such that (1) holds for each T G SO(n). Then

Ff (T) = [ [ foToU (x)du{U )dfi{x)J s ™~1 Jg

= f f foToU (x)d^{x)du(U ), (4)Jg J s ™-1

the change of order of integration being justified since the integrand is a continuous function of (U,x) on Q x Sn~1. The inner integral in (4) vanishes by hypothesis, so Ff(T) = 0 for all T G SO(n). Hence, by (2),

[ Im, z d [ fH da — 0 (H eH m ,m e N).J s n~1 J s ™-1

Recall that z is an arbitrary point 5n_1 and that for each m there exists a finite subset Em of 5n_1 such that {Im,z '• z € Em} forms a basis of Hm- It follows that

[ G d tx f fH d a = 0 (G, H G Hm, m G N). (5)J s ™-1 J s ™-1

There exists a sequence (hj) of harmonic polynomials converging uniformly to / on Sn~1, and hence

[ h jf da -> [ f 2 da ± 0 (j -»• oo),J s ™-1 J s ™-1

so there exists some homogeneous harmonic polynomial H such thati fH da / 0,s™—1

and (5) implies that /i annihilates Hdeg h -

3. Special Cases

The first special case that we discuss is that in which the measure /i in Theorem 1.1 is invariant under rotations about a fixed axis. Recall that a typical point of S'71-1 can be represented in the form (t,£\/l — t2), where t G [—1,1] and £ G Sn~2. We use the standard notation (see, for example, Szego [8]) for ultraspherical polynomials, and we denote (n — 2)-dimensional measure by r.

14 D.H. ARMITAGE

Theorem 3.1. Let be a finite signed measure on [—1,1]. There exists a function f G C(S'n~1 )\{0} such that

[ f foT(t,£y/\ - t2)dr(€)din{t) = 0 (6)[—1,1] JSn~2

for every T G SO (n) if and only if

[ P « ’*-2)/2)(t)d/*i(t) = 0 (7)J[-1 ,1]

for some m G N.

Specializing further, we recover some known results as corollaries. We denote a typical point of Mn by x = (x i ,... , xn) and define

Aa = {x G S'” - 1 : X\ = a}, Ca = {x G 5n_1 : X\ > cn}, (8)

where a G [0,1).

Corollary 3.2. There exists a function f G C'(5n_1 )\{0} such that

[ f oT dr = 0 JAa

for every T G SO(n) if and only if Pmn~2 2\oi) = 0 for some m G N.

Corollary 3.3. The spherical cap Ca has PP if and only if Pm ^ (a) ^ 0 for all m G N.

Corollary 3.2 is due to Schneider [6]. Corollary 3.3 was proved by Ungar [10] for n = 3 and by Schneider [7] for n > 3.

Finally we generalize to higher dimensions the result of Laquer [5] and Ullrich[9] stating that sectors have PP. We call a subset E of 5n_1 a sector if E has the form

E = {((,£^1 - t 2) : -1 < t < 1,£ € D }, (9)where D is a spherical cap (arc if n = 3) given by

D = {£ e S’*"2 : ft > 0 } (10)for some (3 G (0,1).

Theorem 3.4. Every sector in Sn~1 has PP.

4. Proofs of Theorems 3.1 and 3.4

We need to use some results about harmonic polynomials. First we modify the notation used in Section 2. In the case where 2 = (1 ,0 ,... ,0) G Mn, we write Jm,n = Im,z and call Jm,n a zonal harmonic. We also define polynomials Jm,n+2p on Mn by

t m,n+2p(a' l » • • • ixn) ~ Jm,n+2p{x li ■ • • )0);here p G N and there are 2p zeros on the right-hand side. Let H* be the subspace of Hp given by

n ; = {H G Tip : dH/dxi = 0}.

THE POMPEIU PROPERTY ON THE SPHERE 15

The elements of H* can be identified with homogeneous harmonic polynomials of degree p on En_1 in an obvious way. If 0 < p < m and H G Hp, then HJ^_pn+2p G Hm (see e.g. Kuran [4, Theorem 2]). In the case where 0 < p < m, we call such a function HJ}n_p n+2p a tesseral harmonic; in the case where p = 0 this function is, up to a multiplicative constant, simply the zonal harmonic Jm,n• It is known (see e.g. [4, Theorem 3]) that Hm, where m > 1 , has a basis consisting of tesseral harmonics together with the zonal harmonic Jm,n- Note that if G = HJ^_p n+2p is a tesseral harmonic (with H 6 Hp), then for all t G [—1,1]

This follows from the mean value property of harmonic functions applied to the harmonic polynomial on Mn_1 given by h H(0,y). We shall also need the following equation relating zonal harmonics to ultraspherical polynomials:

(-1 < t < 1,£ G Sn~2). (13)We can now prove Theorem 3.1. By Theorem 1.1, there will exist a function

/ G C'(S'n -1 )\{0} satisfying (6) for every T G SO(n) if and only if there exists m G N such that

for every G G TLm■ If G is a tesseral harmonic, then (1 1 ) implies that (14) holds. Since Hm(m > 1) has a basis consisting of tesseral harmonics together with JTOjn (and {1} = { Jo,n} is a basis for H0), it follows that (14) will hold for every G G Hm if and only if it holds with G — Jm,n■ In view of (12), this condition is equivalent

Corollary 3.2 follows by taking /ui to be a point measure at a.To prove Corollary 3.3, we choose so that fii = (j)X\, where Ai denotes

Lebesgue measure and

(t, y jl - t2,0, . . . . 0) f H ( 0 , t V l - t 2)dr(0Jsn~2

= 0. ( 11)

(12)

(see e.g. [2, p. 477]). From this it follows that

(

[ - 1 ,1] J s n~2(14)

to (7).

(j){t) = (1 - £2)(n- 3)/2 ( a < * < l ) , <l>{t)= 0 (—1 < t < a).

Condition (6) is equivalent to

16 D.H. ARMITAGE

and by Theorem 3.1 this will hold for some / £ C(S'n_1 )\{0} and every T £ SO(n) if and only if (7) holds for some m £ N. That is to say, Ca fails to have PP if and only if

p^(n-2)/2) ^ _ ^ (n_ 3)/2 dt = Q (15)*

for some m £ N. Rodrigues’ formula (see, e.g. [8, formula (4.7.12)]) states that

n f2U -ip (A )m _ (~2)m r(m + A)r(m + 2A) 2 r +A-i/2(1 - 0 2pm ( t ) _ — _ _ _ _ _ _ _ _ 2^) _ ( i _ t )

(-1 < t < 1, A > —1/2). (16)Applying this formula with A — (n — 2)/2, we see that (15) holds if and only if a is a zero of the function

{ d r - 1/dtrn~1){\ - t2)m—l+(n—1)/2)

and by (16) with A = nf 2, this is equivalent to the condition that Pin'? (a ) = 0. The proof of Theorem 3.4 requires two simple lemmas.

Lemma 4.1. Suppose that n > 2 and that Ca is the spherical cap given by (8) with 0 < a < 1. There exist elements Hj £ Hj(j = 1, 2) such that

[ Hj da ^ 0. (17)Jca

To prove Lemma 4.1, we defineHi(x) = xi, H2(x) = ( n - l ) x \ - ( x l + . . . + x2n).

Then Hj £ Hj (j = 1,2). Clearly (17) holds with j — 1. With j — 2 (17) is equivalent to the assertion thatfJo

((n — 1) cos2 9 — sin2 9) sinn 2 9 dd ^ 0,'o

which is true since the integrand is (d/d0)(cos0sinn-1 9).

Lemma 4.2. If k is a non-negative even integer and A > 0, then

y % f +1)( t ) ( l - i y - i dt ± 0 (18)

and

J dt f o. (19)

Denote the integral in (18) by a By [8, formula (4.7.29)]

ak+2 = ak + A *(& + 2 + A) J P \ t)(l — t2) 5dt = .

Hence a0 — — 04 = .. •, and obviously a0 ^ 0.To prove (19), note that by (16) the integral in (19) is

C f {1 - t2)~^2{dk/dtk){ 1 - t2)k+x~^2dt,

THE POMPEIU PROPERTY ON THE SPHERE 17

where C is a non-zero constant depending on k and A. Integrating the last-written integral by parts k times, we find that, with k even, it is equal to

J 1 (1 - t2)k+x~1/2(dk/dtk)( 1 - t2)~1/2dt. (20)

An induction argument shows that the derivative in (20) is a linear combination with positive coefficients of terms of the form t2p( 1 — t2)~q~1' 2, where p, q € N, so that the integrand in (20) is positive on (—1,1) and hence (19) holds.

We can now complete the proof of Theorem 3.4. Let E, D be given by (9), (10). By Theorem 1.1, it is enough to show that each space Hm contains an element G such that

[ Gda ± 0. (21)J E

This is trivial when m — 0,1, and we now suppose that m > 2. If G is a tesseral harmonic in Hm given by G = -#Vm-p,n+2p> where H EH*, then

f G da J e

= J J G(t, ?v /T ^ ) ( i - *2)< "-3>/2dT(e) dt

= f ,0)(1 - t ^ ~ 3^ H (0 ^ V T ~ fi)d r (O dt

= ,0)(1 - t2Yn~3+p)/2dt J^H {0,O dr(O - (22)

If m is even (resp. odd) we choose p = 2 (resp. p = 1) and take H to be an element of H* such that

f H(0,Odr(O / 0,JD

which is possible by Lemma 4.1 (with n — 1 in place of n). By (13) the first integral in (22) is

( m™ _ 2 x) 1 L p" - 2+2)/2)(<)(i ■ e r ~ l ) l 2 d t {m even)

( m m - l 2) 1 f \ p ^ m - t 2f n- 2)l2dt (m odd).

By Lemma 4.2, these integrals are non-zero, so (21) holds, as required.

References

1 . D.H. Armitage, The Pompeiu property for spherical polygons, Proc. Royal Irish Acad. 96 A (1996), 25-32.

2. D.H. Armitage and U. Kuran, Sharp bounds for harmonic polynomials, J. Lon­don Math. Soc. (2) 42 A (1990), 475-488.

3. S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Springer; New York 1992.

18 D.H. ARMITAGE

4. U. Kuran, On Brelot-Choquet axial polynomials, J. London Math. Soc. (2) 4 (1971), 15-26.

5. H.T. Laquer, The Pompeiu problem, Amer. Math. Monthly, 100 (1993), 461- 467.

6. R. Schneider, Functions on a sphere with vanishing integrals over certain sub­spheres, J. Math. Anal. Appl. 26 (1969), 381-384.

7. R. Schneider, Uber eine Integralgleichung in der Theorie der convexen Korper, Math. Nachr. 44 (1970), 55-75.

8. G. Szego, Orthogonal Polynomials, Amer. Math. Soc.; Providence, Rhode Is­land 1967.

9. D.C. Ullrich, More on the Pompeiu problem, Amer. Math. Monthly, 101 (1994), 165-168.

10. P. Ungar, Freak theorem about functions on a sphere, J. London Math. Soc. 29 (1954), 101-103.

11. L. Zalcman, A bibliographic survey of the Pompeiu problem, in Approximation by Solutions of Partial Differential Equations (In B. Fuglede et al., eds), NATO ASI Series C, Vol. 365, Kluwer, Dordrecht, 1992, pp. 185-194.

D.H. ArmitageDepartment of Pure Mathematics Queen’s University Belfast BT7 INN.Northern IrelandUNITED KINGDOMd. armitage@ Q ueens-Belfast. AC. UK