Lemma 1341 if pz i ajzj is a holomorphic polynomial

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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.
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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 N-oo 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 Ilfr-fIIL2 =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)
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408 13. Special Classes of Holomorphic Functions Here the aj's are the coefficients in the expansion of f . Let E > 0. Since I IajI2-A<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 r-1- 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
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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 Cauchy-Schwarz inequality) IIfT-fIIL2' 1_IzI -40 as r-4 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 measure-theoretic 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
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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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