Lemma 13.4.1. If p(z) = >I ajzj is a holomorphic polynomial and if
0 < r < 1, then
1
27r
Ip(rexe) I
2
d0 =
N
I aj I
Proofi
j=0
By direct calculation, using the fact that
I
27r
eime
dO = 0
Lemma 13.4.2. If f is holomorphic on D, then
1
27r
00
Lj
If(reie)I2dO=
27r
n
j=0
2r2j
where aj is the jth coefficient of the power series expansion off about 0.
13.4. Classes of Holomorphic Functions
Proof. For r fixed,
N
E aj (Teie)j
f (Teie)
7=0
uniformly in 0 as N
oo. So
27r
f
27r
If (rexe) I2 d0 =
1
j2ir
lim
0
27r
N+oo
1
27r
lim
Noo 27r
0
N
lim E I aj I2r2j
AT
(by Lemma 13.4.1)
j=0
N
Eaj(r
j=0
iej
N
1: aj (rei° )
j=0
2
dO
2
dO
407
00

EIajI2r2j.
j=0
Corollary 13.4.3. With notation as in the lemma, f E H2 if and only if
EIaj12<oo.
Proposition 13.4.4. For f E H2(D), let
fr(eio) = f(reZ")
,
0 < r < 1.
Then there is a unique function f E L2(aD) such that
lim
IlfrfIIL2 =0.
Furthermore, IIf
IIL2 = IIf IIH2 = limr.i
IIfrIIL2.
Proof. Let 0 < rl < r2 < 1. Then
1
27r
Ilfr1  fr2IIL2
=
27r
)r
I f
(rleio)

f(r2eio)I2
d0
1
27r
27r
f
Ig(r2ete)I2d0,
where
g(z) = f
(r2z)
 f(z) E
H2.
By Lemma 13.4.2 the last line equals
00
E
j=0
12
\j
rl
I
r2 ///
Ia12T2'l.
(13.4.4.1)
408
13. Special Classes of Holomorphic Functions
Here the aj's are the coefficients in the expansion of f
.
Let E > 0. Since
I
IajI2A<oo,
we may choose J so large that
00
2
Iajl2 <
E
.
j=J+1
Now choose 0 < ro < 1 so near 1 that if ro < rl < r2 < 1, then
2
<2A for 1<j<J.
With J, rl, r2 so chosen we see that Eq. (13.4.4.1) is less than
J
00
2jajl2+
1'IajI2< .A+2=E.
j=0
j=J+1
Thus {fr}o<r<i is Cauchy in L2 and it follows that there is a unique
L2(3D) such that
I f  MI L2 > 0 as r > 1.
The last statement of the proposition is immediate since
IIfIIL2 = lim IIfrIIL2
r1
which by Lemma 13.3.1
= IfIIH2.
f
E
Theorem 13.4.5. If f E H2(D) and f E L2(8D) is as in Proposition 13.4.4,
then
f(z)=1
f(e)d(,
forallzED.
27ri
D( z
Proof. Apply the Cauchy integral formula to the function fr, 0 < r < 1,
which is holomorphic on D: For any z E D we have
f(rz) = fr(z) =
1
d(.
(13.4.5.1)
27rz
j
D (  z
Now for z fixed in D we have
II
1
II
(ZIIL2(aD)
1
1 IzI
L2(aD)

1
1 IzI
So for
I z I < r < 1 we have
13.4. Classes of Holomorphic Functions
409
1
fr 0
27ri
iD
(Z
d _
2
1
7ri
8D (
f(0
z
dS
2r
f2, (fr(eze)

f(eie)I
eie
eiO
z
dB
1/2
127r
I fr(eie) _
f(eze)I2 d0)
27r
C
j2,
(where we have used the CauchySchwarz inequality)
IIfTfIIL2'
1_IzI
40 as r4
Thus, as r , 1, Eq. (13.4.5.1) becomes
f (Z) = 2I iOD
1
eio  z
Theorem 13.4.5 is the sort of result we want to establish for every
HP, 0 < p < oo. A rather technical measuretheoretic result is required first.
Lemma 13.4.6. If cpj, cp are measurable on 3D, cpj, cp E LP (,91?), cpj * p
a.e., and
Ik IILP, then Ilk  PIILP > 0.
Proof. Let E > 0. Choose 8 > 0 such that if the Lebesgue measure m(E) of
E is less than S, then
fE I
(p(eio)IPd0
< E.
By Egorov's theorem, there is a set F C [0, 27r) with m([0, 27r)  F) <
S and cpj * cp uniformly on F. Choose J so large that if j > J, then
IWj(eio)
 o
(eio)I
< E for all 9 E F.
Finally, choose K so large that if j > K, then I IWAP
, 
IIW11LP < E
If j > max(J, K), then we have
Ivj(ezo)

cp(eio)IPdO
0
IF
I
I
vj
(eio)  p(eio) IP d9
F
IF
EPdO+ f 2'I(e°)I" dO
F
I(e°)IdO
+
IF
2
1
410
13. Special Classes of Holomorphic Fbnctions
<
27ru1 +
JF
21Icp(eie) In dO
+
2P IF I pj (e")II cP(eie)IPdO
+°
2(e) IP dO
.
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 Fall '19
 holomorphic functions