A R K I V F O R M A T E M A T I K B a n d 7 n r 25 1.67 06 3 Communicated 12 April 1967 by OTTO FROSTMA~
Overconvergence of sequences of rational functions with sparse poles
By HAROLD S. SHAPIRO
I n this p a p e r we consider B a n a c h spaces S of functions holomorphic in D + = {z II z I < 1} which contain all functions { i / ( z - b)) with b E D - = {z I[ z I > 1}. W e show (Theorem 1) t h a t u n d e r r a t h e r mild restrictions on S, a sequence {/~) of r a t i o n a l functions which converges in the n o r m of S, if t h e poles of a l l / n are confined to a
" s p a r s e " subset E of D - (here the sparseness criterion is d e t e r m i n e d b y the parti- cular space S, a n d we give it only in a n implicit form), necessarily converges uni- f o r m l y on c o m p a c t subsets of
D-\E.
This it seems a p p r o p r i a t e to call a n " o v e r - c o n v e r g e n c e " theorem, a l t h o u g h it has a s o m e w h a t different c h a r a c t e r t h a n o t h e r t h e o r e m s bearing this designation, e.g. those due to Ostrowski [3] a n d t o W a l s h ([5], p. 77). I n general t h e sequence {/~} will not converge in d o m a i n s which inter- sect t h e unit circumference, a n d the limit functions in D + a n d D - are not a n a l y t i c continuations of one another. I n T h e o r e m 2 it is shown, w i t h additional h y p o t h e s e s o n S, t h a t w h e n the closure of E does not include the entire unit circumference we g e t overconvergence in d o m a i n s which intersect the circumference, a n d hence ana- lytic c o n t i n u a b i l i t y of the limit functions. These t h e o r e m s are p r o v e d in w 1.The p r e s e n t p a p e r was m o t i v a t e d b y a s t u d y of the p a p e r [1] of Akutowicz a n d Carleson, f r o m which we h a v e a d a p t e d the m e t h o d of proof of T h e o r e m 2. Conver- sely, m a n y of t h e t h e o r e m s of [1] are deducible f r o m our T h e o r e m 2. I n w 2 t h e relationship of our p a p e r with [1] is discussed briefly. I n w 3 some p r o b l e m s which invite f u r t h e r investigation are p o i n t e d out.
T h e a u t h o r wishes to acknowledge some helpful conversations with D. J. N e w m a n a n d A. L. Shields.
1. The main theorems
W e consider a B a n a c h space S of functions
/(z)
holomorphic inD +,
a n d satis- f y i n g the following conditions:(1) 1 E S, a n d for e v e r y [b I > 1, 1/(z - b) is a n e l e m e n t of S. T h e set of all functions 1/(z - b) with I b[ > 1 is t o t a l in S.
(2) F o r e v e r y l a[ < 1, t h e e v a l u a t i o n functional Ia defined b y
Ia[=[(a)is
bounded.
(3) lira b-.~r z----b = 0 . 1
I-L S. SHAPIRO, Sequences of rational functions
(4) F o r e v e r y
I bl > 1,
multiplication b y 1/(z - b) is a b o u n d e d o p e r a t o r Mb f r o m S t o S.(5) IIMb ]] is b o u n d e d on c o m p a c t subsets of D .
Observe t h a t m o s t B a n a c h spaces of a n a l y t i c functions in D + p r e v i o u s l y studied (in p a r t i c u l a r t h e H v spaces) satisfy these properties, as well as t h e e x t r a ones to be i n t r o d u c e d below in the h y p o t h e s e s of T h e o r e m 2.
Theorem
1. Let E be any subset o/ D - such that the set U o / / u n c t i o n s 1 / ( z - b ) with b E E is not total in S. Let R denote the set o/ /inite linear combinations o/elements of U. Then, i/ /~ is a norm convergent sequence o/ elements o / R , the sequence/n(z) converges uni/ormly on compact subsets o/ D \ E .Remark. I t would be e a s y to e x t e n d this result to allow also [ b [ = 1, if t h e corre- sponding 1 / ( z - b) belong to S.
Proo/ o/ Theorem 1. L e t / E R ,
II/ll
= 1. W r i t e / ( z ) = ~ _ l p k / ( z - - b k ) , b k e E . L e t G be a c o m p a c t subset of D - \ E . T o p r o v e t h e theorem, it is sufficient to show t h a t t h e m a x i m u m of I/(z)[ on G is b o u n d e d b y a c o n s t a n t n o t depending u p o n t h e p a r t i - cular / E R of n o r m 1. Indeed, this t h e n implies t h a t t h e m a x i m u m of I/re(z) - / , ( z ) I on G is less t h a n a c o n s t a n t times I ] / m - / n I], which clearly implies the desired result.Now, since { 1 / ( z - b ) } , b E E are not t o t a l there is a non-null linear functional L E S* such t h a t
L(1)
2(w) = Z ~ (4 is the " B o r e l t r a n s f o r m " of L)
vanishes on K . B y (1), ~t(w)~ 0. Also, it is easily deduced f r o m our h y p o t h e s e s t h a t is holomorphic for I w[ > 1 a n d vanishes a t ~ . N o w fix a point c E D - \ E such t h a t 2(c):t: 0 a n d define
L M c Qc = 2(c);
clearly Q~ is a b o u n d e d linear functional on S, a n d
N o w , for b E E we h a v e
II Q=II IILll. l[ Moll (6)
I (c)l
L 1 c z - b ) - 2 ( c )
Therefore, Qc(1/(z - b)) = 1/(c - b), hence
i n 1
/ ( e ) l = = 1 c - b~ = k = 1
P Qo(- -I=IQ /t<IIQ II.
\ z - ok/I 344ARKIV FOR MATEMATIK. B d 7 n r 2 5 Now, we m a y s u r r o u n d G b y a c o n t o u r on which 2(w) does n o t vanish, a n d which contains no points of E. B y t h e last inequality, t o g e t h e r with (6) and (5), we o b t a i n a b o u n d for / on the contour, a n d so b y t h e m a x i m u m m o d u l u s t h e o r e m o n G. This proves T h e o r e m 1.
Theorem 2. Suppose for some p > 1 S satisfies, in addition to (1)-(5), the additional .conditions
fl(=
(log+log+Hl~ll)Vrdrdt< (a=re~t), (7)d o d o
f
Q (log + log + N(x)) p dx < ~ (for e v e r y 1 < Q < ~ ) , (S) +where N(x) satis]iesII M II < N(I b l)
]or I b l > 1.Then, u n d e r the h y p o t h e s e s of T h e o r e m 1, {]~(z)} converges u n i f o r m l y on c o m p a c t subsets of Z \ E ( Z = R i e m a n n sphere).
Remark. T h e o r e m 2 gives new information in the case t h a t E does n o t include I z I = 1, assuring analytic c o n t i n u a t i o n of lira t~.
W e require first a lemma.
L e m m a . Let F(z) 8ome q < 1
be holomorphic in D +, and write
M(r)=maxt [F(reU)l.
I / / o rf~q[log+log+M(r)]Pdr<~, where p>~l, (9)
t h e n log + log + rdrdt < ~ (10)
Pro@ I t is easy to reduce the general case to t h e case w h e n F(0) = 1, which we assume. N o w
- -1 ~< - log I F(rdt)] + log + M(r).
I n t e g r a t i n g a n d recalling t h a t F(0) = 1, we get
2~r J 0 l~ IF(re't)} dt <~ log + M(r). (11) L e t us first suppose t h a t p ~< 3/2. Now, the function (log x) v is concave for x > e p-1 (and so for Ix [ >~ 2), hence using J e n s e n ' s c o n v e x i t y inequality
Ho S. SHAPIRO, Sequences of rational functions
r [log
~< log ~ 2 + l o g + iF(rdt)l dt , a n d this is, b y (11),
~< [log (2 + log + M(r))] p ~< [log 3 + log + log + M(r)] ~'
~< 2"[(log 3) p + (log + log + M ( r ) ) ' ]
a n d now, observing (9), (10) follows readily, The case p > 3 / 2 involves a trivial modification of the a r g u m e n t , which we leave to the reader.
Remark. I t is clear, b y the change of variables z = 1/w, t h a t the l e m m a is t r u e also u n d e r the hypothesis t h a t F is holomorphic in D , where n o w the range of integration in (9) is from 1 to Q, a n d in (10) over the annulus 1 < r < Q. I t is in this.
f o r m t h a t we shall a c t u a l l y e m p l o y the lemma.
Proo/ o/ Theorem 2. We shall base the proof on a t h e o r e m of Beurling (see [2]) according to which, if we have a family of functions holomorphic in a d o m a i n H a n d there exists a function J(z) such t h a t J(z) >~l/(z)l for all / in the family, w h e r e for some p > 1
f f [l~176 (12)~
t h e n the functions of the family are u n i f o r m l y b o u n d e d on c o m p a c t subsets of H.
Again, we consider the family R o of all /E R of n o r m one. N o t e first that, for
aED*,
I/(a)l II If. (13)
Now, when I c[ > 1, as we have seen
AiN(Ict}
It(c)1 I z( )l ' where
A I=IILll.
{14)I n w h a t follows A~, A s . . . . denote positive constants. Define J(a)to be [l/all for- [a[ < 1 a n d g(c) = r i g h t - h a n d side of (14) for Icl > 1. Clearly I/(z)] <g(z) f o r / E R o . Moreover, for [c[ > 1,
1 log + log + g(c) <~ A 2 + log + log + N(] c I) + log + log + 12(c)l' therefore
1 P
L e t n o w H be a n y b o u n d e d open subset of Z \ E . All / E Ro are holomorphic on H._
ARKIV FOR MATEMATIK. Bd 7 nr 25 Moreover (7) guarantees tha~ the contribution to the integral (12) coming from H N D + is finite. Likewise the contribution coming from H N D - is finite, as we see from (15), together with (8), providing we can show, for any Q > 1
0 J1 l ~ 1 7 6 r d r d t < ~ (c=re~t). (16)
But,
I~(c)]= (~-~)
L 1~IILII'I]M~]I~AIN(Icl).
Again observing (8), and applying the lemma (note the r e m a r k following its proof) we see t h a t (16) holds. Therefore, the functions in Ro are uniformly bounded on compact subsets of Z \ E , and this implies Theorem 2.
2. T h c A k u t o w i c z - C a r l e s o n m i n i m u m p r o b l e m
In [1] the following type of problem is studied: One has a Banaeh space F of functions holomorphic in a domain D, and two given complex sequences {zn}, zn E D and {an} such t h a t the interpolation p r o b l e m / ( z , ) = an is underdetermined, i.e. there exist two distinct functions (and hence infinitely many) in F which satisfy/(zn) = an, n = 1, 2 . . . . Then, under suitable hypotheses about F, which are moreover sufficient to guarantee the existence of a unique solution of the interpolation problem which has m i n i m a l norm (for instance, uniform convexity of F guarantees this), it is shown in [1] t h a t this minimal solution is analytically continuable across each boundary point of D which is not a limit point of {zn}. Moreover, the analytic behaviour of the function thus continued on the whole Riemann sphere is obtained.
From Theorem 2 one m a y deduce m a n y of the theorems in [1], notably those where F is a Hilbert space. We content ourselves with sketching the method of deduction in one typical case only. Let B denote the Hilbert space of analytic functions in D + normed b y ][]]12=SloS~l/(re~t)]2rdrdt. Let {zn} be points of D +, and a n complex numbers such that the interpolation problem/(zn) =an(n = l, 2 . . . . ) has more than one solution in B. Let E = (l/Sn}, and suppose/~ is an arc of I z I = 1 disjoint from the closure of E. Then, it was shown in [1] t h a t the (unique) solution ]* of the interpolation problem which has minimal norm is analytically continuable across/~ and into all of Z\/~.
To deduce this result from Theorem 2, observe that B has the reproducing kernel K(z, ~)= 1/(1 - ~ z ) ~, and that the minimal interpolating function (as is known from general principles concerning Hilbert spaces with reproducing kernel) is that solution which is spanned by the functions K(z, zn) (this set is not total b y the underdeter- minedness hypothesis). Now, we can't yet apply Theorem 2 in this situation since the K(z, zn) have double poles. However, let H be the Hilbert space of functions g(z) = ~ cn z n normed by [[ g [I 2 = ~ (n + 1)[ C n [2. The correspondence of orthonormal bases
(n + 1 ) 8 9 § 1)-89 ~
induces an isometry between B and H, under which K(z, zn) corresponds to 1/(1 -~nz) in H. The transform g* o f / * is, by Theorem 2, continuable across fl into
n. s. SHAPIRO,
Sequences of rational functions
Z \ E , and finally the corresponding continuability for /* is readily inferred from the fact t h a t / * is the H a d a m a r d product of g* a n d 1/(1 - z ) 2.
We wish to r e m a r k t h a t throughout [1] the authors assumed t h a t the closure of the set {zn} doesn't include the whole b o u n d a r y of the domain. Our Theorem 1, together with the technique just employed (or, suitable modification of the dis- cussion in [1]) enables one to assert something also a b o u t the case where the zn cluster at all b o u n d a r y points. F o r instance, returning to the example of the space B just discussed, if ]* denotes the unique solution of the/inite interpolation problem
/(zk)=ak, k = l, 2, ..., n
having least norm, then t h e / * converge in norm to/*: and what is more
(/*(z))
con- verges uniformly on compact subsets of D - \ E to a certain function g meromorphic in D - . I n other words, we still have overconvergence of the solutions of the cor- responding minimization problems with finitely m a n y interpolation conditions.W h e t h e r the functions /* and g bear to one another some simple relation in the case when E includes the whole unit circumference (in which case
]*
will in gen- eral be nowhere continuable), for instance whether they are connected b y a func- tional equation, or exhibit some kind of matching b o u n d a r y behaviour, seems wor- t h y of investigation.3. Concluding remarks and open questions
3:1. I n Theorem 1 we showed t h a t i f / n e R and H / n - I l l -+0, then (under suitable hypotheses)
{fn(z))
converges on compact subsets of D - - \ E to a certain function, say f, which is then meromorphic in D - (indeed, the hypothesis of non-totality implies t h a t E has no limit points in D - ) . Moreover, as the proof of Theorem 1 showed, / = 0 implies ] = 0, t h a t is, f uniquely determines ].I t seems plausible t h a t the converse is true, i.e. t h a t ] = 0 implies / = 0, b u t we have not been able to show this, not even under stronger hypotheses on S. We wish also to raise the following more general questions:
(i) If f is analytically continuable across some point of ]z I = 1, m u s t the contin- uation coincide with ]? (The answer is
yes
in the special case / = 0, as we just remarked.)(ii) If ] is analytically continuable across some point of I zl = l, m u s t the contin- uation coincide with ]?
Of course, the answers to these questions are trivially yes under the hypotheses of Theorem 2, i f / ~ doesn't include the whole unit circumference, since then / a n d ] are analytic continuations of one another. These questions are suggested b y the a u t h o r ' s speculations about generalizing the notion of analytic continuation [4].
3.2. I t seems of some interest to generalize the results of this paper to open sets more general t h a n the unit disk. For smoothly bounded domains this is quite rou- tine, b u t for general open sets serious technical difficulties appear, for instance in extending the l e m m a needed for Theorem 2.
I n like manner, one could a t t e m p t to generalize the results to topological linear spaces (even the unit disk) of holomorphic functions other t h a n normed spaces.
Here again some new ideas are needed. T h a t the overconvergence theorems are 348
ARKIV FOR MATEM)~TIK. B d 7 n r 2 5 t r u e for some i n t e r e s t i n g n o n - n o r m e d spaces m a y p e r h a p s be considered p l a u s i b l e i n view of t h e r e m a r k t h a t t h e y are t r u e (rather t r i v i a l l y ) for t h e topology of u n i - f o r m convergence on c o m p a c t sets. Here t h e n o n - t o t a l i t y h y p o t h e s i s is e q u i v a l e n t t o t h e finiteness of E.
University o] Michigan and University o] Stockholm
R E F E R E N C E S
1. AKUTOWICZ, E. J. and CARLESON, L., The analytic continuation of interpolatory functions, J. d'Analyse Math. 7, 223-247 (1959/60).
2. Do~.~R, Y.. On the existence of a largest subharmonic minorant of a given function, Ark. fSr Mat. 3, 429-440 (1957).
3. OSTROWSKI, A., ~ber vollstiindige Gebiete gleichm/~ssiger Konvergenz yon Folgen analytiseher Funktionen, Abh. math. Sem. Hamburg 1, 327-350 (1922).
4. SHAPIRO, H. S., Generalized Analytic Continuation, Matscience Symposium, Plenum Press (to appear).
5. WALSH, J. L., Interpolation and Approximation by Rational Functions in the Complex Domain A. M. S. Coll. Pub. X X , 3 ed., 1960.
Tryekt den 29 december 1967 Uppsala 1967. Ahnqvist & Wiksells Boktryckeri AB
