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NEW YO?.K UNIVSRSiTY

INSTITUTE OF MATHEMATICAL SCIENCES

LIBRARY

[4 Washington Place, New York 3, N. Y.

IMM-NYU 289

JANUARY 1962

COURANT INSTITUTE

OF

MATHEMATICAL SCIENCES

I

Reproducing Kernels and Beurling's Theorem

HAROLD S. SHAPIRO

IS PER ir:

OF THE I

PREPARED UNDER

CONTRACT NO. NONR-285(46)

WITH THE

OFFICE OF NAVAL RESEARCH

UNITED STATES NAVY

^ 0). Passing to the Fourier transform

00

\

\

J.

F(z) = ) f{t)e ^ dt (Im z > 0) the problem similarly reduces

to identifying closed subspaces of the "Paley-^''iener space" of

the upper half -pi ^me vhich are mapped into themselves upon

multiplication by bounded analytic functions.

.rroi JfO{

â€¢v

â€¢ (.J- /â– -,f r,

. Sit J fa.' iU i â–

Â» â– X90 -r

c.no/joaj'l â– â– â– sri;^ nsqe .nYloq e *1ii 0x^7.:;

â€¢^ -â– .:t)'^ â– 'â– â– * >:-r'- ' > T â– ; -'â– } ''â– -. â€¢.â– ^-â– ,,..i- r.-f-: _ â€¢ â€¢. â–

r2c-r5tfD8'Â£ '^^j.-ul.c.^j Â» msXc-o'io erii (0 â– J) ;fb * ''Tict)',

Tfj ' '^'dcf 'to Eoo besolo r

:!n-? h' >ld-j' err* lo - .

X'

...^fRO^i- â– â– '!'-

In this paper we obtain a product representation for the

closed invariant (in the above sense) subspaces of a class of

Hilbert spaces of analytic functions which includes both Ii and

the Paley^'lener space of the half-plane. The class of Kilbert

spaces in question (vhich we define axiomatically ) is not very

2

general: intuitively it is a weighted H space in some simply

connected domain; but the more obvious generalizations of the

Beurling factorization are simply not true in any greater general-

ity, as simple examples show. The most interesting special cases

of our theorem could be deduced by conformsl mapping fron the unit

circle case, but ^-e prefer to give ^ direct proof based on a

systematic exploitation of the notion of reproducing kernel (r.k.).

That the Beurling factorization stands in close relationship with

the notion of r.k. is suggested by features in Lax's proof, which

operates however with Fourier analysis and does not use properties

2 -

of the r.k.

In Â§ 3 we show how, analogously to the discussion in Bergman

[2], chapter VII, and Garabedian [bj, important extremal properties

of bounded analytic functions, generalized Blaschke products, etc.

can be b?9sed on the r.k. and its properties by use of the theorem

of I 2. Thus the existence of a Hilbert space of analytic fvmc-

tlons which satisfies certain axioms can be made to yield proper-

ties of bounded analytic functions which are usually deduced from

the existence of Green's function, or the Rlemann flapping theorem.

These approaches are closely allied, but we believe our approcich

r

n , Ve assume moreover the following axioms :

nipfnob o-o-zosrij- â– ':'

-J c ;;> AÂ» .;

'.r- ilT.XfT J J-'S'

V L * Jt

sio:^i'tl;fos

riW- ^â™¦â€¢ri^'^rj'^ i^Ot"'^

^^Bfj.! YCf'XÂ«>MO'x-:r e-n^t er-

sfict ;en">.'

.â– XAPri'''ll

Tauo&dj

' fol erfd-

A 'â–

Al. For every X e J\ the linear functional

L,f =Â» f(X) Is bounded.

A2. To every f e H there is associated s "boundar;v'- function",

which T.re denote by f(t), defined a.e* on | ' ; and f(z) tends to

f(t) as z â€”> t non-t^ngentially, for almost all t e j â€¢

A3* The norm of f is uniquely determined by the values of If(t)I

a*e*

Alj.. (Maximum principle) If |f(t)| < M a.e., then |f(z)| < M,

z e -^ 1 â€¢ i

A5Â» Let B = B( i i. ) denote the class of functions ^(z) analytic

and bounded in J i ; then for every e B the transformation

f -> f msps H into Itself.

A6. For every f, , f^t g^* So ^ ^ which satisfy for every e B

the two relations

(i) < f^, g^ > = < f^, g2 >

(il) < f-L* g;i_ > = < ^2* ^2 "^

we have

(iii) f^(t) i^Ttl = f2(t) i^tT a.e. on P

Remarks on the axioms *

From Al we infer that H possesses a uniquely determined

reproducing kernel (r.k.) vhich we write K^(z) or occasionally

K(z, ^)Â» ^-'e assume known to the reader the following properties

of reproducing kernels (vihose proofs are immediate; the properties

,K > \ir)^\ fie. :i . .-.? H

'-. ( , .': I'i > r^ < 'J. . '"^ K > if '' )

dViirf dW

.err-o.:Kfi eri^ â€¢. = f (^) , for every f e H, ^ e A

(3) K^(z) = K^

(5) For every f e H we have the inequality

|f(^)|^ < K^(^) llfll^

As for A2 it could be weakened into a context where | is no

longer assumed locally rectifiable and "almost every^-here" is

defined in a modified way eÂ«gÂ« in terms of sets of linear measure

zero. The re'^der vill see from the proof to follow that the

existence of non-tangential limits is used only in a very weak

way*

Prom A3 it follows that multiplication by a ^ e B for which

||Zi(t)| = 1 aÂ»eÂ» preserves norms, hence also ixoner products.

It also follows that f(z) is uniquely determined from the a.e.

values of f (t ) .

The axiom A^ is necessary for the very formulation of the

problem of "invariant subspaces" to be meaningful. A5 Implies,

since the transformation f â€” > ^ f has a closed graph, that it

is bounded. A6 is perhaps the only axiom which does not seem

altogether natural; it is clearly satisfied by IT and the Paley-

'â€¢'iener space of the half-plane, because the real parts of bounded

; ( [I] 6Qa iablori -Ik

Â«i-,

-Ilenp&ni edd" .. 3 1 r,iBve i'.

S

111 ii i;-i^i' 2: lU)"!

it.

2

analytic functions, restricted to the boundary, are dense In L

norm in the space L {| }â€¢ It is here that we need in an essen-

tial way that ~- L is simply-connected (see Â§ 1^ below) â€¢ ^''hen -^ -_

is a Jordan domain, the space 'S.A \) (the analog of IT In the

unit disk; see Privalov [9]) satisfies AS. when, and only when,

-i ^ satisfies the Smirnov condition ([9], pÂ» l59 ffÂ«)Â» And

indeed, Theorem 1 below is false in 'S.^il I ) when - - fails to

satisfy the Smirnov condition, since the norm closure of the

polynomials ( equivalent ly: of the bounded analytic functions)

are a proper closed Invariant subsp^ce of E^{^ I ), which does

not consist of all multiples of a single function.

Examples of Hilbert spnces satisfying!; our axioms are (1)

Ep( - ^ ), where -ii 1 is a Srairnov domain, and (ii) for any .:. â€”

(Smirnov or not), the norm closure (with respect to square integr-

ability on the boundary) of bounded analytic functions in > ^ .

^'e remark also that the present theory generalizes into the

following context: all functions considered (both those of H and

those of B( - ^)) are restricted so as to be invariant idth respect

to a given group G of conformal self-mappings of - L â€¢ Then

Theorem 1 below continues to hold, r-here â€¢â€¢ (z) is also invariant

with respect to G. ''e shall not pursue further, however, this

line of generalization.

2. The Main Theorem

Theorem 1. Let H be a Hilbert space of analy tic f imc t i on s__a j^

-ns; - '^ '- rJ. bean Â«? - ct-ri:r i-'i^rf -:. :iT. .( H'^J isceqe I'd'i rer ej 00 a fl OB jjcJ o.ei.c;ae'X -^j'-c-t U ' *. -vr.o

'ipstlBiensp to aa-M

cjeT-'oe r ..' ,,^ ,' ,. .;.^^_;! â€¢ "^

described In S 1, satisfying axioms Al Â«Â«- A6Â» Let 3 de n ote a

closed subspace of H vlth the property that gf f e H whenever

e B, f e H. Then S a HJ (t)| = 1 aÂ«eÂ«)Â«

Proof of theorem .

1) Let ky (z) denote the reproducing kernel of S. ''e prove

first that for every X, ^ e i I

(1) k^(^) K^(t) K^ (t) = K^(^) k^(t) k^rry

for almost all t e f" â€¢

Indeed, applying A 6 vith

f^(2) = k^(^) K^(z) , g^(z) = K^(z)

f^Cz) = K^(^) k^(z) , g2(z) = k^(z>

we will have proved (1) if 'je verify the tvo relations

(2) k^it^) < K^, K^ > = \{K) < k^ , k^ >

(3) \iK) < K^, K^ > = V^iK.) < k^ , k^ >

for every e B, Using the facts that k^ e S, k^ e S, and

the reproducing properties of the kernels k and K, these relations

are equivalent to

T

,(Â«''^(t) iv (t) , ^ e E are valid. Choosing such

a t-value we then see from (10),

(11) l*.(^)l < 1 , ^ e E .

â– J wis :-'Oj'i BJJ Jy-.'-

nO/.t.'-'X'; .'.:â€¢ ' â– R:{i or.

.,:â– â– [iij^i ~ [^}

ii?

/"

( S ; ^.7T

A(:iI.i'Or:.c'i a'' 7

^i^ - ,

avpfi s. f .1 ^

iTi

r.Jr/-^%

r^ (

1 â€¢Â«:) Â»!'

rrs eiii lo I'sbn â– â– â– â€¢i..C''

(s)

'^D

10

Then ve hnve

(16) I u;^(z)l < 1

and, from (13) -â€¢'nd (lU), by the skew-symmetry of )|f^(z) we have

(17) lim '^'J'^^) = ^x^^o^ ' ^ ^ ^

nâ€” >oo

(18) lim oj^(\) '^^U) = ^^\(K) , X e E, ^ e E

n - >oo

Thus the sequence of functions tJ (z) is uniformly bounded by

1 and converges polntwi.je on a dense set. Hence it converges

uniformly on compact subsets of ^- '~ to a function ^J {z) s^tisfyli^g

(19) \(^) = '-^ C^) ^^ (?)

initially only for \ and ^ in E, and hence for all X and ^ in - - ,

by continuity. From (19) we get, setting X = ^ :

(20) r]f^(X) = 1^-' (X)I^ , and also

(21) 1*^(^)1^ = 1^ - (X)|2 I ^^'(^)l^

holding X fixed and letting ^ tend non-t.Â«in,qentially to the

boundary point t, we see from (6), (20), (21) thrt I - ' (^) | -> 1

for almost all t. Thus, â€¢ â€¢â–

(22) I (â– -' (t) I = 1 a.e.

3) ^'e can now easily complete the proof of Theorem 1. Let S^

denote the closed subspace H ;

â– o-}lr:i:iic

I "^ '? ''V ' "^ '^ O 5 â– ""' i"!C 'J " ' ' 'â– ' ? '^f ^

.fOr-i-TlftOG

â– 'XCilJ.;.

(

\ ' '

iiri!-: _Â»

(^.^^^l; ixO

OB Is Â£â€¢.:

a Lx:

"iC^)' 1 ^l(x)

-'c O'-/ ,3 ^n^^ q T

ri"r- Â«^ 1

'^ nc Tj 'if. It Tii^" r.-T"''^ -.-[.â– 'â– â€¢'â€¢â– o' â– forr â– Tf!? 6

.0"i;2 ;;Â£=-

(:=. )'â– ' (&)./

leva lo'i ^levoO'foM â€¢ ,

11

we have

< g, k^> = {z) f(z), K^(z) i^J (ii) t^;(z) >^

= ^- (K) < -' f, - K^ > = (^) < f, K^ >

= g(^)Â« Hence k^(z) is a reproducing kernel for S,, vhence

by the uniqueness of the r.k. S. = S. Finally, the uniqueness

of u.' (z) is irnmedlnte from (20), which shovs th-t cfort ^-ri* Ms {O- '"- S'-JOIXol r.

ir.Qiili ^.y:b''i'od p cj: 7lSl i-

simply connected) the Rieriann mapping function of -onto |w! < 1

taking a into w = 0. Moreover the solution to problem B is the

same without the assumption 0(a) = rince the extremal function

in the wider class must in fact vanish at a. Here however we do

not presuppose this Information, as we -'ish to discuss problems

A and B from the standpoint of Theorem 1.

Denote by 5 = 3(a) the closed subsp-^ce of H consi.'^ting of

all f e H which vanish at a. This is an invariant subspace

ajid so, by Theorem 1, and (2.23)

(1) k^(z) = K^(z) iJ iK) LW (z) â–

SI

fi-i

, â– :â€¢ ,, )S -J C^ *k iTjeXio'X'-I

â– w

ob J ....â– â– â€¢â– ^rt r.o 'to"

â€¢ â€¢ ;-, â– â– ' â– â– * J . â– " â– ^

(rs.:a) tin . â– â– â–

(s) ^'-C^) â– â– (s)^x - (>')^jf (I)

where ky Is the r.k. of S, and K^ is the r.k. of H. Here

'-vj (z) e B(-'^ i- ) and has modulus one a.e. on f â€¢ Moreover

v> (a) = because of A7Â» "e show that this function -'(z) = "-i(z)

is the unique extremal function for problems A and B ; (we use

the word "unique" to mean : unique apart from n const -nt factor

of modulus one). Consider first problem A, and let )^ be a

"competing function" of norm one. Then 0Ky belongs to S snd so

|0(^) |2 K^(^)2 = I < 0{z) K^(z) , k^(z) > f

< I uJ (^)l^ kAK)^ , by A5Â« and SchvprtzÂ» inequality.

This proves the assertion regarding problem A, the uniqueness

following by the condition for equality in Schwartz' inequality.

As for problem B, we have, for f e S :

f(^) = < f(z), k (z) > , hence

(2) f'(^) = < f(z), -^ k^ (z) >

and from (1) we have

(3) ~ k^(z) = K^(z) -^jHK) - (z)

+ [ ^â€” K^iz) ] L- (^) .J(z)

Let us now substitute in (2) 0{z) K (z) in place of f(z), nnd

3.

a for ^, and apply the Schviartz inequality:

PI

019!1

â– 5 a;i7 a| :' buB ^r fc

Â©ri^ii;.;

â€¢je)VOÂ£>r nox:fpniTl eirfct J-flf't voris o" j'^a lo f .i.ri)^^ = fn)

JB â– .

'irjcfoj'^'l .â– tf'^'?;.rfO!> " nt^T*!. jTr-cTP ^if/ox-^ ^ .i'" od^ ''s'r.'olr'f '" ftr*-' â€¢?.''d'

08 fjn" "^ 07 â– ';:%toi9.'f ^A%

'net 3:

> ^ > ii w

,n" J-orm.oo

n

1 â€¢: â–

'(^:

.^arxij

â€¢'^^-:; S^I-t^

eoneii t <

r

,(::)^ ^^

3).-

&V" n â€¢>

\^A ' ' .T^.T -' {-?^.. V

\,^ * -L,*jimj^* â€¢Â«Â«Â«>. i^v>'j.> .1 ii.,*/,* *y

11;

|^Â«(a)I^ K (a)2 = I Lj'(a)|^ | < jZS{z) K (z) , c^y (z) K (z) > |^

Si a o.

2 2

< |cJ'(a) I K (a) , and the essertlon follo-'s as before*

Now, k^(z) can be expressed in terms of K (z) by the formula

(here it is convenient to write lc(z, ^), K(z,^) for k^(z),

Ky(z) respectively) :

j K(z,?) K(z,a) j

! K(a,^) K(a,a) |

(U) k(z,^) =

K(a,a)

(i^) is evident by the uniqueness of the rÂ»k. since the ri^'^^ht

side belongs to S for each ^, qnd is a r.k. for SÂ«

Comparing this with (1) gives

iK(z,^) K(z,a) .

(5) -^ iK) u'(z)

!K(a,^) K(a,a) '

Kiz.K) K(a,a)

Setting z = ? gives

!K(^,^) K(^:,a) ,

2 iK(a,?) r(a,a): |,w ^. |2

K(^,^) K(a,M K(a,9)K(^,^)

Another useful formula gotten from (5) is

|K(z^,?) K(z^,a) I

'-'(z,) K(zp,^) \k{s,K) K(a,a) !

(7) ^ . = 2

"^ ^^2^ K(z^,?) |K(Z2,?) K(z2,a) |

lK(a,^) K(a,a) !

in which the right hand side is actunlly independent of ^Â«

411

/ â– .

â– 1 u-.-

â– O'lOiQ'l â€¢'"â€¢^.

Cot :?(â€¢â– ; dt -3 KÂ«;-? s-";? ba-} . ""^'O^?; "i 1>^)''.:I

nii-mo'x 3ri7 "ij (i)^;; "1.::^

â€¢â– :?* e-c

.{s)^>f ';'-."^ t^,r'" , ,:S ,W>I O.Jjl,.' â– : 1 :trfe?''-avn:-ri j*.?

\ â– â– %

\ .., ^J.^^

K-r^-

ir;

ir. / Ofji"

V - r

cJs)}r (;>,^/

s.

s

IC:^)

.od^oaii

15

We 9lao obtain, clearing of fractions In (5) ?ind differentiating:

K(z,a)

i-i^i^^ K(a,a)

z = ^ = a

(8) |o;.(a)|2

K(a,a)^

These results may be summarized as

Theorem 2Â« The common solution (unique apart from a constant

factor) of the extremal problems A and B Is '^ (z) = i-'-'^(2) ,

lAjhich is uniquely determined from the formulas (5), (6) once

arg vj (y,) is given at some point. The maximum in problem A is

the right side of (6), the maximum in problem 3 is the right

side^ of (8).

3.3 Let us nov consider a generalization of problem A:

Problem A Â« e B, W 0\\ ^ Â± ^ Â» ^(a^) = ... ^(a^) = .

Maximize |0(^)|Â« Here a., ... a are distinct points of -^ i. â€¢

Denote by S the closed (invariant) subsp^ce of H consisting

of all functions vanishing at a^, â€¢â€¢â€¢ a Â» Let k (z,^) denote.

the r.k. of S . Then as before,

n '

(1) k^(z,^) =. K(z,?) ^OJK) ^-^n^^^

Here ' -^ (z) e B, vanishes at z = a_ , ... a and has modulus

n X n

one a.e. on I â€¢ Just as in Â§ 3*2 T^e see that '-^J ^ Is the

unique extremal function for problem AÂ» . Since, on the other

hand, it is readily deduced that ^(z) = 'â– '-' (2)

(z)

n

t3ni:?.''i-j'-fe'ie1t.'itÂ£' hrtr. {?.; ; â– â– â€¢- ,^v; '/^-r-f';- 7r' r- ?..^>Â»e->

^/:.AM_._. AA,:V,_/

(â€¢3,0;

:. â€¢ ; â€¢:, :

itind.-.n-.

; : 'C'iq â€¢::

"*(%r)/i

:>(â– ' T

set:':Â© or:

'r n..

f , I

." -r

I- ."> r â– ;

series-

(./-Â«)^ '

H'.'f : ! r-

f.-:

^-S

TirrJ

i. 1.." \* CVu..

., ;U \' y-' ^ ^

^ t

'. fi.Q.

â– rtivfV

i^^Cj .-1 -â– ; â€¢, .S , .*< J ^^

,1 ,

once'

â– :!*'â– â– :> â€¢: ^- v^-!-^^â€ž^ Â«s)

â€¢n "n

INSTITUTE OF MATHEMATICAL SCIENCES

LIBRARY

[4 Washington Place, New York 3, N. Y.

IMM-NYU 289

JANUARY 1962

COURANT INSTITUTE

OF

MATHEMATICAL SCIENCES

I

Reproducing Kernels and Beurling's Theorem

HAROLD S. SHAPIRO

IS PER ir:

OF THE I

PREPARED UNDER

CONTRACT NO. NONR-285(46)

WITH THE

OFFICE OF NAVAL RESEARCH

UNITED STATES NAVY

^ 0). Passing to the Fourier transform

00

\

\

J.

F(z) = ) f{t)e ^ dt (Im z > 0) the problem similarly reduces

to identifying closed subspaces of the "Paley-^''iener space" of

the upper half -pi ^me vhich are mapped into themselves upon

multiplication by bounded analytic functions.

.rroi JfO{

â€¢v

â€¢ (.J- /â– -,f r,

. Sit J fa.' iU i â–

Â» â– X90 -r

c.no/joaj'l â– â– â– sri;^ nsqe .nYloq e *1ii 0x^7.:;

â€¢^ -â– .:t)'^ â– 'â– â– * >:-r'- ' > T â– ; -'â– } ''â– -. â€¢.â– ^-â– ,,..i- r.-f-: _ â€¢ â€¢. â–

r2c-r5tfD8'Â£ '^^j.-ul.c.^j Â» msXc-o'io erii (0 â– J) ;fb * ''Tict)',

Tfj ' '^'dcf 'to Eoo besolo r

:!n-? h' >ld-j' err* lo - .

X'

...^fRO^i- â– â– '!'-

In this paper we obtain a product representation for the

closed invariant (in the above sense) subspaces of a class of

Hilbert spaces of analytic functions which includes both Ii and

the Paley^'lener space of the half-plane. The class of Kilbert

spaces in question (vhich we define axiomatically ) is not very

2

general: intuitively it is a weighted H space in some simply

connected domain; but the more obvious generalizations of the

Beurling factorization are simply not true in any greater general-

ity, as simple examples show. The most interesting special cases

of our theorem could be deduced by conformsl mapping fron the unit

circle case, but ^-e prefer to give ^ direct proof based on a

systematic exploitation of the notion of reproducing kernel (r.k.).

That the Beurling factorization stands in close relationship with

the notion of r.k. is suggested by features in Lax's proof, which

operates however with Fourier analysis and does not use properties

2 -

of the r.k.

In Â§ 3 we show how, analogously to the discussion in Bergman

[2], chapter VII, and Garabedian [bj, important extremal properties

of bounded analytic functions, generalized Blaschke products, etc.

can be b?9sed on the r.k. and its properties by use of the theorem

of I 2. Thus the existence of a Hilbert space of analytic fvmc-

tlons which satisfies certain axioms can be made to yield proper-

ties of bounded analytic functions which are usually deduced from

the existence of Green's function, or the Rlemann flapping theorem.

These approaches are closely allied, but we believe our approcich

r

n , Ve assume moreover the following axioms :

nipfnob o-o-zosrij- â– ':'

-J c ;;> AÂ» .;

'.r- ilT.XfT J J-'S'

V L * Jt

sio:^i'tl;fos

riW- ^â™¦â€¢ri^'^rj'^ i^Ot"'^

^^Bfj.! YCf'XÂ«>MO'x-:r e-n^t er-

sfict ;en">.'

.â– XAPri'''ll

Tauo&dj

' fol erfd-

A 'â–

Al. For every X e J\ the linear functional

L,f =Â» f(X) Is bounded.

A2. To every f e H there is associated s "boundar;v'- function",

which T.re denote by f(t), defined a.e* on | ' ; and f(z) tends to

f(t) as z â€”> t non-t^ngentially, for almost all t e j â€¢

A3* The norm of f is uniquely determined by the values of If(t)I

a*e*

Alj.. (Maximum principle) If |f(t)| < M a.e., then |f(z)| < M,

z e -^ 1 â€¢ i

A5Â» Let B = B( i i. ) denote the class of functions ^(z) analytic

and bounded in J i ; then for every e B the transformation

f -> f msps H into Itself.

A6. For every f, , f^t g^* So ^ ^ which satisfy for every e B

the two relations

(i) < f^, g^ > = < f^, g2 >

(il) < f-L* g;i_ > = < ^2* ^2 "^

we have

(iii) f^(t) i^Ttl = f2(t) i^tT a.e. on P

Remarks on the axioms *

From Al we infer that H possesses a uniquely determined

reproducing kernel (r.k.) vhich we write K^(z) or occasionally

K(z, ^)Â» ^-'e assume known to the reader the following properties

of reproducing kernels (vihose proofs are immediate; the properties

,K > \ir)^\ fie. :i . .-.? H

'-. ( , .': I'i > r^ < 'J. . '"^ K > if '' )

dViirf dW

.err-o.:Kfi eri^ â€¢. = f (^) , for every f e H, ^ e A

(3) K^(z) = K^

(5) For every f e H we have the inequality

|f(^)|^ < K^(^) llfll^

As for A2 it could be weakened into a context where | is no

longer assumed locally rectifiable and "almost every^-here" is

defined in a modified way eÂ«gÂ« in terms of sets of linear measure

zero. The re'^der vill see from the proof to follow that the

existence of non-tangential limits is used only in a very weak

way*

Prom A3 it follows that multiplication by a ^ e B for which

||Zi(t)| = 1 aÂ»eÂ» preserves norms, hence also ixoner products.

It also follows that f(z) is uniquely determined from the a.e.

values of f (t ) .

The axiom A^ is necessary for the very formulation of the

problem of "invariant subspaces" to be meaningful. A5 Implies,

since the transformation f â€” > ^ f has a closed graph, that it

is bounded. A6 is perhaps the only axiom which does not seem

altogether natural; it is clearly satisfied by IT and the Paley-

'â€¢'iener space of the half-plane, because the real parts of bounded

; ( [I] 6Qa iablori -Ik

Â«i-,

-Ilenp&ni edd" .. 3 1 r,iBve i'.

S

111 ii i;-i^i' 2: lU)"!

it.

2

analytic functions, restricted to the boundary, are dense In L

norm in the space L {| }â€¢ It is here that we need in an essen-

tial way that ~- L is simply-connected (see Â§ 1^ below) â€¢ ^''hen -^ -_

is a Jordan domain, the space 'S.A \) (the analog of IT In the

unit disk; see Privalov [9]) satisfies AS. when, and only when,

-i ^ satisfies the Smirnov condition ([9], pÂ» l59 ffÂ«)Â» And

indeed, Theorem 1 below is false in 'S.^il I ) when - - fails to

satisfy the Smirnov condition, since the norm closure of the

polynomials ( equivalent ly: of the bounded analytic functions)

are a proper closed Invariant subsp^ce of E^{^ I ), which does

not consist of all multiples of a single function.

Examples of Hilbert spnces satisfying!; our axioms are (1)

Ep( - ^ ), where -ii 1 is a Srairnov domain, and (ii) for any .:. â€”

(Smirnov or not), the norm closure (with respect to square integr-

ability on the boundary) of bounded analytic functions in > ^ .

^'e remark also that the present theory generalizes into the

following context: all functions considered (both those of H and

those of B( - ^)) are restricted so as to be invariant idth respect

to a given group G of conformal self-mappings of - L â€¢ Then

Theorem 1 below continues to hold, r-here â€¢â€¢ (z) is also invariant

with respect to G. ''e shall not pursue further, however, this

line of generalization.

2. The Main Theorem

Theorem 1. Let H be a Hilbert space of analy tic f imc t i on s__a j^

-ns; - '^ '- rJ. bean Â«? - ct-ri:r i-'i^rf -:. :iT. .( H'^J isceqe I'd'i rer ej 00 a fl OB jjcJ o.ei.c;ae'X -^j'-c-t U ' *. -vr.o

'ipstlBiensp to aa-M

cjeT-'oe r ..' ,,^ ,' ,. .;.^^_;! â€¢ "^

described In S 1, satisfying axioms Al Â«Â«- A6Â» Let 3 de n ote a

closed subspace of H vlth the property that gf f e H whenever

e B, f e H. Then S a HJ (t)| = 1 aÂ«eÂ«)Â«

Proof of theorem .

1) Let ky (z) denote the reproducing kernel of S. ''e prove

first that for every X, ^ e i I

(1) k^(^) K^(t) K^ (t) = K^(^) k^(t) k^rry

for almost all t e f" â€¢

Indeed, applying A 6 vith

f^(2) = k^(^) K^(z) , g^(z) = K^(z)

f^Cz) = K^(^) k^(z) , g2(z) = k^(z>

we will have proved (1) if 'je verify the tvo relations

(2) k^it^) < K^, K^ > = \{K) < k^ , k^ >

(3) \iK) < K^, K^ > = V^iK.) < k^ , k^ >

for every e B, Using the facts that k^ e S, k^ e S, and

the reproducing properties of the kernels k and K, these relations

are equivalent to

T

,(Â«''^(t) iv (t) , ^ e E are valid. Choosing such

a t-value we then see from (10),

(11) l*.(^)l < 1 , ^ e E .

â– J wis :-'Oj'i BJJ Jy-.'-

nO/.t.'-'X'; .'.:â€¢ ' â– R:{i or.

.,:â– â– [iij^i ~ [^}

ii?

/"

( S ; ^.7T

A(:iI.i'Or:.c'i a'' 7

^i^ - ,

avpfi s. f .1 ^

iTi

r.Jr/-^%

r^ (

1 â€¢Â«:) Â»!'

rrs eiii lo I'sbn â– â– â– â€¢i..C''

(s)

'^D

10

Then ve hnve

(16) I u;^(z)l < 1

and, from (13) -â€¢'nd (lU), by the skew-symmetry of )|f^(z) we have

(17) lim '^'J'^^) = ^x^^o^ ' ^ ^ ^

nâ€” >oo

(18) lim oj^(\) '^^U) = ^^\(K) , X e E, ^ e E

n - >oo

Thus the sequence of functions tJ (z) is uniformly bounded by

1 and converges polntwi.je on a dense set. Hence it converges

uniformly on compact subsets of ^- '~ to a function ^J {z) s^tisfyli^g

(19) \(^) = '-^ C^) ^^ (?)

initially only for \ and ^ in E, and hence for all X and ^ in - - ,

by continuity. From (19) we get, setting X = ^ :

(20) r]f^(X) = 1^-' (X)I^ , and also

(21) 1*^(^)1^ = 1^ - (X)|2 I ^^'(^)l^

holding X fixed and letting ^ tend non-t.Â«in,qentially to the

boundary point t, we see from (6), (20), (21) thrt I - ' (^) | -> 1

for almost all t. Thus, â€¢ â€¢â–

(22) I (â– -' (t) I = 1 a.e.

3) ^'e can now easily complete the proof of Theorem 1. Let S^

denote the closed subspace H ;

â– o-}lr:i:iic

I "^ '? ''V ' "^ '^ O 5 â– ""' i"!C 'J " ' ' 'â– ' ? '^f ^

.fOr-i-TlftOG

â– 'XCilJ.;.

(

\ ' '

iiri!-: _Â»

(^.^^^l; ixO

OB Is Â£â€¢.:

a Lx:

"iC^)' 1 ^l(x)

-'c O'-/ ,3 ^n^^ q T

ri"r- Â«^ 1

'^ nc Tj 'if. It Tii^" r.-T"''^ -.-[.â– 'â– â€¢'â€¢â– o' â– forr â– Tf!? 6

.0"i;2 ;;Â£=-

(:=. )'â– ' (&)./

leva lo'i ^levoO'foM â€¢ ,

11

we have

< g, k^> = {z) f(z), K^(z) i^J (ii) t^;(z) >^

= ^- (K) < -' f, - K^ > = (^) < f, K^ >

= g(^)Â« Hence k^(z) is a reproducing kernel for S,, vhence

by the uniqueness of the r.k. S. = S. Finally, the uniqueness

of u.' (z) is irnmedlnte from (20), which shovs th-t cfort ^-ri* Ms {O- '"- S'-JOIXol r.

ir.Qiili ^.y:b''i'od p cj: 7lSl i-

simply connected) the Rieriann mapping function of -onto |w! < 1

taking a into w = 0. Moreover the solution to problem B is the

same without the assumption 0(a) = rince the extremal function

in the wider class must in fact vanish at a. Here however we do

not presuppose this Information, as we -'ish to discuss problems

A and B from the standpoint of Theorem 1.

Denote by 5 = 3(a) the closed subsp-^ce of H consi.'^ting of

all f e H which vanish at a. This is an invariant subspace

ajid so, by Theorem 1, and (2.23)

(1) k^(z) = K^(z) iJ iK) LW (z) â–

SI

fi-i

, â– :â€¢ ,, )S -J C^ *k iTjeXio'X'-I

â– w

ob J ....â– â– â€¢â– ^rt r.o 'to"

â€¢ â€¢ ;-, â– â– ' â– â– * J . â– " â– ^

(rs.:a) tin . â– â– â–

(s) ^'-C^) â– â– (s)^x - (>')^jf (I)

where ky Is the r.k. of S, and K^ is the r.k. of H. Here

'-vj (z) e B(-'^ i- ) and has modulus one a.e. on f â€¢ Moreover

v> (a) = because of A7Â» "e show that this function -'(z) = "-i(z)

is the unique extremal function for problems A and B ; (we use

the word "unique" to mean : unique apart from n const -nt factor

of modulus one). Consider first problem A, and let )^ be a

"competing function" of norm one. Then 0Ky belongs to S snd so

|0(^) |2 K^(^)2 = I < 0{z) K^(z) , k^(z) > f

< I uJ (^)l^ kAK)^ , by A5Â« and SchvprtzÂ» inequality.

This proves the assertion regarding problem A, the uniqueness

following by the condition for equality in Schwartz' inequality.

As for problem B, we have, for f e S :

f(^) = < f(z), k (z) > , hence

(2) f'(^) = < f(z), -^ k^ (z) >

and from (1) we have

(3) ~ k^(z) = K^(z) -^jHK) - (z)

+ [ ^â€” K^iz) ] L- (^) .J(z)

Let us now substitute in (2) 0{z) K (z) in place of f(z), nnd

3.

a for ^, and apply the Schviartz inequality:

PI

019!1

â– 5 a;i7 a| :' buB ^r fc

Â©ri^ii;.;

â€¢je)VOÂ£>r nox:fpniTl eirfct J-flf't voris o" j'^a lo f .i.ri)^^ = fn)

JB â– .

'irjcfoj'^'l .â– tf'^'?;.rfO!> " nt^T*!. jTr-cTP ^if/ox-^ ^ .i'" od^ ''s'r.'olr'f '" ftr*-' â€¢?.''d'

08 fjn" "^ 07 â– ';:%toi9.'f ^A%

'net 3:

> ^ > ii w

,n" J-orm.oo

n

1 â€¢: â–

'(^:

.^arxij

â€¢'^^-:; S^I-t^

eoneii t <

r

,(::)^ ^^

3).-

&V" n â€¢>

\^A ' ' .T^.T -' {-?^.. V

\,^ * -L,*jimj^* â€¢Â«Â«Â«>. i^v>'j.> .1 ii.,*/,* *y

11;

|^Â«(a)I^ K (a)2 = I Lj'(a)|^ | < jZS{z) K (z) , c^y (z) K (z) > |^

Si a o.

2 2

< |cJ'(a) I K (a) , and the essertlon follo-'s as before*

Now, k^(z) can be expressed in terms of K (z) by the formula

(here it is convenient to write lc(z, ^), K(z,^) for k^(z),

Ky(z) respectively) :

j K(z,?) K(z,a) j

! K(a,^) K(a,a) |

(U) k(z,^) =

K(a,a)

(i^) is evident by the uniqueness of the rÂ»k. since the ri^'^^ht

side belongs to S for each ^, qnd is a r.k. for SÂ«

Comparing this with (1) gives

iK(z,^) K(z,a) .

(5) -^ iK) u'(z)

!K(a,^) K(a,a) '

Kiz.K) K(a,a)

Setting z = ? gives

!K(^,^) K(^:,a) ,

2 iK(a,?) r(a,a): |,w ^. |2

K(^,^) K(a,M K(a,9)K(^,^)

Another useful formula gotten from (5) is

|K(z^,?) K(z^,a) I

'-'(z,) K(zp,^) \k{s,K) K(a,a) !

(7) ^ . = 2

"^ ^^2^ K(z^,?) |K(Z2,?) K(z2,a) |

lK(a,^) K(a,a) !

in which the right hand side is actunlly independent of ^Â«

411

/ â– .

â– 1 u-.-

â– O'lOiQ'l â€¢'"â€¢^.

Cot :?(â€¢â– ; dt -3 KÂ«;-? s-";? ba-} . ""^'O^?; "i 1>^)''.:I

nii-mo'x 3ri7 "ij (i)^;; "1.::^

â€¢â– :?* e-c

.{s)^>f ';'-."^ t^,r'" , ,:S ,W>I O.Jjl,.' â– : 1 :trfe?''-avn:-ri j*.?

\ â– â– %

\ .., ^J.^^

K-r^-

ir;

ir. / Ofji"

V - r

cJs)}r (;>,^/

s.

s

IC:^)

.od^oaii

15

We 9lao obtain, clearing of fractions In (5) ?ind differentiating:

K(z,a)

i-i^i^^ K(a,a)

z = ^ = a

(8) |o;.(a)|2

K(a,a)^

These results may be summarized as

Theorem 2Â« The common solution (unique apart from a constant

factor) of the extremal problems A and B Is '^ (z) = i-'-'^(2) ,

lAjhich is uniquely determined from the formulas (5), (6) once

arg vj (y,) is given at some point. The maximum in problem A is

the right side of (6), the maximum in problem 3 is the right

side^ of (8).

3.3 Let us nov consider a generalization of problem A:

Problem A Â« e B, W 0\\ ^ Â± ^ Â» ^(a^) = ... ^(a^) = .

Maximize |0(^)|Â« Here a., ... a are distinct points of -^ i. â€¢

Denote by S the closed (invariant) subsp^ce of H consisting

of all functions vanishing at a^, â€¢â€¢â€¢ a Â» Let k (z,^) denote.

the r.k. of S . Then as before,

n '

(1) k^(z,^) =. K(z,?) ^OJK) ^-^n^^^

Here ' -^ (z) e B, vanishes at z = a_ , ... a and has modulus

n X n

one a.e. on I â€¢ Just as in Â§ 3*2 T^e see that '-^J ^ Is the

unique extremal function for problem AÂ» . Since, on the other

hand, it is readily deduced that ^(z) = 'â– '-' (2)

(z)

n

t3ni:?.''i-j'-fe'ie1t.'itÂ£' hrtr. {?.; ; â– â– â€¢- ,^v; '/^-r-f';- 7r' r- ?..^>Â»e->

^/:.AM_._. AA,:V,_/

(â€¢3,0;

:. â€¢ ; â€¢:, :

itind.-.n-.

; : 'C'iq â€¢::

"*(%r)/i

:>(â– ' T

set:':Â© or:

'r n..

f , I

." -r

I- ."> r â– ;

series-

(./-Â«)^ '

H'.'f : ! r-

f.-:

^-S

TirrJ

i. 1.." \* CVu..

., ;U \' y-' ^ ^

^ t

'. fi.Q.

â– rtivfV

i^^Cj .-1 -â– ; â€¢, .S , .*< J ^^

,1 ,

once'

â– :!*'â– â– :> â€¢: ^- v^-!-^^â€ž^ Â«s)

â€¢n "n

1 2

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