THEOREMS ON POWER SERIES OF THE CLASS

Transcription

1 THEOREMS ON POWER SERIES OF THE CLASS GEN-ICHIRO SUNOUCHI (Received August 22, 1955 and in revised frm, Octber 29,1955) 1. Intrductin. Let be a functin regular fr [2 < 1. If, fr sme p >, the expressin is bunded as r -> 1, then the functin f(z) and its pwer series are said t belng t the class H p. It is well knwn that, if f(z) belngs t the class H p, then f(z) has a bundary functin (1.3) /(O = limfir^), ^ θ ^ 2π fr almst all θ and f(e iθ ) is integrable L ι \ Mrever if p > 1 a necessary and sufficient cnditin fr the functin f(z) t belng t the class H p is that the series (1.4) 2«is the Furier series f its bundary functin f(e iθ ). Hence, in virtiίre f M. Riesz's therem, if p > 1, the class H p is ismrphic t the class L p. In this case, the series (1.4) is summable (C, 6), 6 >, t the bundary functins /( * a > at almst all θ. The prblem whether in this result we may replace summability (C, ) by rdinary cnvergence remains pen, but if p = 1, the answer is negative (Sunuchi [7]). On the behaviur f pwer series f class H D n the circle f cnvergence, imprtant results were btained by Littlewd and Paley [6] and Zygmund [11] [12]. The main tl f Littlewd and Paley was an auxiliary functin (1.5) g*(θ) = g*(θ f) = (j (1 - r) X%r, θ) drj", g θ ^ 2π where X(r,θ)= ( *> Presented t the Meeting f Mathematical Sciety f apan n 23 May 195$ (Tky).

2 126 G. SUNOUCHI and But they prved an inequality therem cncerning t g*(θ) nly fr H p, p = 2k (k = integer) and their prf is very difficult. Later Zygmund [13] gave a cmplete and simple prf. In the ther papers [1 [11], he used anther inequality therem f Littlewd and Paley cncerning t (1.β) g(θ) = g(θ; f) = (f (1 - and gave a simple prf f the main result f Littlewd-Paley and many interesting generalizatins. The purpse f this paper is t give the generalized therem n g*(θ) and systematic treatment and generalizatin f therems n the pwer series f the class H p. 2. The functin g*(θ) fr the class HP ( < p ^ 2). The definitin f the functin g*(θ) is slightly less simple. It is given by the frmula 1 2Λ (2.1) t&ff) = &θ : f)= ( (1-rrdr ^ ^ ϊ If a = 1, gl{θ) reduces t the functin g*(θ) f Littlewd and Paley excepting cnstant factr. S we dn't distinguish between gf (θ) and g*(θ). It is knwn that g*(θ) is a majrant f many imprtant functins. Especially intervenes fr the partial sums f the series (1.4). Let us dente S n {θ) = 2 <(θ) = ^τ2 A«n z\ s v {θ\ (a > - 1) where <{θ) - ~r 2 At:] t v {θ\ {a > ) then (2. 2) τ%(θ) = n{<τ%(θ) - σ«_ Ύ {θ)} = rtίσ ;- 1^) cr ()}. Further put (2.3) ft«(β) = W(9; f) = (2 " 7 """ MΛ

3 THEOREMS ON POWER SERIES OF THE CLASS H p 127 then we have the fllwing relatin. The right half was given by Chw [1] and the left half fr a = 1 was remarked by Kizumi [5]. LEMMA 1. Fr the functin f(z) H P (2.4) Aι«{θ) % g (θ) < B Λ h«(θ). (Q<p< ), REMARK 1. Thrughut this paper, A*, B a, are psitive cnstants depending nly n a, and may be different frm ne ccurence t anther. REMARK 2. Fr the definiteness f cnjugate series, we put $c =. Remark 3. In prving therems in this paper, we can suppse withut lss f generarity that f(z) has n zers inside the unite circle. And in many case, it is enugh t prve therems fr f(z) regular n 2 <Ξ 1. Fr, prving therem fr f(rz) ( < R < 1) and making R tend t 1, we get the therem. PROOF. If we write (2.5) φ a (r, θ) = 2 ( A n? I τj() I * r*\ then (2.6) _ n = l 1 = A a ί (1 - rfvφccir, θ) dr. Since (2-7) we have, by ParsevaPs therem s that <2.9) <UW - B.f (1 - rrdrf by the definitin (2.1). On the ther hand, fr ^ r < 1/2 X-rT are limited abve and belw by psitive numbers, and

4 128 G.SUNOUCHI I \f'w**>)\* dφ. is nn-decreasing functin f r. Thus Cnsequently ϋ = / + / 1/4 by (2.9). Thus we have (2.4). LEMMA 2. If a > 1/2 <md < r < 1, then ί21) PROOF. Since /"iϊ^ίs (2.1D % where δ = 1 r, are bund abve and belw by psitive numbers, ί dc P <A f dc P - A f +A f -7t δ

5 fr 1-2a <. THEOREMS ON POWER SERIES OF THE CLASS H p LEMMA 3. If we put (2.12) f*(θ) = sup if h 129 (2.13) f*(θ) = sup ft ^ 1 /' h (2.14) ^ /δ) /r (2.15) l/ϊr^β+^>) I ^ B/*(^) (1 + I φ I /δ)^fc /r /f^) U (k > 1) wter^. S = 1 - r, flwrf /ϊ(ί) Z 2, ί//(^θ) # flwrf Λ < 2. This is essentially due t Hardy and Littlewd [3]. Since r lίt + <p) )=, L ί f(ei «)+u> >)P(r,u-φ)du, where P(r,t) is the Pissn kernel, by partial integratin, we can easily get (2.14). T deduce (2.15) frm (2.14) it is enugh t nte that, by ensen's in equality (2.16) H i l/(έ?ίc * +t)) ' P(r 'V dt \' k S i"/ IΛ«(ί+t> )I*P(r,t)dt. THEOREM 1. Iff(z) H p ( < S 2), and a > 1/p, then This was given in my previus paper [9]. We shall give a slightly simpler prf. PROOF. We shall begin with the case p = 2. (1) p = 2. Then fr 2α > 1,

6 13 G.SUNOUCHI ^ C* 2 n *M* [ d - rf{l - r) 1 " 2 * r**"- 1 * dr (by Lemma 2) i Hence the case = 2 is prved. (2) Suppse that < p < 2 and put F(z) = then F(2) is regular fr \z\ < 1 and belngs t H\ Since s that we have, by ParsevaPs relatin By Lemma 3, fr k < 2 this is smaller than α - rr Let α = (1 + 6)/ί (6 > ), /8 = (1 + θ)/# - l/λ(2/ί - 1) and k be sufficiently near t 2, then and the abve frmula is ^ f (1 - rf'^rf 'g^

7 THEOREMS ON POWER SERIES OF THE CLASS H p 131 Thus dφ (2/3 > 1). {g* a (θ f)}»dθ S F^ {F*(θ)}*-»{g*(θ F)}* dθ 2τc 2* ί Γ Λ{2-P)j2 ( Γ ^ FC a { I (F*(θ)y dθ\ II (g*(θ \ ί \ by the inequality f Hlder. Frm the maximal therem f Hardy and Littlewd, we have Γ I {Fΐ(θ)} 2 dθ t^a^l I Fie 16 )\ 2 dθ and s by the case (1), Γ P.- / Thus we prved Therem 1 cmpletely. COROLLARY 1. Iff(z) belngs t the class H p ( < p S 2), then W)\ 9 dθ. zjhere 2 [fit > IIP) H*(6) = { sup 2 ki'w) - This is a maximal therem cncerning with strng summability. PROOF. Let us put nβ is the index n when Hζ(θ) attains the supremum, then /* lay (2.3). Thus Crllary is immediate frm Therem 1 and Lemma 1. COROLLARY 2. If f(z) belngs t the class H p (<p<l 2), then' V7 K()l a

8 132 is cnvergent fr almst all θ. G SUNOUCHI This is immediate frm Therem 1 and Lemma 1. This Crllary has been ever prved by Chw [1]. Frm this, we can get the fllwing Crllaries by the well knwn methd. COROLLARY 3. Iff{z) belngs t H p (<ps 2), then the series 2 *>nc n e nίθ, w^ere^l (λw) a /w cnverges, is summable \C, a\,(a > lip) fr almst all θ. Fr the prf, see Chw [1]. COROLLARY 4. If f{z) belngs t H p ( < p<, 2), then fr almst all θ, the sequence {n} can be divided int tw cmplementary subsequences {rik} and {m/c}, depending in general n θ, and such that σf^iθ) tends t f(θ) and the series 2 */ w fc cnverges, where a >l/p. Fr the reductin f this Crllary, see Zygmund [1]. 3. The functin g*(θ) fr the class II P ( > p > 2). Fr the class H p ( > p > 2), we have the fllwing therem. This is essentially due t Zygmund [13]. The prf is als repetitin f his argument. THEOREM 2. If f(z) H p ( > p > 2) and a > 1/2, then (3.1) (tfxθwdθ^ap,.] \f(e iθ )\»dθ. PROOF. Let μ be a psitive number such that pβ μ and let ξ{θ) be any psitive functin such that (3.2) ί ξ»(θ)dθv' μ ^1. Then it is knwn that - (3.3) / (g*(θ))*dθ\ = {/ (g* 2 (θ)) Pi2 dθ\ and the inner integral = sup

9 THEOREMS ON POWER SERIES OF THE CLASS H p 133 (gz(β)ϋ ()dθ^a a l ζ[θ)dθ j δ 2Λ dr ί V t PA d< P- (δ = 1 - r) Let us put («, Ψ) = and (3.4) then the last integral say. Then It 1 - rβ"*-* Tt δ»-p - > +») + ι δ :^ - «)> = /+/, + fi) + f (<? - «)} -^ and S δ - 1 {ξ{φ + U) + ξ(φ - δ U)}^ = δ'^- 1 Γ B(«, ^)w 2Λ I 7 " + 2Λ δ 25-1 Γ B( L δ δ ί u' ιa 2a δ 25-1?*(^) ί u' ιa - 2 "- 1 du du

10 134 G. SUNOUCHI Thus the left-hand side f (3.3) f {^(θ)}' j i B(θ)dθ^F a j Bdrj \f(n»)\*ξ*{φ)dφ \fμ.f Γ [ ϋ frm the maximal therem f Hardy-Littlewd and the therem f Littlewd-Paley [6] (simple prf; Zygmund [14]). Thus we get 2ic ( Γ 2τr Γ and the therem is prved. Frm this, we can derive easily that if f(z) belngs t H p ( >p > 2), and a > 1/2, then 2τr ^ / and / I sup V I σ-ίf-w - /(β ίθ ) I a i d<9 g 2 >. β I /(β ί9 ) I * dθ 4. A prf f the therem f Littlewd and Paley. Frm Therems 1 and 2, we have especially, THEOREM 3. If f(z) belngs t H p (Kp< ), then (4.1) I {g*(θ)) v dθsapl \f(e iθ ) p dθ. Frm this, we have the fllwing and {S(θ)} p dθ S BP l/(^ίa ) I p ί», (ί > 1)

11 THEOREMS ON POWER SERIES OF THE CLASS HP 135,a* 2* / {μ(θ)} p dθ^c P I Λe ίθ )\*dθ (P>D where S(θ) is the functin f Lusin and μ(θ) is the functin f Marcinkiewicz. The functin f g\θ) is essenitally a majrant f these functins. The reductin f (4. 2) is dne by a mment's cnsideratin, but the reductin f (4.3) is smewhat difficult. Fr the detailed definitins and prfs, see Zygmund [13]. The main therem f Littlewd-Paley is cndensed in the fllwing therem. THEOREM 4. If f(z) belngs t P (1< p< ), then (4.4) A P f (4.5) B P<a, β 2/r l j l ^ W f (4.6) C P^β \f(^ψdθ^ {Σ\MΘ)\*W>dθ^C paβ (4. 4) is a cnsequence f Therem 3 and Lemma 1*>. The left-halves f (4.5) and (4.6) are prved by the fllwing results f Zygmund [11]. That is / " p/a fr 1 < p <. Fr the prf f the reverse inequalities, we need LEMMA 4. Let {f n (z)} (n = 1,2 ) δe «sequence f the functin f H p (Kp< ), dwd /^ί sn, C(Θ) dente the k-th partial sum f the bundary series ffniz). Then w = 1 A cmparatively simple prf was given by Zygmund [1], using Rademacher's functin. PROOF OF THEOREM 4. Frm Lemma 4, we have *> We suppse that the left-half f (4.4) is prved by anther methd.

12 136 G. SUNOUCHί / 2 "* 2 l/v*)~" dθ v=n k +l (Σ Σ 4 Σ" 1^^' dθ Thus (4. 5) is prved. On the ther hand «fc-»-l lg π; { 2* by (4. 2). Since 2 ^ s» Λ+1 (β) - σ. t+1 (ί) «+ \s nk (θ) - <r^() a + <r» M ( we get the required ineqnality 2 IΔ.I s iw 2 iw - w«ι + B Λ.,β 2 \*M-«* and get the frmula (4. β). Thus the therem is prved cmpletely. The fllwing crllary is deduced by the well knwn methd frm Therem 4. COROLLARY 5. If f(z) belngs t the class H p > a > 1, then (p > 1), and if β > fh+ifih \ sup I s*β) \Ydθ^A,,*,β f l/t«*) I 'dθ

13 THEOREMS ON POWER SERIES OF THE CLASS H p 137 COROLLARY 6. If { &} is any sequence f numbers f which each has ne f the values 1, 1, and if f ^ H p (p > 1), β > n^λjn k >a>l, then Γ"* ί l, p re= - 5. The pλver series f ϊ-elass. Cncerning with the pwer series f ϋγ-class, A.Zygmund [11] prved that (5.1) hι(θ) dθ^aj \f(e iθ )Ilg + \f{e iθ ) dθ + A and (5.2) {h(θ)) μ dθ<b μ ( 1/(^)1^), (</»<l). S, frm Lemma 1, we get THEOREM 5. Iff(z) belngs t the class H, then (5.3) g*(θ)dθ^a /(«") lg + /r*") rf + A' r 27C r~ π (5.4) (g*φ)rdθ^b^ \f(«p)\dθy, ( < μ < 1) The present authr has nt ever a simple and direct prf f this theprem. THEOREM 6. If f(z) belngs t H, then (5-5) {μ(θ)}dθsλ I(fe iθ ) lg + \f(e iθ )\dθ + A' (5. 6) {^F dθ S B r (j \f(e«) I dθj, (<r< 1) and j 11/2

14 138 G. SUNOUCHI This is immediate frm the fact μ(θ)<cg*(θ) which was prved by Zygmund [13]. 6. Pwer series f the class BP(O < p < 1). Fr the class H p φ < p < 1), we have a mre precise results than Therem 2. (6.1) (6.2) THEOREM 7. If f(z) belngs t H p f ί ( < />< 1) and a = lip, then \f(e«)i * lg- \f(e iθ )\dθ + A; PROOF. This case is reduced t the case p = 1. If we take < p < 1, a^ljp and p G(z) = {f(z)} p then fix) = a{g(z)y-ig'(z), and (l-ry«drί _ Frm (2.14) f Lemma 3, AG {Θ) ίl + L ll (S = 1 - r) where G*()= sup the right-hand side is smaller than (θ)}*«-vj (1 - ry " drj (1 _ ^fiΐ)^"^)^

15 ϋ THEOREMS ON POWER SERIES OF THE CLASS H» 139 -^g iθi Gψ. <sz(β' f)y dθ ^ A" { g*(θ;g)dθ\ P by Hlder's inequality. Frm the maximal therem and ^Therem 5, we get WΘ) p dθ<b P ] \G{<*»)\lg+\G(e*»)\dθ + B p Analgusly f \A<*») I' lg- /(e«>) I dθ + B p {g*"(θ f)y» dθ^a P I {G%θ)}^1-1 Hg*(θ G)}»» dθ -τt 2* Thus Therem is prved cmpletely. THEOREM 8, Iff(z) belngs t H*> ( < ί g 1), and a = IIp, then (6.3) {h a {θ)y ϋ (6.4) This is immediate frm Therem 7 and Lemma 1. This frmula was given by the present authr [9] by mre cmplicated methd. 7. Maximal therems f the Cesar mean f the pwer series f the class UP ( < p < 1). THEORFM 9. Iff(z) belngs t H p {<p^ 1) and ct>l/p- 1, then

16 14 G SUNOUCHI <7.D / {snp\σl{θ)\ydθ<a th <n< THEOREM 1. Iff(z) belngs t H p ( <p^ 1/2), and a=l/p- 1, then <7.2) { sup I σ %β)\ydθ%b p \ \fi<f*)) * lg + \f(e iθ )\dθ + B' p 2τc (7. 3) Γ { sup I <()»}^ <# Si C, f \f{e«)\» dθx, (<μ< 1). <n< \ / Therem 9 is a generalizatin f classical results f Hardy-Littlewd {4] and Gwilliam [2]. (7.2) f Therem 1 is an affirmative answer f a prblem f Zygmund [12. Frm (7.3) we can easily see that σ%(θ) (a = 1/p 1) cnverges t f(e iθ ) fr almst all θ. This was prved by Zygmund [12] fr < p < 1. Fr the case 1/2 < p < 1, the maximal therem is left pen, but cnvergence f σ%{θ) is prved in the next sectin. The present authr [9] deduced Therem 9 and Therem 1 frm Lemma 1 and Therem 8 with the aid f the fllwing lemma. LEMMA 5. Iff(z) = g\z) and a >, then,.,. where σ? τ (θ; f) is the a-th Cesar mean f the bundary series f f(z). Fr the detailed argument, see my previus paper [9. 8. Strng summability and rdinary summability f the pwer series. In Crllary 1, we have prved the maximal therem f the strng summability f σ%-\θ) (a > 1/p) fr H p (Kp<L 2). But if we give up the maxinal therem, then we can prve the mre precise result. THEOREM 11. If f{z) belngs t H p (1 <Lp < 2) and a = 1/p, then fc= fr almst all θ, where <q<p/(p 1). This was prved in the authr's paper [8]. The methd f the prf depends clsely upn the paper f Zygmund [12]. In his paper, Zygmund prved the strng summability therem f the functin f L and the Cesar summability therem f the series f the class IP ( < p < 1). The prfs

17 THEOREMS ON POWER SERIES OF THE CIASS H p 141 f bth therems have many features in cmmn, but the the details f prfs are different. After prving Therem 11, we can deduce frm that the fllwing Cesar summability therem. THEOREM 12. If f(z) belngs t H» (1/2 <p<l) and a = l/p± 1, then the series 2 c n e inθ is summable (C, a) t f(e ίθ ) fr almst all θ. PROOF. If we put then g(z) belngs t H\\ = 2p, 1< λ < 2), and (a + l)/2 = l/2p = 1/λ. Frm the frmula in [8], p. 225, we have that is n 2 Al> Ial 1 *- 1 {θ g) - a-lhθ g) * = ('* +1 ). a. e. fc = l n 2 ; g) - <r^+1)/2 (; g)\* = (w α+2 ), a. e. fc=l By Abel's transfrmatin, Frm Lemma 5, we have Since evidently σ * +1)/2 (#; r/) tends t g(e iθ ) fr almst all θ, if we take plynmial f e (z) near t f(z), then we can cnclude <r%(fi ί /) -*f(β ίθ ), a. e. as /2 ->-. Thus we get Therem 12 frm Therem The affirmative answer t a prblem f Zygmund. On anther cnjecture f Zygmund, we can.prve the fllwing therem. THEOREM 13. Iff(z) belngs t H p, then a ΪP/2 Γ (9. 2) / I Qsup L^»Wl-. ^ B p /(^) I * <*, ( < ί ^ 1, α = (9. 2) is deduced frm (9.1) by the usual methd. (9. 2) is an answer t the prblem raised by Zygmund [12]. But there is a slip in his riginal paper, s I prved (9. 2) in the case 1/2 < p<l, since this case is better than Zygmund's riginal cnjecture. After-that, I nticed his crrectin [I4 in such a frm as (9.2), s we will prve Therem [13 cmpletely. Fr the

18 142 G. SUNOUCHI prf f therem we need a lemma which was given in [9]. LEMMA. (A) Fr psitive a, Π "{lg («+!)}** " \lg(l-r)\*«l-rβ»» ψ <B) If we put integrals f {f**(θ)}' z and {f(θ)y z are majrated by the integral f PROOF OF THEOREM. The case 1 < p <; 2 f (9.1) was prved in my previus paper [9]. S we begin with the case p = 1. (1) p=l. Let us put F(z) = {/U)}i/* then F(z) 2*. Dente by s*- 1 /^^) and τ^\θ) the crrespnding (C, -1/2) sums etc. f the bundary series f F\z). If we put = / '"[j^yc'tf*»)) «F(r^+>> )) a l-rβ" l-rβ* = 2 2 s = 2 Σ ί Σ n=q *Ί/= dφ v- Applying Minkwski's inequality and Hlder's inequality successively, we btain l/2 (-2 ls*: ^ nn -» 1/2

19 THEOREMS ON POWER SERIES OF THE CLASS H" 143 ^ A a 2 (v + I) 1 ' 2 1 τ?l*(θ) 11 lg (1 - r) 1»* r* F+«?) (by (B) f Lemma 2) ( 2(^ + i) 1/2 1 / 2 S Φ(r, fl) ^ CΛFKΘ)y\l - r)-m lg(l By the frmula f Lemma (A) κ- Cnsequently ϊit (F\θ)f dθ 1 ' 1 ί Γ {F*\θ)f dθ V " by Lemma (B). The case 1/2 < p < 1. put then G(<ε) -ίγ. Dente by s*(^), σ*(θ) and τ*(^) the crrespnding partial sums, Cesar means and their differences f the bundary series f G(z). Then we have

20 144 G.SUNOUCHI r " by the case prved. Mrever we have and let us put then we can prve analgusly t Lemma 3 in [9] and r r Wt(θ)\ ^lg( W + " I I(θ)dθ^ \G(έ»)\dθ. On the ther hand, since a = 1/p, and G(z) = {f(z)} p, we have f(z) =. and by applying Hlder's inequality, W n{lg(n-\- l)} 2lP ) dy dcp \ [ \ \l re tφ X λ j l- r^ a(a-«), ί ( \G(r<?<-»)\*. y«(l - r) -r) 3 '2τr \ l_ r^ 2/(2-«) ^)

21 THEOREMS ON POWER SERIES OF THE CLASS HP 145 where and Thus we get \ H(θ)dθ<G\ \G(e iθ )\dθ. ϋ ^A P \G(e* θ )\ ^ Fr the case p = 1/m and l/(m -\-l) <p < 1/m (m = 2, 3, ), we can prve similarly, cf. Chw [1. The reductin f (9. 2) frm (9.1) is identical t the prf f Therem 6 in [9]. That is frm Lemma 5, where W n φ,f)\ ^A ana 2, ik+ίy^ This is smaller than C α (lgny>». A.I. ^\ή a+^- θ L8}]l_ a n ύ (k + iy-* fiz) = g\z), and a = l/p- 1. 4^4:1^11^1 + B;f SUP wr^θ; S) \ I s (k+ΐ) l<>»<» Cnsequently

22 146 G SUNOUCHI / ( I a (θ f)\ 1 p ) sup * σn^ '-^'* \ dθ } ^] *i '! + C' < sup cr (βf+1^z(; ") > lθ</i< ' ϋ ^ D«I^^ I'xdθ^Daj IΛOI» ^ by (9.1) and a = l/ - 1. Thus we get the frmula (8. 2). LITERATURES [ I ] H. C. CHOW, An extensin f a therem f Zygmund and its applicatin, urn. Lndn Math. Sc, 29(1954), [ 2 ] A. E. GWILLIAM, Cesar means f pwer series, urn. Lndn Math. Sc, 1 (1935), [ 3 ] G. H. HARDY AND. E. LITTLEWOOD, The strng summability f Furier series, Fund. Math., 25(1935), [41 G. H. HARDY AND. E. LITTLEWOOD, Therems cncerning Cear means f pwer series, Prc. Lndn Math. Sc, 36(1934), [ 5 ] S. KOIZUMI, On integral inequalities and certain applicatins t Furier series, ThkuMath. urn., 7(1955), ]. E. LITTLEWOOD AND R. E. A. C. PALEY, Therems n Furier series and pwer series (1), urn. Lndn Math. Sc, 6(1931), , (II), Prc Lndn math. Sc, 42(1937), 52-89, (III), ibid. 43(1937), [ 7 ] G. SUNOUCHI, A Furier series which belngs t the class H and diverges almst everywheie, Kdai Math. Seminar Reprts (1951), [ 8 ] G. SUNOUCHI, On the strng summability f pwer series and Furier series, ThkuMath. urn., 6(1954), ] G. SUNOUCHI, On the summability f pwer series and Furier series, Tόbku Math. urn., 7(1955), 96-19, [1] A. ZYGMUND, Prf f a therem f Paley, Prc. Cambridge Phil. Sc, 34 (1938), [11] A. ZYGMUND, On the cnvergence and summability f pwer series n the circle f cnvergence (I), Fund. Math., 3(1938), [12] A. ZYGMUND, On the cnvergence and summability f pwer series n the circle f cnvergence (II), Prc Lndn Math. Sc, 47(1942), ] A. ZYGMUND, On certain integrals, Trans. Amer. Math. Sc, 55(1944), [14] A. ZYGMUND, Prf f a therem f Littlewd and Paley, Bull. Amer. Math. Sc, 51(1945), MATHEMATICAL INSTITUTE, TOHOKU UNIVERSITY.