on banach spaces of absolutely and strongly convergent fourier series

Acta Math. Hung. 55 (1--2) (1990), 149--160.

O N B A N A C H S P A C E S O F A B S O L U T E L Y

A N D S T R O N G L Y C O N V E R G E N T F O U R I E R S E R I E S

I. SZALAY (Szeged) and N. TANOVIC-MILLER (Sarajevo)

Introduction, definitions and preliminaries

For p=>l, let L v denote the Banach space of all real or complex valued 27z- t 1 - ~l/p

periodic integrable function f with the norm Ilfll L~ = [ ~ - j I fl~) �9 Here, the integ-

ral is taken over any interval of length 2re. Let C denote the Banach space of all real or complex valued 2re-periodic continuous functions f with the norm Ilfllc= -- sup If(x)I.

Considering a trigonometric series

(1.1) ao/2 + Z a k COS kx + b k s in kx k=l

let sn(x ) and an(x ) be the ordinary and Ces~ro-1, n th partial sums of (1.1), respectively. If (1.1)is a Fourier series of some function f E L 1 we shall write s ,f , o-,f and fc(k), fs(k) for the partial sums s,, o-n and the coefficients ak, bk, respectively.

Let I be the ordinary and IIJ the absolute convergence. The classes of uniformly and absolutely convergent Fourier series are described by

q / = {fEC: s , f ~ f I uniformly}, ~r = {fEC: s , f - ~ f ] I I } .

Clearly qZc{fCL~: s , f ~ f I a.e.} properly, while

sr = {fEL~: s , f ~ f [ I I a.e.} = {f: s, -~f[I[ a.e.}. We define

(1.2) IIflTm = sup llsnfllc and

(1.3) Irfl[~r = I.~(0)1/2+ ~ IL(k)l + IL(k)!. k=l

We collect the well known facts about these classes in the following statement, see [1], [3], [4] or [11]:

TI-~OREM A. (i) q/, [I lie and d , [I l[ ~ are Banach spaces. (ii) d c q / properly and llflloSIIfll~,.

(iii) d is not a Banach space under the norm (1.2) and ql is not a Banach space under the norm II IIc-

In this paper we discuss similar properties of classes of functions, described by replacing ordinary convergence I, or the absolute convergence [I[, by the strong

150 I. SZALAY and N. TANOVI~-.-MILLER

convergence of index 3~->1, denoted [I]z, or the absolute convergence of index A>I , denoted Illx. These concepts have already been applied to trigonometric and Fourier series in [5] through [10]. The obtained results put many classical statements or questions into a new perspective. We believe that this paper wilt help pursue that goal further.

Let 2 > 0 and let (sk) be a sequence of real or complex numbers. We say that (sk) is strongly convergent of index 2, to a number t, and we write sk--*t [I]4, if

(1.4) 1 ~ [ ( k+ l ) ( s~ - t ) - k ( s k_~- t ) l 4 = o(1) (n ~,~). n + l k=o

Here and in similar expressions s_~=0. The strong convergence of index 2 = 1 is called simply the strong convergence and is denoted by [I].

We say that (sk) is absolutely convergent of index 2_->1, to a number t, and we write s k i t 1114, if s k i t I and

(1.5) ~ k ~-1 ISk--Sk_l[ 4 < co. k=0

I f 2=1 IIl. This however is not true for 2>1 , see [7].

The following properties show the relationships types. Here C~ denotes the Ces~ro method of order 1.

1) Let 2_->1. The following are equivalent:

then (1.5) implies sk-*t for some t, and this is just the absolute convergence

(i) sk --* t[I]z

(ii) sk -* t I and

between these convergence

lz

(1.6) n+l l k~=ok4lsk--Sg_l] 4 = o(1) (n -*~) ,

(iii) sk --* t C1 and (1.6).

2 ) [ I ]4=~[ I ]~ for 2 > # > 0

3) [ I1~=*[114=~I=~Ca for 2=> 1

4) Let 2=>1. Then s k i t lI[4 is equivalent to sk--*t CI and (1.5). For the simple proofs 1), 2) and 3) see [7], [9] and the references cited there. We now verify 4). By definition clearly sk--*t lII4 implies s k i t C1 and (1.5). Conversely, suppose that s k i t C a and (1.5). Now (1.5) clearly implies (1.6) and therefore by 1) sk--*t [114. Consequently by 3) s k i t I, which together with (1.5) means that sk-~t 1114.

Finally we should remark that 1114 and ]Ilu are incomparable for 2r This can be illustrated by simple examples, and it will be discussed in Section 3 for the Fourier series.

In view of 1), 3) and 4) we restrict our attention to [I]z and [I1~ convergence of trigonometric and Fourier series for indices ) .~1. We consider the following

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A B S O L U T E L Y A N D S T R O N G L Y C O N V E R G E N T F O U R I E R S E R I E S 151

classes of functions : Y~ = {fCC : s , f ~ f [I]~ uniformly},

~r = {fCC: snf ~ f [Ilx uniformly}.

It is obvious that in this definition fCC can be replaced by fEL 1. Moreover by the above S ex and d a are precisely the classes of all sums of uniformly [I]z, respectively lIl~, convergent trigonometric series, i.e.,

SeX = {f: s , ~ f [I]z uniformly} and d x = {f: s,,-~/lIlx uniformly}.

For 2 = 1 d z = d clearly. However, as it can be seen from our earlier results, [5], [7] and [10], the classes Sex for 2=>1 and d a for 2:>1, behave more like q / than like ~r Namely, we have the following proper inclusions :

Sex c {fELl: s ~ f ~ f [ I ] ~ a.e.} for 2 => 1, and

d z c { f E L ~ : s ~ f ~ f l l [ ~ , a.e.} for 2 > 1 .

The class Sex for 2 = 1 will be denoted by Se. From Theorem 4 in [7] it follows that ~r properly. We have showed in [6] and [10] that Seand Sez for 2>1 are Banach spaces. In the following theorem we collect some of the properties of these classes established in [6] and [10].

THEOREM B. Let 2=> 1. Then

(i) fESe z i f and only i f fEC and

II

1 .Z__okx(lf~(k)l+lf~(k)l)X = o(1) (n ~oo). (1.7) n + 1 =

(ii) SeacSe" for 2:>/~. Sezcq! properly and Sez is not a Banach space under the norm 11 [1~.

(iii) I f fEC then (1.7) hoMs i f and only i f

[ 1 " ,,~-~ =o )~'~ (1.8) "-:7:-7-k~k~'ls~f-s~-lfl~ c = 0(1) (n -_,-oo).

(iv) Sea is a Banach space under the norm

[Iflls~,0-sup -~-~=o[(k+ l)skf--ksk-~fl~J ]to (1.9)

and (111o) [ i f It < FIfIl < ][f l l C ~ [ q[ ~ 6 a z , 0 "

(V) I f f E Y a then [Isnf-f]]~o = o(1) (n-v o@

Statements (i) through (iii) are contained in Theorem 1, Lemma 1 and Remark 1 in [10]. Statements (iv) and (v) follow from Theorems 2 and 3 in [10]. The corresponding results for the case 2= 1 were obtained in [6].

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152 I. SZALAY and N. TANOVI~-MILLER

The object of this paper is to examine other natural norms for the classes 5 "z, 2 _ -> 1 and consider similar questions for the classes ~r 2 >1. Inspired by these equivalent norms for 5e~, in Section 3, we introduce the corresponding norms for the classes sr x, 2>1 ; show that a # are also Banach spaces, discuss other properties of these classes and examine their relationship to the classes 5ca.

2. Banach spaces 6 a~ and some equivalent norms

Although the essential properties of the classes 5 a~ are established by Theorem B and other results proved in [6] and [10], it is natural, in view of (i), (iii) and (iv) of Theorem B, to investigate other norms expressed in terms of the coefficients and the q/-norm.

For f C ~ ~, 2 ~ 1 let

( 1 " ~11~ [ 1 n ~1I~11 (2.1) 1,,/1>~,1--su 2 t-~-g-7-2~is~/-s~_~lt~J +t-~mrfolSJl"J IIc

1 ll.;. (2.2) lISting,,- s~p (-7-4q-k~ls~S-~-~Sl ') +IISlI~,

1 n -r (2.3) ilSil,:.~ -- sup (74T,,~ k~(Ig(k)l + ig(k)lYJ + IlSii~.

REMARK 1. That these expressions are finite and do define norms on Sa~ follows immediately from Minkowski's inequality and Theorem B. Moreover, by Min- kowski's inequality

1 " 'dl~ (2.4) ( -~T]- 2o I(k+ 1)s~f-ks~-,fi~J <--

<= k ~ l s k f - s ~ - a f l ~ + / - - - : - 7 ~ ' I sk f l ~ <= k n--t- 1 k=O #

n x:Wx t 1 ~11,~ f {, 2k~(IL(k)l+ J -~ _<-- IL(k) l ) * is.si*J t n S - I k=o k=O

and consequently

(2.5) Ilflls,~,o <-- llflb'*,l -<- Ilflls,<~ <- llfll~,-,3.

Hence Ilfl l~a are finite for each rE<9 ~ and i=O, 1, 2, 3 by (2.5) and (i) of Theorem B. To establish the equivalence of these norms it is useful to write the following

simple lemma.

L ~ t A 1. For any sequence of real numbers (ek)~=o with Icos sol =1/2 and 2_->1,

1 fi~ 1 2zoo [cos(kX+~k)l ~ d x > - ~ T for all k - - 0 , 1 , 2 . . . . .

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ABSOLUTELY AND STRONGLY CONVERGENT FOURIER SERIES 153

PROOF. For k = 0 the conclusion is obvious by assumption that [cos s0[ = 1/2. By HSlder's inequality we have

1 Icos (kx + ek)l a dx => t,--~-~ o lcos (kx + ek)[ dx . 2n o

The inequality for k = 1, 2 . . . . is also obvious, noticing that by the above,

Cos(kX+~k --> 1+ c o s 2 ( k x + ~ k ) d x = -~T.

THEOREM 1. Let 2>=1. The norms II Ily~,i i=o, 1,2 ,3 are equivalent.

PROOF. We first establish the equivalence of [[ 1[~,~ for i=0 , 1, 2. By (1.9), (2.2) and Minkowski's inequality, clearly,

= + II/ll,, <= 1l/[l~,o+2llfll~,. k=0 C

But IIfllm<_-Jlfl[~ 0 by (1.10) and consequently from (2.5) we see that

(2.6) [ifll~,0 -<- I[fl[~,l <-- [Ifll~,2 <= 3llflls-~,0.

Thus II I[~-,i i = 0 , 1, 2 are equivalent. Since Hf[I s,~, 2 -< Ilf[ls~,z by (2.5) it remains to be shown that there exists a positive

constant K such that

(2.7) Ilf[Is'~,3 <= g l l f l l~ , z .

Casting a glance at (2.2) and (2.3), i.e., the expressions defining I1 I[s,~,2 and I! 11~,3, it is clearly enough to prove that

(2.8) sRp( !, k OL(k)l k n - f - 1 k=O

"< K ' su ~ [ s ~ - 1 �9 = ~ t n + l k=o ) Ic

The inequality (2.7) will then follow by taking K = m a x (K', 1). To verify (2.8) we first notice that

( 1 k=~O ) l/~ll k~lskf - -sk- l f l~ >~ C

= n + 1 2 k~ l sk f ( x ) - - s k - , f ( x ) l ~ dxJ . k=O

Writing skf(x)--Sk_ff(X)=r (kX+~k) where e =tL( ~ k2+)l lf~(k~)l, from the

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154 I. SZALAY and N. TANOVI~-MILLER

above inequality and Lemma 1 it follows that

1 n 11~

1 " x icos (kx+c~k)lx dx) 1/x 1 x x => -5-Ti- e~-~ =>7 -5-Ti- k e~

Consequently, by Minkowski's inequality

ZkX(lfc(k)l+lf~(k)l) ~ ~ 4 ~ k = O C \ 7- k=0

so that (2.8) holds for K ' = 4 . Hence (2.7) is true for K = 4 and therefore by (2.5)

(2.9) [If[ls~,~ <= Ilfl[~,3 -<- 4 l l f l l~ ,z -

�9 From (2~6) and (2.9) we conclude that II [Is~,i i=0 , 1, 2, 3 are equivalent.

REMARK 2. For 2=1 , the equivalence of the norms II IIs~,0 and II ~-,~ was already established by Do N in [2].

TrIZOREM 2. Let 2>= 1. (i) For each i = 0, 1, 2, 3, 5 ~ 11 II ~ , i is a Banach space and

(2.10) II/llc <-- Ilfl[~ <= l l f l l~ , i .

(ii) For every fESp~ and for each i=0 , 1, 2, 3, [[s,f-flls~,~ =o(1) (n-,-~). (iii) 5 r 11" I[ou is not a Banach space and

(2.11) sup llfll~-',dllfll~ =~ , i= O, 1,2, 3. , f E S ' ,~ , f #O

PROOF. Statements (i) and (ii) are direct corollaries of Theorem 1 and the statements (iv) and (v) of Theorem B, i.e., Theorems 2 and 3 in [10]. Moreover, the inequality (2.10) follows from (1.10) and (2.5). We now verify (iii). Clearly 6ezcq / and by (ii) of Theorem B q l \ S f z D J # \ 6 e for each 2=>1. By Theorem 4 in [7] ~ Consequently we can take gE~#\5 ~ . Clearly then [[s,g-gtl~=o(1) (n~ ~), s, gE5 ~ for each n and (s,g) is a Cauchy sequence in ~z , [1 I[~, Thus 5ez is not a Banach space under the #g-norm. Moreover, if (2.11) is no t satisfied then for some i=0 , 1, 2 or 3 there exists a constant K such that l[/'[Is~,~<:gl[fllou for each rE5 ax. Consequently (skg) is a Cauchy sequence in 6 pz, II lls~,~ for that index i and therefore by statement i) IIs, g-hlls~,~=o(1) ( n ~ ) for some hESez. But (2.10) implies that also I[s,g-hll~u=o(1) ( n ~ ) , so that by above g=h, which is a contradiction to the assumption that g E # / \ 5 a~'. Thus (2.11) is true.

3. Banach spaces d a and their relat ionship to ~

We begin this presentation of the properties of the classes d z by considering the analogues of (i) through (iiJ) of Theorem B. The theorems we prove here are based on the results on absolutely convergent trigonometric series obtained in [5] and their


ABSOLUTELY AND S T R O N G L Y C O N V E R G E N T F O U R I E R SERIES 155

natural connection to strongly convergent trigonometric series implied by 3) of Section 1.

THEOREM 3. Let 2=>1. Then fE sg z i f and only i f fEC and

(3.1) Ilflli~l "- ( ~ k'~-l(If~(k)l + IL(k)l)~) ~#~ < c o . k=O

Moreover, i f fE C then (3.1) holds i f and only i f

(3.2) I1(. k:-lis.s-s - fl )V llc < ~ .

PROOF. Suppose fE~r x. Then fEC and s,f-*tlb, uniformly. Consequently by (1.5)

(3.3) ~ kZ-llskf(x)--sk_lf(x)] ~ <oo uniformly in x. k=O

By Theorem 1 (i) in [5] (3.3) implies that

k~-l(lf~(k)[ + ]fs(k)O :~ < k=O

i.e., (3.1) holds. Conversely, suppose that f-(C and that (3.1) holds. Then (3.3) is clearly satisfied

because I skf(x)--sk_lf(x)i <= ]f~(k)l + If~(k)[ for every x and k. By the Fej6r--Lebes- gue theorem fEC implies that s n f ~ f C1 uniformly. Consequently by property 4) of Section 1 we see that s , f ~ f fIlx uniformly, that is f E d ~.

Moreover, (3.1) clearly implies (3.2). Conversely, (3.2) implies (3.3) which in turn implies (3.1) by Theorem 1 (i) in [5].

THEOREM 4. (i) For each 2 => l, d ; ' c 6e~ c ql properly. (ii) The classes s~ ~ and ~r are incomparable for all 2>2 '~1 .

REMARK 3. The case 2 = 1 of (i) is statement (ii) of Theorem 4 in [7].

PROOF. (i) By (ii) of Theorem B, 6azcSecq/ properly for each 2=>1. By the above Remark 3, the case 2= 1 has been proved already, i.e., d c 6 e c q / properly. It remains to be shown that also ~r a~ properly for 2>1.

We first verify the inclusion. Let fE sr z. Then by Theorem 3, fEC and (3.1) holds. But dearly

12 )0 (3.4) k ~ k)l + lf,(k z = t t k = l

1 . - 1 = - , = 1 l - (IL(l)l+lX(OI)

for each n and every 2_->1. Hence (3.1) implies (1.7) and therefore by (i) of The- orem B fE5 "a. Thus ~r176

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156 I. SZALAY and N. TANOVI~,-MILLER

We show now that this inclusion is proper for a l l 2 > 1. To verify this let 2 > 1 and let us consider the series

(3.5) ~ ak cos kx k=0

(3.6) and

with ak=j-1/;~2 -zu for k = 2 j, j = l , 2 . . . . and ak=O otherwise, where 1/2+1//~=1. Then clearly

2 z~ lakl = j-1/~2-Z# <oo k=0 j = l

1 1

L2, (3.7) k ~ U l a f = o(1) (n ~ ) n + l = n + l j=l j

where j , is that integer for which 2J~_<-n<2J-+L From (3.6) we see that the series, (3.5) converges absolutely and therefore uniformly to a continuous function f . Moreover (3.5) is the Fourier series of f and f ( x ) = ~ a k cos kx. From this, (3.7) and (i) Theorem B, it follows that fC5 e~. However f ~ ar ~, because by the definition of the series (3.5)

k~-llaki~= ~--:-=co, k=o j=l J

that is, (3.1) does not hold. (ii) Let 2 '-> 1. To show that d a and ~)~' are incomparable let

and

A 1 . 1 1 f (x ) = 2-~7-;cos2Jx where - - + - ~ - = 1 .= ]A

g(x) = = k In 1/~' (k + 1) sin kx.

Clearly f and g are well defined, f , gCC and each of the series is the Fourier series of the corresponding function, see [3] Vol. 1 or [11] Vol. 1.

By assumption 2>~" and consequently the Fourier coefficients of f satisfy the following relations :

kZ'-llfc(k)lZ'= 2m_a,/a) < ~ k=O "=

and

= 1 k=O j=O

Thus, observing that f , ( k )=0 for each k, we see that (3.1) holds for 2' and that it does not hold for 2. Since fCC, by Theorem 3 we conclude that f E d z' and f r ~.

Similarly, by above, the Fourier coefficients of g satisfy:

1 ZkZ' - l[gs(k) f = k ln ( k + 1)

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ABSOLUTELY AND STRONGI~Y CONVERGENT FOURIER SERIES 157

and

k ~ - ~ l g ~ ( k ) l ~ = k l n ~ / ~ ' ( k + 1) < ~ " k = l =

Noticing that ~c(k)=0 for each k, it follows that (3.1) holds for 2, but does not hold for )V. Consequently by Theorem 3 again, g~ ~r and g~ ~,x,.

Thus ~r and o~r are incomparable. In what follows we will show that ~r 2 > l are Banach spaces with properly

selected norms and examine other properties of these spaces. For fE ~r 2 => 1 let

(3.8) l l f l l a . < , l - k~-~[s~f--sk-l f[x) ~1~ + I .fIllc,

(3.9) ~ 0 2 - 1 z z/,z HSiJ ,., - I!(.:: Iskf-s -if[) Ilc+ lSH,,

(3.10) l l f l l~- ,3 -

REMARK 4. It is trivial

( Z k~-~ (lf~ (k)l + IL(k)l)~)'~ + l lfl l~. k=0

to see that

(3.11) ~ Ilfll~,~,~ < I l f l l~- ,~ = I l f l l l~ l+ I l f l l~ , l = = Ilfll~.

Consequently by Theorem 3 these expressions are finite for each fE ~r By Min- kowski's inequality they are clearly norms.

THEOREM 5. Let 2-->1. The norms II I1~,~ i=1, 2, 3 are equivalent. Moreover, for i = 0 , 1 , 2 , 3 ; j = 2 , 3 and i<=j

( 3 . 1 2 ) llfll~, < Ilftls,~, < I l f l l~,s

PROOF. It was already established that I[ I1~,~,~ i=1, 2, 3 define norms on ~r and that the inequality (3.11) holds. Thus to show the equivalence it suffices to prove that there exists a constant K such that

(3 .13) Ilfll~,,.,3 <- K I I f l ! ~ , ~ .

By (3.8) clearly Ilfll+,=sup Ils.fllc<=llflld~,~ s o that by (3.10),

(3 .14) I l f l l ~ ,~ <= Ilfll ~ + I l f l l ~ , ~ .

Hence it is enough to prove that there exists a constant K' such that

llflll~l - ( ~ k~'-l(Ifc(k)l + I f , (k) l )~) v~ ~-- k~0

K'I I (2 k -ll kS-sk-lSl )l"ll = c ~< K'tlfll~<~,l.

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158 I. SZALA.Y and N, TANOVIt~-MILLER

By monotonicity of the power function

er

( 3 . 1 5 ) I It~ k~-'i~s-~_~si~)l,~ll~ = I1~ k~-~i~'.s-*.-.si~lll# �9

Moreover clea~ly,

~_ ll,~f_s~_lfl~No _-> ~1 i~: ,,=0z" k~-~ls~f-*~-~fl~ax.

Using this, Lemma 1, letting

(Skf--Sk-lf)(x) = 0k COS (kX+~k) where Ok ~ - - - - I L ( k ) l % l L ( k ) l ~

and arguing similarly as in the proof of Theorem 1, we have

II @ 'ff,cos(kx+ m< xr = ka-llskf--Sk-lfl*) lsa C >= ka-*O~"~o

"2"1 ( k:OZ~ k2-1 ~Ok~) 1/)" =:" -4-1 (~=o "~ k~-~(iL(k)l + Ik (k)i)~)v~"

Consequently Ilf l l i , i~=4llfl l~. ,1 and therefore from (3.14) we conclude that Ilfll~.,a<= <--5llfll~*a- Thus (3.13) holds for K=5, which completes the proof of the equivalence of the norms.

We now establish the inequality (3.12) for i=0, 1, 2, 3; j = 2 , 3 and i<-j, The inequality clearly holds if f([ d ~. So suppose that f E d z. Then by (i) Theorem 4 fC5 a*. Moreover by (2.5) and (2.10)

Ilfll~' < I l f i lJ*, < I l f l l~<s

Thus it only remains to be shown that I l l ti,-.,j <--Ilf II ~ , s for j = 2, 3. Taking a glance at (2.2), (2.3), (3.9), and (3.10) it suffices to show that

(3.16)

and

(3.17)

( 1 n ~.lia ( ~ 0 s~p [7~- - i -~ k~l~"S-'~-~sl~J ~ ~ II = k~-~ls~f -'~-,si~)i'~ll~

n ~ 2 supf. , n + ~ !" ,~ k~(IL(k)' + iL(k)O~Y '~ ~- ( Z~=o k -~(IL(k)l + IL(k)OW ~.

By partial summation for any sequence (ck)

(3.18) ~ k~lC'kl ~= ~ ~ ia-llC~[Z. k=O k=O i=k

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Taking Ck=skJ--sl,_~Jl using the monotonicity of the power function and (3.18) we have

1 n Z Z I/Z <

<= ~i i a-* s,:-,,_ <= :-* Is, S-s,-,/IZ)*:Zllc /c=O = =

which clearly implies (3.16). The inequality (3.17) is even more obvious.

RI~MARK 5. I1 I!~,~, i=1, 2, 3 are equivalent to [1 lid where II I1~, is the ordinary norm for ag= d 1 defined by (1.3). Namely, by Theorem 5 the norms I[ 1[ ~,~, i= 1, 2, 3 are equivalent. By (3.10) and (1.3) we have, taking )~=1,

llft[~r <= llf[l~c,a -<- 21lfl],r and the claim follows.

THEOREM 6. Let 2=>1. O) For each i = 1, 2, 3, a'r I] [[ a, i is a Banach space.

(ii) For every .fEd;" and i = 1 , 2 , 3 , Hs, f - f [ ]d~ , i=o(1) (n--,~). (iii) d z, [] []~;. ~ is not a Banach space for i= O, 1, 2, 3 and

(3.19) sup l]fild~,/I]flF~%~ = ~ i = 0, 1, 2, 3; j = 1, 2, 3. f C ~l *%, f ~ D

PROOF. (i) By Theorem 5 it suffices to prove that a# , [f [[~,~,a is a Banach space. By (3.10) or (3.11)

(3.20) !lfll~: - !lflld~.a = llfll [~j + Ilfll~-

So suppose that (fro) is a Cauchy sequence in ag z, [] l[~*. By (i) Theorem 4 and (3.12) clearly (fro) is a Cauchy sequence in 5 ~z, IT lie;-,0. Consequently by (iv) Theorem B there exists rE5 Pz such that

(3.21) l]f,,-fl[:;-,0 = o(1) (m -+~).

Moreover, by (1.10) or (3.12) it follows that

(3.22) I[f,<-fiI~u = o(1) (m -+~).

Thus bY (3.20) we have to show that .rE s/;" and that

(3.23) l[f,,,-fllla I = o(1) (m -+co).

Now from (3.22) clearly I[f=-fm'Hoe=o(1) (m, r n ' - ~ ) . Hence, by the assumption that (fro) is Cauchy in d z, [] tlw~ and by (3.20) it follows that

(3.24) ]]f,,,--f,'l]lzj = o(1) (m, m' -+~).

Using the definition of r[ I[I;.I given by (3.1) and (3.24) we see that: given ~>0 there exists rn~ such that

(3.25) ( ~ k,.-1 (! L,,c (k) --L,.~ (k) l + I L : (k) -L , . , (k) l)~) m < ~/2 k=0

A cta Mathematica Hungarica 55, 1990

160 I. SZALAY and N. TANOVI(~-MILLER: ABSOLUTELY AND STRONGLY CONVERGENT...

for all n = 0, 1, 2, ... and m, m ' => m,. But by (3.22) IIf~,-fll~=o(1) (m'--~) and consequent ly fm,,~(k) ~f~(k) ( m ' ~ ) and f , , ,~ (k )~ f~(k ) ( r n ' ~ o ) tmi formly in k. So letting r n ' ~ in (3.25) we conclude tha t

(3.26)

for all that

n

( Z kz-~(lf~,~ (k)-f~(k)t + if~ ,~ (k)-f~(k)l)z)v ~ <= 5/2 k=O

n = 0, 1, 2 . . . . and m => m, . Let t ing n - ~ in (3.26), by (3.1) it follows

0 .27 ) [If,,-f[]l~.l ~ 5/2 < e for m ~ rn,.

Thus (3.23) holds and therefore by (3.22) Ilfm-flld~--o(1) (m~o~). I t remains to be shown tha t f E d ~. This is immedia te f rom (3.27). Namely ,

by (3.27), T h e o r e m 3 and the fact tha t f E 6 e ~ c C we see tha t f m , - f E d z and consequently tha t f E s # , since free d z for each m. This completes the p r o o f o f state- men t (i).

(ii) Le t f E ~/~. By Theo rem 3 then f E C and (3.1) holds. Consequent ly by (3.1) clearly I[f-s,f[]lzl=o(1)(n-~). By Theo rem 3 again llf-snfll~=o(1) ( n ~ ) . Hence by (3 .20)we see tha t IIf-s, flld~.a=o(1) (n~o) and therefore by (3.11) I[f-sJIl~,~=o(1) (n-~oo) for each i=1 ,2 ,3 .

(iii) By Theo rem 4, d z c S ez properly . Using (v) o f T h e o r e m B and arguing similarly as in the p r o o f o f s ta tement (iii) o f T h e o r e m 2 we conclude tha t (3.19) holds.

References

[1] N. K. Bary, Trigonometrideskie Rjadi, Gosudarstvennoe Izdateljstvo (Moscow, 1961). [2] J. Dobi, A remark on the norm of the Banaeh space of uniformly strong convergent trigono-

metric series, Acta Math. Hung., 51 (1988), 201--203. [3] R. E. Edwards, Fourier Series, Modern Introduction, Vols. I & 2, Holt, Rinehart and Winston,

Inc. (New York, 1967). [4] C. P. Kahane, S~ries de Fourier Absolument Convergentes, Springer-Verlag (Berlin, Heidelberg,

New York, 1970). [5] I. Szalay, Ob absolutnoi summiruemosti trigonometrireskih rjadov, Mat. Zametki, 30 (1981),

823--837 (in Russian). [6] I. Szalay, On the Banaeh space of strongly convergent trigonometric series, Acta Math. Hung.;

46 (1985), 39--45. [7] N. Tanovid-Miller, On strong convergence of trigonometric and Fourier series, Acta Math.

Hung., 42 (1983), 35--43. [8] N. Tanovid-Miller, On a paper of Bojani6 and Stanojevir, Rendiconti Cir. Mat. Palermo, 34

(1985), 310--324. [9] N. Tanovid-Miller, Strongly convergent trigonometric series as Fourier series, Acta Math.

Hung.. 47 (1986), 127--135. [10] N. Tanovi6-Miller, On Banach spaces of strongly convergent trigonometric series, to appear. [11] A. Zygmund, Trigonometric Series, Vols. I & II, Cambridge University Press (New York, 1959).

BOLYAI INSTITUTE UNIVERSITY OF SZEGED 6720 SZEGED, HUNGARY

DEPARTMENT OF MATHEMATICS UNIVERSITY OF SARAJEVO 71 000 SARAJEu YUGOSLAVIA

(Received June 25, 1987)


on banach spaces of absolutely and strongly convergent fourier series

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