Best uniform approximation by analytic functions
LENNAI~T CAP~LESON and SIGVARD ~ACOBS
Introduction
L e t F C LP( - st, ~), 1 < p ___ oo, and consider the extremal problem inf
lip + flip, f
in Hp. For notations and basic results on H~-spaces, see [6]. This problem was extensively treated b y i%ogosinski and H. S. Shapiro in [10] and also b y Kavinson in a series of papers. Havinson studied the corresponding problem in general domains. We refer to the survey [13] b y Toumarkine and I-Lvinson, which also contains a quite complete bibliography.The case p = oo is of a special interest. I t can also be formulated as a problem on so called I-[ankel matrices and is in this w a y of importance in probability. We wish to mention in particular the papers b y Nehari [9], I-Iartman [4], I-Ielson and Szeg5 [5] and Adamyan, Arov a n d Krein [1]. We shall here consider continuous
~' and look for results on the best approximation f, whereas in the above mentioned papers (except [5]) the results are expressed in terms of t h e matrix. I t is easy to see t h a t in this case a unique best approximation f exists in H ~. One might ask to what extent do F a n d f have the same regularity. The investigation of these problems is the main object of the present paper. We shall see t h a t the answer is about the same as for conjugate functions. We shall also restrict ourselves to the case of the u n i t disc U. I{owever, the function-theoretic proofs in sections 2, 3 and 4 are all of a local character, and so all the results can easily be carried over to a n y region which has in each case a sufficiently regular boundary.
I n section 1 we have stated the dual problem and collected some well-known material. Theorem 1 is originally due to Bonsall [3] and Shapiro [11]. Section 2 is devoted to the s t u d y of the extremal functions of the dual problem in H 1. I n the case F E L ~, de Leeuw and i%udin [8] have examined the question of uniqueness for the corresponding extremal function in H I. We give a complete solution of this problem, provided F is in C. I n section 3 we t r e a t our main problem and in section 4 we give an example which shows t h a t the conditions in Theorem 3 a can not be weakened as long as the regularity of _~ is expressed only in terms of
220 L E i N ~ q A ~ T C A R L E S O ~ N " A I ~ D S I G V A R D J A C O ] ~ S
its m o d u l u s o f c o n t i n u i t y . I n p a r t i c u l a r we c o n s t r u c t a n F E C whose best a p p r o x i m a t i o n f is n o t in A ---- H + N C. This was f i r s t done b y A d a m y a n , A r o v a n d K r e i n [1], R e m a r k 3.2 p. 13.
1. The dual problem
W e s t a r t w i t h t h e following simple b u t useful c o r o l l a r y o f t h e ~ a h n - B a n a c h t h e o r e m , a t e c h n i q u e w h i c h has n o w b e c o m e s t a n d a r d .
L ~ A 1. Let B be a Banach space, M a linear subspace and denote by M ~ the annihilator subspace of B* corresponding to M. Then for any L E B* we have
i n f l l L - - 1 ] l = sup IL(x) I,
1 E M ~ x E M , I]x]] <_ 1
and there is always an l E M ~ for which the i n f is attained.
F o r a p r o o f see e.g. [12], t h e o r e m 4.3-F, p. 188. I f we a p p l y this l e m m a to B ---- L 1, M z H 1 (the subspace o f H 1 o f f u n c t i o n s v a n i s h i n g a t t h e origin), we g e t
m = i n f lip + fl]~ = sup F(e~~176 ,
fEHOO hEHo~llh]h < 1 - - ~
since in this case /~* ---- L +, M ~ ---- H *. T h u s , t h e r e is always a n f E H + for w h i c h [[~P + f l [ , ~ m. Since we a v o i d t h e t r i v i a l case m -~ 0, we m a y suppose m ~- 1 a n d t h e sup can o f course be t a k e n o f t h e real p a r t of t h e i n t e g r a l as well.
T h e n e x t q u e s t i o n is w h e t h e r a m a x i m i z i n g h E H01 exists. W i t h t h e aid o f F e i d r ' s t h e o r e m a n d t h e t h e o r y o f n o r m a l families, it is e a s y t o p r o v e t h e following, cf [3]
T h e o r e m 4 or [8] T h e o r e m 10 a.
TgEORE~ 1. I f f ' E C ~- H ~176 in particular if 2' E C, then at least one h in the unit ball of H~ exists maximizing
R e ~
1/
F(e'+)h(e~+)dO.COROLLARY. Under the above conditions on F the minimizing
f
E H ~ is unique.This follows f r o m T h e o r e m 14 in [10] a n d also s i m p l y f r o m t h e following fact. F o r a n y p a i r o f e x t r e m a l s f a n d h, we h a v e
B E S T U N I F O R M APPROXIi~CIATION B Y A N A L Y T I C F U N C T I O N S 2 2 I 1 = R e ~
1/
(F(e i~ -]-f(e~~176 < 1, a n d so- - x r
(F(e ~~ + f(d~ ~ -~ Ih(d~ a.e. (*)
F r o m this v a r i a t i o n a l e q u a t i o n a n d t h e f a c t t h a t av E C, we shall o b t a i n all o u r i n f o r m a t i o n .
2. Properties of the extremals in the dual problem
E v e r y t h i n g in this section will d e p e n d on t h e f a c t t h a t i f F E C a n d if /~(e i~) = 0, t h e n b y (*), f h has its v a l u e s in a small sector for 0 close t o T a n d we c a n m a k e use o f t h e following t h e o r e m .
LwMMA 2. I f v is real valued and harmonic in the simply connected domain D and Iv(z)[ < #, z C D, then the analytic function g =-e u+~€ is in HP(D) for
p < ~/2#.
A p r o o f w h i c h easily carries o v e r to a n y s i m p l y c o n n e c t e d d o m a i n D m a y be f o u n d in [7], T h e o r e m 1.9, p. 70.
B e f o r e a n n o u n c i n g t h e t h e o r e m s we w a n t t o m a k e a d e f i n i t i o n .
Definition. L e t f be a n a l y t i c in a region D h a v i n g a s m o o t h b o u n d a r y 0D, D n o t necessarily s i m p l y c o n n e c t e d a n d a s s u m e e.g. f E HI(D). W e s a y t h a t f is o u t e r in D i f for z E D
1 s
OG(z, ~)
log If(z)l - - 2:~ J anO o
G being t h e G r e e n ' s f u n c t i o n of D.
- - log ]f($)lds,
TI~nOREM 2. I f aV C C, then we have for the corresponding h E H~
a. h E H ~ for every p < oo.
b. h is outer in some annulus t l o~- {z]r o < lzI < 1}.
_Remark. :For l a t e r use we o b s e r v e t h a t t h e a s s u m p t i o n on h can be localized t o h e H i ( R ) for some a n n u l u s /~ = {zIr < [z I < 1}. T h e conclusions are t h e n v a l i d in some _R 0 w i t h r o > r .
T h e following corollaries are consequences of t h e p r o o f of T h e o r e m 2 b r a t h e r t h a n o f t h e result.
COROLLAnr 1. I f f~ = f + F(ei~), then f~ is outer in subregions D r of D o = {z = rd~ < r < 1, IO - - T1 < 6} with smooth boundaries. We can choose 6 and r o independently of 7.
222 L E N ~ A _ R T C A R L E S O N A N D S I G V A R D J A C O B S
COROLLARY 2. I f h (1) and h (2) are extremals corresponding to the same F E C, then h~ (2) is a rational function.
COROLLARY 3. I f 1~ E C, a necessary and sufficient condition for h E H 1 satisfying (*) to be unique is that h -1 6 H 1, and this implies that h -1 E H e for every p < ~ . Remark 1. I f we r e s t r i c t h to be in HI, Corollary 3 s h o u l d b e f o r m u l a t e d with h r e p l a c e d b y h 0, w h e r e ho(z ) ~ h(z)/z.
Remark, 2. I n t h e ease _P E L ~ de L e e u w a n d R u d i n h a v e p r o v e d , [8] T h e o r e m 8, t h a t a s u f f i c i e n t c o n d i t i o n for h 6 H I to be u n i q u e is t h a t h -~ 6 H ~. N e c e s s a r y is t h a t h is o u t e r a n d t h a t for h~(z) ~ h(z)/(z -- e~a) ~, h~ fails t o b e in H I for e v e r y real ~. W e shall give a p r o o f w h i c h shows t h a t h -1 6 H 1 is sufficient also in t h e general case F E L ~.
F o r t h e p r o o f o f T h e o r e m 2, set ~v = F - - iV(e t~) a n d f~ ~ f + F(e") so t h a t F~(e t~) = 0 a n d (,) r e a d s
(F (#o) + L(e ))h(e ) : to to [h(et~ a.e.
F i x a n a r b i t r a r y p < ~ , choose s > 0 so small t h a t a r c t g e/(1 - - e) < ~r/2p a n d let ~ be such t h a t ]F~(et~ < e for 0 E ~ = {0110 - - z] < 5}. O b s e r v e t h a t this implies ]f~(ele)] > 1 - - e a.e. on y~. P u t f f l = u + i v . F o r 0 E v ~ we t h e n h a v e u(e t~ > (1 - - e)]h(#~ a n d Iv(#~ < e]h(#~ a.e. T h u s , if (1 ~- fi)e : 1, 1 ~ u(e ~~ =~ flv(e ~~ ~ 1 a.e. on 7~, a n d since f~h 6 H 1, this implies t h a t 1 + u(z) ~ flv(z) > I in D~ = {z]lO - - z[ < 81 ( 8, r T < Iz] < 1}.
H e n c e , log (1 + f f l ) is a n a l y t i c in D~ a n d [arg (1 +A(z)h(z)l < a r c t g 1/fl <
< z / P p , z E D . B y L e m m a 2 , f f l E H q D ) , a n d since If~(et~ 1 - - e a.e. on r~, ]h(#~ PdO < m. N o w F is u n i f o r m l y c o n t i n u o u s on t h e u n i t circle w h i c h can t h u s be c o v e r e d b y a f i n i t e n u m b e r o f y,'s. H e n c e f ~ [h(e ~~ fdO < ~ .
N e x t , choose p = 2 a n d d e f i n e ~ b y ~ ( 0 ) = - - i a r c t g v(e~~ ~ i f 0 E V~
a n d h(e t~ ~ O, a n d ~ ( 0 ) = 0 elsewhere. F o r m
7~
1
fle
t~+
z= e x p 3
B y L e m m a 2, k E H 2. Thus, for G~ ~-- f f l k we h a v e b y T h e o r e m 2 a t h a t G~ E H i, a n d b y t h e c o n s t r u c t i o n of G~, t h a t G~(# ~ ~ 0 a.e. on y~. This implies t h a t G~
can be a n a l y t i c a l l y c o n t i n u e d o v e r 7,, a n d in p a r t i c u l a r t h a t G~ is o u t e r in D~
for a n y (~1 < ~ i f o n l y r~ is s u f f i c i e n t l y close to 1. T h e f a c t o r s o f G~ are t h u s all o u t e r in D r T h i s p r o v e s T h e o r e m 2 b a n d also C o r o l l a r y 1, e x c e p t t h a t we h a v e to p r o v e t h a t t h e r e is an r 0 w h i c h will do for e v e r y ~. I n T h e o r e m 2 b this is e v i d e n t , since [ - - 7~, ~r] c a n be c o v e r e d b y a f i n i t e n u m b e r of i n t e r v a l s o f l e n g t h 2~ 1 = ~. If, h o w e v e r , r~ is s u f f i c i e n t l y close to 1, we also f i n d f r o m t h e r e p r e s e n t a t i o n f o r m u l a for
B E S T U I q l F O I ~ N I AP:PROXIlVIATION B Y A N A L Y T I C F U N C T I O N S 2 2 3
o u t e r f u n c t i o n s t h a t [f~(z)] > 1(1 - - e) in D . Since f~ = f~ -4- F(e ~) - - -F(e~), f~ is also o u t e r in D i f l a - - v l < ~ - - ~ . T h e same r w i l l t h e r e f o r e do for la - - v[ < rain (~x, 6 - - ~x), a n d b y t h e same a r g u m e n t as a b o v e t h e p r o o f o f Corollary 1 is finished.
T o p r o v e Corollary 2, suppose t h a t h r a n d h (2) C H ' b o t h s a t i s f y (,). T h e y m u s t t h e n h a v e t h e same a r g u m e n t on 7 . T h u s ~(') = w~(2), k~l) = k~(2) a n d so t h e f u n c t i o n ~ = h(~)/h(2)= G!~)/G~ 2) is a n a l y t i c in a n e i g h b o u r h o o d of 7~
e x c e p t for possible poles on 7~ a n d R takes p o s i t i v e values there. I t n o w follows t h a t R is a r a t i o n a l f u n c t i o n .
I f F E C, we f i n d a l m o s t i m m e d i a t e l y b y t h e p r e c e d i n g a r g u m e n t s t h a t a s u f f i c i e n t c o n d i t i o n for t h e u n i q u e n e s s of h is t h a t h -1 E H i. L e t us, h o w e v e r , for t h e m o m e n t suppose o n l y t h a t (,) is satisfied w i t h F in L ~. I f h -1 E H 1, t h e n h = k ~, k o u t e r , k -1 E H e. L e t hi = glk~ be a n y e x t r e m a l f u n c t i o n c o r r e s p o n d i n g to the same F , w h e r e gl is a n i n n e r f u n c t i o n a n d k 1 is o u t e r w i t h k 1 E H e . B y (*), glkl k-1 = ]~1 k-1 a.e. ]-Ience gxk~k -1 = const., w h e n c e gl = ela, k l = e-la/2]~
a n d hi = h.
I f c o n v e r s e l y h is k n o w n t o be u n i q u e it can h a v e no zeros in U, cf [8], p. 479.
N a m e l y , suppose h(a) = O, a E U, a n d f o r m (z - b ) ( 1 - - g z )
h ( 1 ) ( z ) ---- 9 h ( z ) ,
( z - a ) ( 1 - a z )
w i t h b # a. T h e n h (~) E H 1 a n d h (1) also satisfies (*).
Now, assume a g a i n iv E C a n d f i x a n a r b i t r a r y p < ~ . Since, b y L e m m a 2 again, ffl~ E H P ( D ) , we will be t h r o u g h i f we can p r o v e t h a t G~ -~ E H~(D~), a n d this is fulfilled i f G~ is free f r o m zeros on 7~. I f G~(e ~') = 0, t h e zero is o f e v e n o r d e r a n d so w i t h G ~ ) ( z ) = - d % G ~ ( z ) ( z - d~) -~, G~ 1) is p o s i t i v e a n d a n a l y t i c o n 7~. T h u s h (1) = G(~)/f~k~ E H i ( D ) a n d t h e n h (1) E H ~ since also h(~)(z) ~- -- e~zh(z)(z - - e~) -2. ~Vforeover h (~) satisfies (,), a n d we h a v e p r o v e d t h a t for a u n i q u e e x t r c m a l h we n e c e s s a r i l y h a v e h -x E H y for e v e r y p < ~ , p r o v i d e d t h a t F is c o n t i n u o u s .
3. The regularity of the minimizing f in relation to that of the given F L e t C~o d e n o t e t h e class o f f u n c t i o n s being D i n i - c o n t i n u o u s . T h e y f o r m a
1 1
B a n a e h a l g e b r a u n d e r t h e n o r m []F]]~ ---~ m a x IF(x)[ -~ f o CoF(t)t- dt, w F s t a n d i n g for t h e m o d u l u s o f c o n t i n u i t y o f F . A s is t h e class o f f u n c t i o n s w h i c h s a t i s f y L i p s c h i t z c o n d i t i o n o f o r d e r a, a n d b y / v E C n+a, h E N , 0 < a < 1, is m e a n t t h a t F E C ~ w i t h F (n) in A s.
T ~ E O ~ n ~ 3. a. 2' E C~o ~ f E A and in general F (n) E Co., ~ f(") E A . b. I f 0 < ~ < 1 , then F E A s ~ f E A s and also F E C n+~ ~ f E Cn+%
224 L E N N A ~ T C A ~ L E S O ~ A N D S I G V A ~ D J A C O B S
Consequently F in C + implies f in C ~,
cf. [1], p. 17and we have moreover e. There is a q independent of +n such that
(Hf(n,H~ 11/n (l[[?,k,t[r ~l/k
V v - . ] <- 9
I f thus F E
C(M,)
where the sequence(M,/ni) 1/"
is non-decreasing, then f, regarded as a function of 0, belongs toC(M~+I).
A rough estimate b y the Cauchy and Bessel inequalities also gives(++) + +
n n n
whence f, regarded as a function of z, is in
C(M~+2).
The following theorem has been proved b y H. S. Shapiro [11] in the case M~ = n!COROLLARY.
I f the sequence (M,/nt) 1I~ is non-decreasing and if
log M , = O(n2),then F in C(M~) implies f in C(M~).
That log M n =
O(n ~)
is a sufficient condition forC(M~)
and C(M,+t ) to coincide is proved in [2], Theorem VII.The proof is based on the result in Corollary 1 in the preceding section. Now if f is outer in D~ and if
g(z) = f(ei~),
ther~ g is outer in a corresponding region D: in the upper half-plane. Accordingly, it is sufficient to prove the theorem for the upper half-plane under the conditions t h a tIF(x) +
f(x)] ~ 1 a.e. and thatF ( T ) -~ 0 implies f outer in a region D+ whose boundary contains the interval
(v -- ~, ~ + ~), F and f being periodic with period 2z. B y doing so we reach a typographic simplification but, above all, we do not have to distinguish between derivatives with respect to arc length and derivatives with respect to
z,
when studying the behaviour of f on the boundary. Before proving the theorem we shall state a simple lemma on Cauchy integrals.fl ~,(t)
LEMMA 3. I f Ilc(:J:
t)] ~og(t)
a.e.,where - ~ - dt < ~ and if f ~ lc(t)
= - - dr, then f o r z e So(~) = {z = x + iyrlxl < y, lz[ <
~),
~ t - - z
/ ++ f ~
Ix(z) - ~ ( 0 ) l < A - 7 - dt + A l z l ~ d t .
0 I~[
~(z) =
I f instead
[k(:~t)t ~_ Itineo(t), then
] 3 E S T U N I F O I ~ 4 : APPX:~OXI1M:ATIO12q B Y A ~ A L Y T I C FU/VCTIOI'ffS 2 2 5
Izl a
lu(~)(z) - - x(~)(O)] < Ann! ~ dt +
A~n!Izl T -
dt forW e o n l y h a v e t o e s t i m a t e t h e i~tegrMs
[=1 a
}) fl
- zl -~- m(t)dt a n d t - - z t
o M
u s i n g t h a t I t - - zl > AIt[ i f z Es
z e & ( 6 ) .
I f co r 0 is a m o d u l u s of c o n t i n u i t y , t h e n for e v e r y e > 0 t h e r e is a n V such t h a t Ix(z) -- ~(0)[ < Aco(s) if z E So(T), for
f f f -(')
Izl - ~ dt < re(e) whil~ - 7 - dt a n d lzl ~ - dt
b o t h t e n d t o zero w i t h [zl. I f o~(t) = t ~, 0 < ~ < 1, we f i n d t h a t lz(z) - - z(O)l <
A(~)Izl ~, z e $o(6 ). Also let us o b s e r v e t h a t t h e r e s u l t s h o l d as well for
f ~ § tz k(t)
x ( z ) = t ~ z 1 q - t 2 - - d t
since this i n t e g r a l differs f r o m t h e C a u c h y i n t e g r a l o n l y b y a c o n s t a n t .
F i x a n a r b i t r a r y 3 E_R a n d f o r m F~ a n d f~ as in t h e p r o o f o f T h e o r e m 2.
W e k n o w t h a t f~ is b o u n d e d a w a y f r o m zero in a r e g i o n D r of l e n g t h 26. L e t L~ ---- log f~ a n d let z e Dr, t h e n
f
a 1 q - t zlog
If~(t) IL (z) = i~ t - - z 1 q- t 2
--3
dt q- X~(z) = ~ ( z ) -f- ,~(z),
w h e r e 2 r can be a n a l y t i c a l l y c o n t i n u e d o v e r ( 3 - 6, 3 - r N o w 1 - [fr(x)l~ = IF~(x)l ~-~ 2 R e F~(x)f~(x) a,e. a n d so ]1 - - I f d x ) l ~ l <
A~,F(ix--
31) a.e. T h u s also ilogrf~(x)][ ~ Ave(Ix - - 3[) a.e. in (3 - - 6, 3 -~ 6). B y L e m m a 3, it is t h e n possible for a n y e > 0 to f i n d a n V such t h a t Iz~(z)--x~(3)t < e for z in a s e c t o r S~(V) w i t h r a d i u s 7. T h e a r g u m e n t s a f t e r L e m m a 3 Mso show t h a t ~ can b e chosen so as t o be i n d e p e n d e n t o f 3. Since 2~ is a n a l y t i c o v e r (3 - - 6, z q- 6), [2~(z) - - 2~(3)[ < A l z - - 3[, a n d since the b o u n d o f ~ in D~ a n d t h e l e n g t h a n d h e i g h t o f D~ are all i n d e p e n d e n t o f 3, so is A. T h u s [L~(z) - - L~(3) Z < 2s f o r z e S~(V), i.e. t h e same ~ will do for e v e r y 3. A c c o r d i n g l y[exp (L~(z) - - L~(3)) - - I r < 3e, fL(z) - - L(3) J < 3e a n d If(z) -- f(3)[ < 3e
2 2 6 L E ~ A ~ T C A ~ L E S O ~ A - N D S I G V A R D ffACOBS
for z in S,(~). I f l a - - 3 I < ~ we c a n t h e n f i n d a z 0 E S . ( v ) N S . ( ~ ) such t h a t If(Zo) -- f(3)] < 3e a n d If(Zo) - - f ( a ) ] < 3e, a n d we h a v e If(a) - - f(3) l < 6s for [a - - 3f < v.
T h e p r o o f of t h e f i r s t p a r t o f T h e o r e m 3 b can be c a r r i e d o u t on e x a c t l y t h e same lines. L e m m a 3 will n o w give [n,(z) - - ~ ( 3 ) 1 <
Aiz
- - 31 ~, z E S , ( 6 ) . T h u s I f ( z ) - - f ( 3 ) l < A ] z - - ~ 1 ~ f o r z in S~(V) a n d A a n d ~/ are i n d e p e n d e n t o f T.As [ % - - 3 1 ~-4- 1 % - - a l ~ < 2 1 a - 3[ ~ for a suitable z 0 E S ( ~ ) N S o ( ~ ) we h a v e ]f(a) -- f(3)[ < A la -- 3 [~ for I a ~ 3[ < ~/, a n d this i n e q u a l i t y t h e n holds g e n e r a l l y w i t h a n A i n d e p e n d e n t o f a a n d 3.
Also t h e proofs o f t h e second p a r t s o f T h e o r e m s 3 a a n d 3 b are a l m o s t identical.
W e p r e f e r t o c a r r y o u t t h e second of t h e m w h i c h is s o m e w h a t m o r e i n v o l v e d since we h a v e t o check t h a t all t h e c o n s t a n t s o c c u r r i n g are i n d e p e n d e n t o f 3.
N o w f o r m
,, F(k)(~:) . F~k~(3)
F,(x) : F(x) -- ~0 k! (x -- 3) ~ a n d L(z) ~--f(z) + ~'o ~.. (z -- "~)~.
As long as we r e s t r i c t x a n d ~ to some c o m p a c t set we h a v e I1 --lf,(x)12/ <
< ~ A l x - - 3 I ~+~' w i t h A i n d e p e n d e n t of 3. F o r L , t h u s
a n d
., f
tog IL(t)l dt + 2~")(z) L~~ = i-~ (t - - ~)"+~--3
[loglL(t)ll < A i t -- zl "+~ if it - - ~] < 0. B y L e m m a 3 we get [~!=)(z) - - ~!")(311 < A I z - - 31 ~ for z E S,(61 ,
I f!")(z) f!n)(3) I
L(z) L(3) < A I z - - T [ ~, z ~ S , ( ~ ) .
F u r t h e r , ~G(3) a n d L!")(3) a r e bo~h b o u n d e d for 3 E [ ~ ~, ~], a n d t h e n so is f~)(~). K e n e e [f~")(z) --f~')(~)l < A1 z -- ~l ~ a n d If~O(z) -- f(")(~)[ < AI z -- z] ~ for z E S~(~) w i t h A a n d V still i n d e p e n d e n t o f v. ( T h e y o f course d e p e n d on n.) An a p p l i c a t i o n o f this r e s u l t in t w o sectors S~ a n d S~ gives ]riO(z) - - f(")(3)[ <
< A l a - - 3 1 ~ for l a - - 3 1 < ~ / .
a n d b y t h e same a r g u m e n t as a b o v e it follows t h a t IL!")(z) - - L!")(3)[ < AIz - - 3[ ~ for z i n S,(V), w i t h A a n d V i n d e p e n d e n t o f 3. A s s u m e t h a t we h a v e p r o v e d t h e t h e o r e m u p t o a n d including n - - 1. T h e n also If~k)(z) - - f!k)(3)] < A t z -- 3[ ~, 0 < k < n - - 1 , w i t h A i n d e p e n d e n t o f ~ as long as Iz] a n d 3 b e l o n g t o some
= = ~(~- ~)~ w h e r e
compact set. Form
~ L!"~--f!~
Since ~ ~ ( L , . . . , ~ ,,R(xo . . . x,_O is d i f f e r e n t i a b l e w i t h b o u n d e d d e r i v a t i v e s if Ix01 > m > 0 a n d
~'~-~ [x~L 2 < M , i t follows t h a t [~(z) - - ~v (3)[ < A[z -- 3I ~, w i t h A i n d e p e n d e n t o f T i f zES~(V) a n d v E [ - - ~ r , ~ ] . T h u s also
BEST UNIFORM APPROXIMATION BY" A2,TALYTIC FUNCTIONS 227 I n the p r o o f of T h e o r e m 3 c we suppose t h a t lim~_~ (]lF('Ollo)/n!)~/" = oo. The case w h e n F is a n a l y t i c is more easily t r e a t e d w i t h t h e aid of t h e principle of reflection as IF(x) + f(x)[ = 1, a n d we o m i t it. L e t 8 -1 = max0<k_<~
(llF(k)lloJk!)xz~,
where n is to be t a k e n so large t h a t ~ is sufficiently small. I t will m e a n no restric- tion i f we choose ~ = 0. The construction of F o a n d T a y l o r ' s t h e o r e m yield
x
1 f (F(~ -- ~(~)(0))(-- tydt.
t0(x) - (~ _ ~)!o
W i t h o~ = oF("), we h a v e ]F0(x)l <
]x[~%(lx])/n!
This implies t h a t [F0(x)[ _< 89 i f Ix] ~ 8. F o r o~(Ix[) < o,(~) < ~. j ~ o~(t)t -~ dt <_ 1liFO)lie ~ l n ! 8 - n iflxl < 8 . tIence 1 < Ifo(X)l < ~ and so I~--Ifo(x)?l
<Alxl%~(Ixl)/n!
a n dfloglfo(x)I 1 < A t x l n ~ o = ( t x l ) / n ! if Ixl _<8 w i t h A i n d e p e n d e n t o f n. ~ o r 1 f 1 + tz
log If0(t)[
z~ = i z
J
t - - z 1 -~ t ~ dt thusAj! f l ~n(t) 8 n
f~J)(O) < ~-. J tn-J t at <_ A j ! 8-J . ~ [IF(~ < Aj! 8-J ,
0
i f 0 < j < n. B y i n d u c t i o n it is n o w easily p r o v e d t h a t ](noP)(J)(0) [ __< A P ( j + 1)P-lj!8 - j for e v e r y p E i V a n d 0 < j _ < n . W i t h k 0 = C ~ this gives
j~
i~J)(0)l < j +----~ e ~(i§ 9 ~-J < j! qJS-J, 0 _< j < ~ .
T h e m a x i m u m principle shows t h a t [Re no(Z)] < A. I f for some a < ~ we can p r o v e t h a t ]t~eLo(z)l_<A for [z[__<a, I m z > _ 0 , we f i n d t h a t ~ o = L o - - n o is a n a l y t i c in t h e disc tz] ~ a w i t h IRene(Z)] < A there. L e t l o = e z~ The C a u c h y e s t i m a t e t h e n gives [l(oi)(0)] ~ A j ! a - J , a n d for f o = k o l o we get
lf(on)(0)l
< n!q"a -~. W e will f i n d t h a t a = 8/3 will do. F o rI F ( k ) ( 0 ) ]
ak < ~
[[F(k)ll~ ak< ~
(a(~--l)k <~ 1i f a = 8/3. Thus, i f n is s u f f i c i e n t l y large, T h e o r e m 3 a tells us t h a t 1 9 .< Ifo(Z)l < 8 8 . i f [z] < . (~/3, . I m z > 0, a n d so [ReLo(z)] < A . in this region.
As f~o")(0)= f ( " ) ( 0 ) + F0)(0), we also h a v e If(")(0)[ < n ! q " a - " w i t h a -1 = 3 m a x
(llF(k)llo~[k!)X/k.
0 ~ k ~ < n
This proves T h e o r e m 3 c, since all constants are easily seen to be i n d e p e n d e n t o f t h e choice of ~.
228 L E N N A I % T C A R L E S O N A N D S I G V A R D J A C O B S
4. The weakest condition on ~F which guarantees ] to be in A
THEOREM 4 . . F o r every continuous, non-decreasing and subadditive function ~o with
~o(0)
= 0 andf o ~(t)t-idt
1 divergent, there is an .F ~ C with o~(t) ~ (9(t) for which the best approximation f ~ H ~ is not in A .F o r t h e p r o o f we r e q u i r e t h e following l e m m a .
LEM~A 4. Let co be as in the theorem and p u t F(e i') : m(t -- ~) in [% ~ + ~], F(e/') --~ 0 in [v - - ~, T) and continuous elsewhere. I f the corresponding f e H ~ is in A , then f(e ~ ) = =~= i. We assume F to be defined in such a way that Il~ + fIl~ = L
W e suppose f C A w i t h R e f(e i~) =/= O. L e t u(O) = log
if(d~
Since F(e ~) = 0 a n d lF(e ~~ + f(e~~ = 1, u ( r ~- t) < -- Ao~(t) or u('~ d- t) > Aco(t) for t E ( 0 , ~ l ) a n d some A > 0, d e p e n d i n g on w h e t h e r l~ef(e ~) is p o s i t i v e or n e g a t i v e . F o r t E ( - - @ 0), on t h e o t h e r h a n d , u(~ + t) e q u a l s zero. T h u s1
f u(~ + t) - u(~ - t)
Jim dt = oo
~-~0+ t '
E
whence, see e.g. [14], T h e o r e m 7.20, p. 103, lim,_~l_ ]argf(re~*)] = oo. This c o n t r a - dicts f in A w i t h lf(d~)[ = 1, a n d t h e l e m m a is p r o v e d .
N o w let d ~ = z . 2 -~, n = 0 , 1 , 2 , . . . , a n d p u t co~(0) = w ( 0 - - ~ ) for 0 in
~ , , o)~(0)= co(~_ 1 - - 0 ) for 0 in 2 , d~_l a n d o ~ ( 0 ) = 0 elsewhere. L e t c~ = 0, 2-~ or i . 2-89 d e p e n d i n g on w h e t h e r n is even, n ~ 1, rood 4 or n - - 3, m o d 4 r e s p e c t i v e l y . P u t F(e ~~ = ~.~ c~c%(O). I t is easily seen t h a t 2' E C w i t h coF(t ) _< co(t) if t < ~/4. Assume t h e c o r r e s p o n d i n g f t o be in A. L e m m a 4 t h e n tells us t h a t f(sd~+x ) = 4 - m - i a n d f(ed,+a ) = 4 - m , w h e r e
e n = e id~ a n d m = Ill ~ -4-fl[o~ > 0, as -F(e ~~ = 0 in ( - - ~, 0). Since lim d~ = 0,
this is, h o w e v e r , i n c o n s i s t e n t w i t h f in A. ~-~
References
I. A D • V. M., A ~ O V , D. Z. a n d KREIiV, iV[. G., Infinite H a n k e l matrices a n d generalized Carath6odory~Fej~r a n d F. Riesz problems. Funktsional'nyi Analiz i ego Prilozheniya.
2, 1968, p. i.
2. B A N G , T., O n quasiranalytic functions. (In Danish.) Thesis. C o p e n h a g e n 1946.
3. BONSALIJ, F. F., D u a l e x t r e m u m p r o b l e m s in the theory of functions. Journal of the L o n d o n M a t h . Soc. 31, 1956, p. 105.
: B E S T IYl~IFO:B1V[ AI~I~:BOXL~CIATIO~N- :BY A I ~ A L Y T I O : F U N O T I O ~ S 229
4. tIA:BT~AN, Pro, On completely contbmous t I a n k e l matrices. Prec. of the American Math.
Soc. 9, 1958, p. 862.
5. HELSON, H. a n d SZEGS, G., A problem in prediction theory. A n n a l i di Matematica pura ed applicata 51, 1960, p. 107.
6. 1LIOFF~AN, K., Banach spaces of analytic functions. Englewood Cliffs, N.J. 1962.
7. KATZl~ELSO:N, Y., A n introduction to harmonic analysis. New Y o r k 1968.
8. DE LEEUW, K. a n d I~UDIN, W., E x t r e m e points a n d e x t r e m u m problems in H I. Pacific Journal of Math. 8, 1958, p. 467.
9. N E ~ I , Z., On b o u n d e d bilinear forms. Annals of Math. 65, 1957, p. 153.
10. I~OGOSII~SKI, W. W. a n d SHAI~IRO, I~. S., On eartain e x t r e m u m problems for analytic functions. Acta Math. 90, 1953, p. 287.
11. S~APmO, It. S., E x t r e m a l problems for polynomials a n d power series. Dissertation. M . I . T . 1952.
12. TAYLOR, A. E., I,r~troduction to functional analysis. New York 1958.
13. TOU/CIAICKINE, G. a n d HAVI>rSOX, S., Propridtds qualibatives des solutions des probl@mes extrdmaux de certains types. Fonctions d'une variable complexe. Probl~mes contemporains. Paris 1962, p. 73.
14. ZYGMVND, A., Trigonometric series. Vol. I. Cambridge 1959.
Received February 2, 1972. Lermart Carleson a n d Sigvard Jacobs Inst. Mittag-Leffler
Djursholm
