ZERO INTEGRALS ON CIRCLES AND CHARACTERIZATIONS OF ...

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 317, Number I, January 1990

ZERO INTEGRALS ON CIRCLES AND CHARACTERIZATIONS OF HARMONIC AND ANALYTIC FUNCTIONS JOSIP GLOBEVNIK ABSTRACT. We determine the kernels of two circular Radon transforms of continuous functions on an annulus and use this to obtain a characterization of harmonic functions in the open unit disc which involves Poisson averages over circles computed at only one point of the disc and to obtain a version of Morera's theorem which involves only the circles which surround the origin.

1.

INTRODUCTION

Suppose that 1 is a continuous function on the open unit disc .1. in the complex plane. For each simple closed curve r c .1. bounding a domain D, OED, let Fr be the function which is continuous on D U r, harmonic in D and which coincides with 1 on r. It is known that if Fr(O) = 1(0) for each smooth curve r bounding a strictly convex domain then 1 is harmonic in .1.. This is a special case of the main result of [5]. The starting point of the present investigation was the fact that the function 1 is not necessarily harmonic if one assumes only that Fr(O) = 1(0) for each circle r c .1. surrounding the origin. In fact, for each kEN there is a function 1 of class ~k on .1. such that Fr(O) = 1(0) for each circle r c .1. surrounding the origin and which is not harmonic in .1. ([5], see also §9). One of the results of the present paper is that if Fr(O) = 1(0) for each circle r c .1. which surrounds the origin then 1 is harmonic in .1. under the additional assumption that it is infinitely differentiable at the origin, that is, if for each n E N there is a polynomial Pn of degree n such that (z E .1.)

where lim=_o Izl-nQn(z) = O. In fact, to get harmonicity it is enough to assume only that Fr(O) = 1(0) for each circle r belonging to a family which is only slightly larger than the family of circles in .1. centered at the origin. Note that 1 is infinitely differentiable at the origin if it is of class ~oo in a neighbourhood of the origin, or more generally, if it belongs to ~oo ({O}) , that is, if for each kEN there is a neighbourhood of the origin in which 1 is of class ~k . Received by the editors February 15, 1988 and, in revised form, June 20, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 30E20, 31 A 10; Secondary 53C65. © 1990 American Mathematical Society 0002·9947/90 $1.00 + $.25 per page

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JOSIP GLOBEVNIK

This characterization of harmonic functions is a consequence of our first main result which describes the Fourier coefficients of the continuous functions on an annulus Q = {' E C: RI < 1'1 < R 2 } which have zero average on each circle r c Q that surrounds the origin. The second main result describes the Fourier coefficients of the continuous functions f on Q such that fr f( z) d z = o for each circle that surrounds the origin. Its consequence is a version of Morera's theorem: If f is continuous on d and infinitely differentiable at the origin and if fr f(z) d z = 0 for each circle red which surrounds the origin then f is analytic in d. Again, one cannot drop the smoothness requirement at the origin since for each kEN there is a function f of class !(!k on d such that fr f( z) d z = 0 for each circle red surrounding the origin and which is not analytic in d ([3], see also §9).

2.

g

THE MAIN RESULTS

Let 0 :s RI < R2 and let f be a continuous function on the annulus Q = E C: RI < 1'1 < R 2}. For each r, RI < r < R 2 , and for each n E Z , let

I,,(r) =

2~ fo27C e-inof(reiO)dO.

Thus, for each r, RI < r < R 2 , 2:::'00 I" (r)e inO is the Fourier series of f(re iO ). Let r = {' E C: I' - zl = r} and let e > O. The e-neighbourhood of the circle r is the family of all circles of the form {' E C: I' - wi = p with Iw - zl < e and Ir - pi < e, p > O. Let JJ be a family of circles in Q which surround the origin such that JJ contains a neighbourhood of each circle r c Q centered at the origin. Note that JJ may be only slightly larger that the family of all circles in Q centered at the origin. 7C f(a + eiob)dO = 0 then we will If r = {a + eiob: O:S 0 < 2n} and if say that f has zero average on r.

fg

Theorem 1. Let f be a continuous function on Q. Thefollowing are equivalent: (i) f has zero average on each circle

r E JJ,

(ii) f has zero average on each circle r c Q which surrounds the origin, (iii) fo(r) = 0 (RI < r < R 2 ) andfor each n E Z, n =f. 0, there are numbers

ano,a nl , ... ,an,lnl-1 such that

f.n(r)=r -Inl (ano+anir 2 +···+an,lnl_I r2(lnl-l) ) Theorem 2. Let f be a continuous function on Q. The following are equivalent:

fr f(z) d z = 0 for each circle r E JJ, fr f(z) d z = 0 for each circle r c Q which surrounds the origin, (iii) LI(r) = 0 (RI < r < R 2) and for each n E Z, n =f. -1, there are (i) (ii)

numbers bnO,bnl , ...

fn(r)=r

-Inl

b

b

2

,bn,ln+II-1

such that

2(in+II-I)

(nO+ nl r + ... +bn,ln+ll-i r


)

ZERO INTEGRALS ON CIRCLES

315

Let 71' be a family of circles in d which surround the origin such that 71' contains a neighbourhood of each circle red centered at the origin.

Corollary 1. Let f be a continuous function in d which is infinitely differentiable at the origin. If Fr(O) = f(O) for each circle r E 71' then f is harmonic in d. In particular, if Fr(O) = f(O) for each circle red surrounding the origin then f is harmonic in d. Corollary 2. Let f be a continuous function in d which is infinitely differentiable at the origin. If frf(z) dz = 0 for each circle r E 71' then f is analytic in d. In particular, if fr f(z) d z = 0 for each circle red surrounding the origin then f is analytic in d. An easy consequence of Theorems 1 and 2 is the following support theorem for the circular Radon transforms f 1-+ frf(z) ds and f 1-+ frf(z) dz which probably has a simple direct proof:

Corollary 3. Let f be a continuous function on Q such that for each kEN the function z 1-+ (R2 -Izl)-k f(z) is bounded as Izl-+ R 2 . If f has zero average on each circle r E:§' then f vanishes identically on Q. If fr f( z) d z = 0 for each circle r E:§' then f vanishes identically on Q. The paper is organized as follows. We first prove that Corollaries 1-3 follow from Theorems 1 and 2 (§3). Then we show that frf(z) ds = 0 or frf(z) dz = o for each circle r E Q surrounding the origin if and only if the same holds for each function re i(} 1-+ fn(r)e in (} (§4). This happens if and only if in each case fn satisfies a Volterra integral equation of the first kind whose kernel has a weak singularity on the diagonal (§5). We look at the properties of these equations and iterate the kernels to get equations for the functions r 1-+ In (r)(ro _ r)-1/2 with analytic kernels having zeros of order In I and In + lion the diagonal (§6). Since only the structure of bounded solutions of such equations has been studied in detail [9, 12] with only a remark being made in [9] about the general case we revisit [9] to show that the approximation procedure used there for bounded analytic solutions can be used also for unbounded smooth solutions (§§7, 8). We then present examples of functions satisfying frf(z) ds = 0, fr f( z) d z = 0 and show that using .these examples one gets all solutions of the original integral equations and thus complete the proofs of Theorems 1 and 2 (§9). 3. PROOFS OF THE COROLLARIES

Proposition 1. Let Dee be an open disc, 0 ED, and let r be its boundary. Assume that F is a continuous function on D U r which is harmonic in D. Then F(O) = 0 if and only if the function z 1-+ F(z)/lzI 2 has zero average on r.


JOSIP GLOBEVNIK

316

e

Proof. Let r = {w + Rei : 0 ::; < 2n}, w f/J < 2n then the Poisson formula gives ~

o ::;

.

=

re la . If 0

< P < 1 and

1 1211 F(w + Re it )(1 - p2) d F( w+pe R iffJ) = t. 2n 0 1-2pcos(t-f/J)+p2 Putting p

= rlR

= a + n we get F(re ia + (rIR)Rei(a+rc)) _1 (211 F(re ia + Re it )(1

and f/J

F(O) =

=

2n

10

- p2)

dt

1-2(rIR)cos(t-a-n)+r2IR 2

= (R2 _ r2)_I_ (211 F(re ia

2n

10

Ire la

+ Rei!) dt.

+ Re ll 12

This completes the proof.

I is a continuous function on L1 which is infinitely differentiable at the origin and which satisfies Fr(O) = 1(0) for each circle r E J1t'. It is enough to prove that I is harmonic in RL1 for each R < 1 so we may assume with no loss of generality that I extends continuously to the closure Ll of the unit disc. Let g be the function which is continuous on Ll, harmonic in L1 and which coincides with I on the unit circle bL1 and put h = I - g. The function h is continuous on Ll and vanishes identically on bL1. Since g is harmonic in L1 it follows that h is infinitely differentiable at the origin and that Hr(O) = 0 for each circle r E J1t'. Proposition 1 now implies that the function Z 1-+ w(z) = h(z)/lzI 2 has zero average on each circle r E J1t'. By Theorem 1 it follows that wo(r) = 0 (0 < r < 1) and that for each n E Z, n f= 0, there are numbers ani' 0::; i ::; Inl - 1 , such that Proolol Corollary 1, assuming Theorem 1. Suppose that

W

n

(r) = r

Since hn(r) r

-Inl

-tnt

(a nO

= r 2w n(r)

h n (r) = r

-21nl

+ a nl r 2 + ... + a n .Inl-l r 2(lnl-l) )

(O (t) = In(tR) satisfies (5.4)

1)

(t)Pn(s,t)dt = 0

(R)/R o (r)e- iO (RI < r < R 2 , 0::; e < 2n) and if ~'I=r \f'(z)dz = 0 for each r, RI < r < R 2 , then cf>(r) = 0 (RI < r < R 2) . This shows that we will only have to consider (5.2) if n

n::f. -1.

::f.

0 and (5.3) if

Let Pn and Qn be as in Lemma 3.

Lemma 4. Let 0 < r < 1 and let cf> be a continuous function on [1 - r , 1] . (i) If n E Z, n::f. 0, and if cf> satisfies

(6.1 )

11

cf>(t)Pn(s, t) dt

= 0

(1-r satisfies (6.2)

11

CI>(t)Qn(s, t) dt

= 0

(l-r 0 is small enough then the following are equivalent (8.1 ) (i) g is a smooth solution of (7.4) such that g(S)SI/2 is bounded on (0, r), (ii) g = go + LngO + L~go + ... , where go satisfies Dgo = 0 on (0, r) and is such that go(X)X I/ 2 is bounded on 1:r and where the series converges uniformly on 1:r . Proof. Let r > 0 be so small that Lemma 6 holds and that in 2rd every Pj(x)(logx/ where Pi' 0::; j ::; m, solution of Dy = 0 has the form xr are analytic in 2rd. Suppose that (i) holds. By Lemma 6, go = g - Lng is smooth on (0, r) and such that gO(S)SI/2 is bounded on (0, r). By the definition of Ln we have

2:7=0

(Dgo)(x)

=

(Dg)(x) - (D(Lng))(x)

=

(Dg)(x) -

fox Kn(x ,s)g(s) ds = 0

(0 < x < r)

since g satisfies (7.4). Since go(X)X I/2 is bounded on (0, r) it follows that it is bounded in 1:r • Since go is analytic there Lemma 6 implies that the series (8.1) converges uniformly in 1:r and since go = g - Lng it follows by Lemma 6(ii) that the sum of the series (8.1) is g. Conversely, suppose that (ii) holds. Write L~go = gi so that g = 0 gi' By the uniform convergence of the series in 1:r we have D g = D gi on 0 (0, r) and

2::

for Kn(x ,s)g(s)ds = ~ fox Kn(x ,s)gi(s)ds

2::

(0 < x < r).

Since for each i, (0 < x < r)

it follows that g is a solution of (7.4). That g(S)SI/2 is bounded on (0, r) follows from Lemma 6. This completes the proof.

Lemma 8. If r is small enough then the dimension of the space of all smooth solutions g of(7.4) such that g(S)SI/2 is bounded on (0, r) does not exceed n. Proof. Let r > 0 be so small that Lemmas 6 and 7 hold. If go is a solution of Dgo = 0 such that go(X)X I/ 2 is bounded in 1:r then by Lemma 6 the series (8.1) converges uniformly in 1:r • Let y I 'Y2' ... ,ym be the basis of the space


327


of all solutions of Dy = 0 for which y(X)X 1/ 2 is bounded on 1:,. For each i, 1 :::; i:::; m, let Rl/R then it follows by Lemma 3 that on [q, 1] the function t ~ fn(tR) is a lbear combination of k' 1 ~ k ~ n. As this holds for every R, R) < R < R 2 , the proof is complete. Proposition 5. If n E Z, n ;::: 0, then each of the functions Fk (z) = zn-k z-k ,

n , satisfies If Fk (z) d z = 0 for each circle r surrounding the origin. If n E Z, n ~ -2 then each of the functions Gk(z) = zk zn+k, 0 ~ k ~ -n - 2, satisfies If Gk (z) d z = 0 for each circle r surrounding the origin. Proof. Let n ;::: 0 and let 0 ~ k ~ n. Let 0 < lal < Ibl and let r = {a + iO n-k -k e b: 0 ~ e < 2n}. If Fk (z) = z z then

o~ k

~

i

Fk(z) d z =

!a 27C (a + /0 bt- k(a + e -iob) -k ibe io de

=b =b

1

27C

o

r

1M

e ikO( a + e iOb)n-k(-ae iO

wk(a + wb)n-k(aw

+ b-)-k.Ie iO de

+ b)-k dw

=

0

since the integrand in the last integral is analytic in a neighbourhood of X. Let n ~ -2. Write n = -m and let Gk(z) = zkz-m+k, 0 ~ k ~ m - 2. We have

i

Gk(z) d z

=

!a 27C (a + e-iob)k (a + e iO b)-n1+k ibe io de _!a 27C (a + e iO b{ (a + e -iob)-m+k ibe -iO de _!a 27C (a + e iO b{ /(tn-kjO (ae iO + b)-m+k ibe -iO de

-l (

- b

M

a+w b)k w m-k-2(_aw + b-)-m+k d w = 0

since the integrand in the last integral is analytic in a neighbourhood of X.

Example [3]. If n

E

N then the function rp(z)=

{

z-lzn+2 (z=l=O), 0 (z=O)



329

is of class ~n on C. By Proposition 5 we have fr qJ(z) dz = 0 for every circle r that surrounds the origin, yet qJ is not analytic in Ll. Proof of Theorem 2. It is enough to prove the equivalence of (ii) and (iii). Let

f be a continuous function on n. If f satisfies (iii) then by Proposition 5 for

each n E Z the function re iO I--> '¥n(re iO ) = J,,(r)e inO satisfies fr '¥n(z) dz = 0 for each circle r en surrounding the origin so by Lemma l' f satisfies (ii). Suppose that f satisfies (ii). By Proposition 2, LJ (r) = 0 (R J < r < R 2 ). Let n E Z, n =I- -1. If n 2: 0 then by Proposition 5 and Lemma 2 each of the functions k(t) = t-lnl+2k, 0 S k S n, satisfies (5.3). As in the proof of Theorem 1 it follows that if R J < R < R2 and if q > RJ/R then on [q, 1] the function t I--> fn(tR) is a linear combination of k' 0 S k S n. As this holds for every R, R J < R < R 2 , (iii) follows for n 2: O. If n S -2 then by Proposition 5 and Lemma 2 each of the functions Cl>k(t) = t-lnl+2k , o S k S Inl - 2, satisfies (5.3) and again (iii) follows for n S -2. This completes the proof.

Acknowledgements. The author is indebted to Professor Ivan Vidav for helpful

and stimulating discussions. In particular, when the author told him that he proved Theorem 1 for functions belonging to ~oo ({O}) Professor Vidav proposed the appropriate smoothness assumption at the origin. A major part of this work was done during the month of November 1987 which the author spent at the Research Institute for Mathematical Sciences of Bar-Han University, Ramat Gan, Israel. He wishes to thank Professor Lawrence Zalcman for the invitation and for several very interesting discussions. This work was supported in part by the Republic of Slovenia Science Foundation. REFERENCES

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II. V. G. Romanov, Integral geometry and inverse problems for hyperbolic equations, Tracts in Natural Philos., Vol. 26, Springer, Berlin, 1974. 12. V. Volterra and J. Peres, Theorie generale desfonctionelles, Gauthier-Villars, Paris, 1936. INSTITUTE OF MATHEMATICS, PHYSICS AND MECHANICS,

E. K.

UNIVERSITY OF LJUBUANA,

LJUBUANA, YUGOSLAVIA

Current address: Department of Mathematics, University of Washington, Seattle, Washington 98195
