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MATHEI\,I.4.TICÀ

-

1{E\¡ UE D, ANAI,Y SÐ NUì{]trRIQUE ET THÉORIE IT'APPROXIÙIATION

L,iTNALYSE NUMÉRIQUD

BT LA TIIDOIIIE DE I,'APPROXIMATION

Tome 9, No 2, 1980,

pp. 233-243

ON soME SBQUBNCE TO F'UNCTTON

TRANSFORMATIONS

by

R.N. IIOHAPA'I'RA ancl G. DAS (Santa Bar'bara, California)

l.

Suppose

tinal

q,,

2 0 and

(1,,

+ 0 for i[Tinitely Ílany valeus

of

r?,. We.shall

ict x aíd z

stand, throughout,

for

a real and complex nnrlber fespectivelv

. I,et r

d.enote

the radius of

cont,ergelÌcè

of the

power series

æ

D,ln" (, < -). The analytic lunction

reprcsented 1ly

1¡it power

series

'ioi

1rl

< r is give'

by

(1

1)

q(z)

:

är,,r" (lzl <

r).

Given an inïinite

series

) a,,

wibn

partial sum

{s,,}

wc

sa)¡

that

the rnethocl

(J,q)

is,applicable

tof-

n,,,if

the

series

þor*r,,r'

corl\¡er€les

for lzl{

{ r, say to q,(z), and the

sequence

to function tlansforrnation J'@) : -

q"(x)lq@)

exists for

0

( x 1t'. Further, if

Jo(x)

'l' (x

--'

r-),

Then

the

series

2a,is

said.

to

be surnûrab:ì

(J,q) to

t.. IL

is

said

to be

absolu- teh, strnrnable

(/,

q) or summable

i/, qltl

Jq(x)

e BV

(0,r) i.e.

S rOf

far a

<

oo. BoRwnrN

[1]

has shown

that the

method

(J,Ð

is regular

if

and. only

il

q(x)

- co

as

x --+./-.

BoRwErN

[1,2, 3]

considered,

the

inclusion relati-

G - L'analyse numóriclue et la thóoric de l'apÞroximalion

- Tome 9, No 2. 1980.

(2)

234 R. N. MOHAPATRA anct G. DAS

2 J ON SOME SEQUENCE TO FUNCTION TRANSFORMATIONS 235

Proof.

(i). Consider the series

for

rvhich the

nth partial srlm is (-1)'"

'l'his serics is rlot convergeut but its Borel transfolm

B(x)

: c-ÌD t-

Ø

l)',x,,fnl :

¿-2'

e BI/

t0,.o). Thus the series is surnmable lB l.

tiii:i,* eu:0

(n

: O),

arrd.

o,:

(sin nt)|rc

(n:1,2,...).

t'he series

Zan

cottt,ergcs for ¿r11 I. After sinrplification

it

can lte seen

that if this

series

\\¡ere s111n111ab1e lB I

for I :

J/

then

the intcgral ons betwee" (J,

Ð

and .(J, 17)

'rethods of su'rmability. Das l7l

has obtai-

ned inclusion relation betrveen

lJ, þl

and.

lJ,glmethods.

As

*'ell-k'own partic.lar

cases

of the

(J,

q) nethod,

',e

have the Aber

methocL

(A)

rvhen Ç,,

:

(n

+ 1)-'

(see [9, 51,

-ifr"

rnethod An, when

ç, : _ln+ aì,^-

,,

: I o, J (t"" [l, 5]

(40

is the

same as

the

Abel method) and

the

meblLocr

Bo u'hen r1,,:

(l(n,

I u.i l))-t (8, is the

sarne

as the

Borc,l rncthod) (see [9-],

p.

222).

A real

method

oi s'n'ration 'r is totarly

regrrlar

il

s,,

* s

ir'¡-riics

that T-lirlit of

s,, -+ s

for all fi'ite

and

infi'it"

.,

Ã

n,

+ @. rt is k'ow'

that

a llecessary ancl sufficicrrt condition Tol a real triangnlar

nratrix

tr¿rns-

formatio' to be totally

regular

is that it

shorrld

bc r{ular a'd

posiri.rc (sec

l9], and

[10]

for

a geueral resnlt on

the

subject).

'Ihroughout the paper *,e shall use the

follorvì'g

rrotatiorrs :

For tr¡'o

sunrmabirity processes

A and u, A ¿ B

u,ir1

.rea'

thab ail

seqrlences (series) srrrunable

(,{)

are surnnable

(Bfl c rvill

denote

the

space

of

convcrgent secluences.

2n&,,

t

(A)

will

mean

that

the sedes z,,a,,iÀsurnnable

by the

rnet¡ocr

(A)' rf in this

statement rvc replace

(A) bt c then it rvill

rnean

that

the series )r,a,,

is a

convergent series.

r,, = (4, B) u'ill stand

'or the staterne't that

,,s'rnnrabi1it1. (A)

of \ø,,

irnplies

sumnability (Il) oI

Ea,, e,,,,.

::,T

ä'Tl,:. [o,],1'*'1""

J,:^

iÂï ./ oj

at point.

Thus lve can conclud.e

that nrability

lA*

| of infiuite

series a¡c

:

0. This horvever raises

the

foilo- conversellce

a'd

lA* I arc

not indepe'äà,

",t"jlil iåi:l \Tå åfT:'Jt;Ì.,,î,j the

answcl'.

Wlrcr q,,: (tt,l) ',

so

that

(rI,

q) is thc

Bor.cl mcthocì

(B),

r,ve shu*.

that thc

Ir

olre'ties

or' conr c r g,,nË*

',iocr .ì,",r"ãi;ìiiv-'¡ï¡ ro, a, inlinirc

scr-ies are ìrrclcpe'de'nt of cach

ätn"r. ou,

,..rrrak is

.uipåitea

b_.,

pRolo*srrr(tn l. (i)

7-lterc i.s (¿ sct',tcs surnrn,abl,c

¡n¡

w,nictt

is ttot

t.on_

acr gent.

(ii)

T'hcrc

is

ct sct'íe s u,Jtitrt, i,s ctnacrg*tt but not sr.tnrndtrclts l.

(2.1)

I:

æ

,t :0

@

r- "

5-

/J\ (x,,sirr (¡¡-.1_ l)-r,)/(ø

* l)

! d,x

will

be conr¡ergcnt. I)cnoting

thc

term iuside

the

nodulus sign

by I{(x,

1t),

'we have

/t(r,

r') ==

nr i.-"

D

(x"¿ít"rth'lþ1,

+

1) l)

:

I r .0

: N,1,_2,si"'yl2 Si¡(tsinl,l) _

^i 1e'.

Thus

(2.2)

lt

tc-2r sitr't'tz sin (x sin y) ldx

Choose

I > 0

so small

that

sin

(r

sin v)

is

non-ncgative. 'I'hen

the

sccond integral

of p.2)

is

not

greater than

{a'-2tsitt't'lz sin

(r

sin¡r)i:v sin ),} dx

I

y 1C-2:v sí\1 ylZ dX :

,,\

T c

'tlx I

s1n

ô

ô

I

1' 5

4 +o

æ

¿ -2r sill¡ t'12 ,l

8-r

e-2v slt\2 y12 (lX

:

O((sin' ),12)-')

Choosing J/

to

be different

from

an even multiple

of

æ, lve have

the

altove

integral

bounded.

But the

clivergcuce

to infinity of the first integral

of (2.2) shou's

that I is

clivergent. Ilence rvc establish

thc

assertion.

3. In

rriew

of

$

2

a natual question is to obtain rrecessary

and

suffici-

ent

conditions on a seclrletlce {e,,} srrch

that

2ø,,e,,

is

either

surlmable

I B I

or

lA"

I

(o-

> - 1)

rvhenever'

)øo is a

con\¡er!îent series.

Along this

line

is the follorving

result :

TrrrJoRDì{

A [15],

eo

= (c, IAD iÍ

an,d only

if

(3

1) )lAe"l<oo,

(3.2) Xl.,ltt,-'r<cx¡.

(3)

236 R. N. MOIIAPATR.A. and c. D,{S

4

THEoRErvr

1'

Let qn >-

0

ønd (-1, q) ruethocl be totøil,y regurar. Then

e,e

-

(c,

ll, ttù

onty

if

(3.3) :i ; ) lA.,,l < co,

,

ønd.

(3.4) hold, ahere

Zlu^l q,lnnl 1æ,

(3 5)

ir

0

] {*-l,t{4)eqx¡}

ax

for

euery measurable, c.ssent,ially bounclcd, real futoction q(x).

THEonEu

2. tt¡

tr,

r> 0.

Then the sufficíent cou,cl.itions

for

zan

e, e -lJ,ql

wheneaer

r,=Ð ao:O(l)

øre (B.S) and

(3

6) i , zl r,lcl,f

ü,1

<

.o,

zahere

(3.7)

þ*

R e m a r

k.

xn

of

(3.5) a1wa1.5

exists

for q,,

> 0

(see [g]).

d.

We sha1l need the follo.w,ing lemrnas :

.

r,piriurr

1 (i61,

lernma

B). If

Zg,(x)s,, cotl.aergcs

for

0

< x <y

and.

its

stt'no tend,s

to a tirnit

q.s

x

--+

r - o

taheneaer

,,, i,

,Lrrrr.gent, then trøere

are numbers

M, X

such,

that

ZlS,@)

I <

r'l{

for X:q x <r.

r,ÐMMA

2 (lr3l,

see also t15l). ry ct sequcnce {þ,,) o"f eren'tentsin ct Bcmack sþace

fi

hìeß the þroþerty that there

is a

,untber

H

suclt,

rr.rllà+l,ff

,::,,::,,1::":euery

set of s'igns

!, tttenrT,f(p,)l<q for

eacry tineør func-

T,EMMA

3.

Let

q* >0

øncl (J, q) methrct be totalty rcgular. Th,en Za,r=,ç

and

Zanen

=

(J,

Ð

,imþlics u,

- O(l).

237

Proof.

If

under

the

hypotheses

of the

lemrua

e, is not

tr-ouirded, then

l$t"n

Iu, I

: f

co. Then there exists a seçlrlence of positive

, non-decreasing sequence

of ltositive integrcs tru,

v

:

1,

2,

. .

.,

such

that

1.,,u I

)

ur.

Choose &,, such that

au

:0

þo

¡ nu) and {i.r,:

y-z sg11-le,,u

6t :"").

Thtrs

)

lø,,1

( Ð r-' q

oo. ISut rvhen ø

:

1xv, øt,e,,

: arr€,.y:

y:21.,,u1

;'

1,

and. so

the

seriËs )a,,e,,dir.erges

to f

oo. Since (J,

q) is

assumed

to

Jre a

totally

a regular methocl ),u.,,e,,is l1o11-srlm1nab1e

(/

q)

.

Thus

u,:

O(l).

r,Br\,Ir{a

4. Lct I" ? 0 qnd

(.1,

q)

nrethocl be totøll1t

regulør.

Tþert

e,e

= þ,

(J,

fl)

ortty

if

(3

3)

h,otd.s.

Proof. Writing J@)

f.or

the (/, q)

nrean oT

the

series )ø,,e,; rve have (4 1)

where (4.2)

I@):q,(x)t

q@),

,

Sincc,

by

hypothesis and I,enrma

3,

a,,

is

ccrrtvergent

and e; :

O(1), we hatte

5 ON SOME SEQUENCE TO FUNCTION TRANSFORMÀTIONS

a

We

sha1l

first obtaiu e, such that

e,,

e (c, ll,ql) and

th.en obtain

results

for lA' I and

lB

I by

assig'ing

particula¡ v¿1¡", to qn. oat

resurts

wiil

be

the

follor,ving theorerns :

%"t

,t

q,(x)

:\Q,,r"

oo.u.

f,ln,,*"É nr.,l

,-01 À--o

I

:o (f w,,t.r') o(r),

lq@))

dx

d xn dr

whenever lxl <r

since

thc

raclius con\¡ergence

ol

}q,,x,,

ts

r clrangc

ol order of

surnrnation

in

(4.2), we obtain

I

uoroDqux"

Þ:0 1t:. h

'Ì'h.us by

(4.3) ;i ,1,(x)-

By

Abel's transformation

(4.4) tt"@):

å,,n0(.0 är,.',)

provided

that

(4

5)

,,]3¿ (s..,,ø,,x'')

,,'

q(x,)

:

o.

Ilut by

h¡'pothesis

s,

converges,

by

I,ernma

3,

since

er: O(1)

aud since Xqr,ø"

is

convergent

for 0 < * 11,,

Ç,,x" --+

0

as n, --+ @, Hence (4.5)

holds and therefore (4.a) is

valid.

Norv

write

(4.1)

in

the from

(4

6)

J

"@)

:

)go(.t)so where

(4.7) eo@)

: (q("))

'|l,u (1"xo

æ

D

(4)

238

(5.2)

provicled

that

(s.3)

R. N. MOHAPATRA and c. DAS

6 I ON.SOME SEQUENCE TO FUNCTION TRANSFORMATTONS 239

'since, by_hypothes-is,-/(ø)

existsfor0 ( x<rand. J(x) +ølimit

as )í--+

'-+

¡-

whener¡er {.s,,} conr.erges,

by

r,emrna

2, theie' exists

nurnbers

M

and

X

such

that

(4.B)

Zol1o@)l

< M (forX <x<r

and

for ail

z).

It

folows

that

(4.9) lirn sup

zleo@)

I <

r.lu/.

But since, as

x

--+ r

- ,

\q,,x,,--+ oo, and.

,"

q,¡x,,f c7þ;): 1

- (í^ ø,,o'øfa)

which tends to

1,

it follows frorn (4.7) that

g*@)

-

A7,e¿. Ilence

(4.10)

|[usol :

)o

llin'

go@)

| ç liruinf

l1o@)l

(

tim

sup

lgoçç)l

< M, by

(a.e).

-

ó._Proof of.theorent,

1.

Since

lU, q)l C (I,

q), rreccssitl,

of

(3.3) foltor,vs

{ronr

r,emma

4. rt.proving the nlcêssiÇ õr

(s

rve sliall ube áoiations of I,emma

4 lor

J@) etc. r,vithout restatement.

.

.Sin¡e

,f(r)

exists

for 0 ( x <

1'

a'd.

so

is differe'tiable i'

10,

z),

we

obtain by

straightforrvard calcrrlation.

,

(s.1)

J'@)

: _

ä,.,l,*((:,*D,

q,,*o t,t@)),

By

Abel's transformation,

J'@): - å ,"o,, (,,,*{(p, ."),rrrl)

ft con be

checkèd.

that

(s

s) *l(þ^r*')rot,r) : (Ë" v,*')høt)'

where

I/ ¡

:

ÇtQt¡

2qrq*-t+

. . .

+

mc),c1e-u+r

-

+

l)qt t-tqn

-

l¡QeT'

-

(h

- n I \)q,Qu-n+t: - @f

l)qerr 4o

-

(h

- l)hqoTt- ..' - -

(h

-

2n

-l l)qe_,ttQ,

it

being understood That

q,: 0 if r is a

ne-gative integer. Nor'v

we

sepa-

ratelyionsider the

cases

0 <

/¿

<n, n <å < 2n, h>2n. In

each case

itcanbecheckedf:hal-Vo ( 0for allnandlor0 ( x1r

rvhenever t1n2O

(see

McFadden tl1l).

Ilence it

follows

frorn

(5.5)

that

(s

6) jl * tip, ,rør\øøt)lo. : - to*) h@) : t,

since q(r) -' oo as x

--,

r-.

Norv

fron

(5.4), we have b1'' (5.6)

,*

\ l.l"@)lttx { Ð lr,,llAu,l

o:":0

jl* tn or*\oø)l a.:

@

Bnt

since

{r,,} = t*,

{u,}

et,n

and.

*{nq,,xhfq(x)} -

:Ð, lr"llae,,l a ta,Ð l1\e,l (

1(,

by

(3.3) and the

fact that

s"

:

O(1)

We are

given

(5.4) that

(5 7)

lJ'@)ldr (

co. Since

ll(x)l

clx

1Ø, it follows

from

l/r(r)

I

dx 1cÐ, which is the

same as

1im S,,,

tn+ú €tt -

*l(ä,*r

lq@)

:0

I

jt¿

s,e,q,{(df dx)(x'lc1@)))ctx

I

@,

(5)

240 R. N. MOFIAPATRA and c. LI-AS

I

B ON SOME SEQUENCE TO FUNCTION TRANSFORMATIONS 24lL

THDoRrtlvf

3.

Ða,,.

is

conuergcnt int'þlics }a,,e,,

= lA"lþ- >

-

j 1) if

ancl ort'l1t

it P.1) ønd

(3.2) ,'olcl.

Proof.

Su'fficierccy.

In this

case

!

d):

@'lq@)):

(1

- x)(ríil-t((l - x)ti'- x (l

-l- ")).

Thus for-every.c-onvergent seqllence s,,.

Bnt

(5.7) holds (see

llsl,

lemma

2) it

and only if

(s.B)

r

ll "', ,,q,*

@" lq@))

dx <

Hbctls)

(,

for

sorne atjsolute

positive

constant

H. rn particular,

(5.8)

implies that

(5.e)

!t

¡

\ll n,,,q,,+

@^lq@))

I " n

ólr:0 a* I d

tl* x" lq x

>0(x {nl@-tr|_n);

<0(*>1t,l@+1-ln).

for

each h, and ever)¡ seçlllence of signs. Hence,

by

l,emrna 3, lve have

@

Ð

d

If

rve rvrite

f

@)

for

@"lq@)), n,e hat,e,

I

rLl\u-l-a1-11

(5.l0)

x' clxqq

ds

for

every bounded, real

functior

g@).

This

completes

the

proof

of

'Iheorem 1.

Proof of theorent,2. We are gir.en

that

s,,

: O(i).

Since

r il-7 ¡ il _1

Ð-trt*: Ð

¿\eosof e,,s,,,

we have D

noEÀ <sr1p

f+f p iArelf +

O(1)

:

O(1), so

that q,(x):

O(1)

Ë q,,xt.

f1ettceq" (ø) exists for

lxl<.r.

Since r7(ø)

* 0 for lxl <r, jt foltowJihat J@)

exists

fo¡ lxl <r

as a

power series expansion an<1 therefo¡e

J@) is

diffe¡entiable

in

[0,

r).

Now

we

have

J'@) :

Jr@)

l-

J"@) as

in

(5.4).

But

rvhenever

)lAe,,l <

co, rve ha\.e,

dx: \ *

@"lq(x))ctx

-,u,,1,*,, fr {*"/,t{ù)a* :

:f@lþ.

-l-

1+ n)) -f(0) -Í(r) lfþøl(ø, + 1+ n)):2f(nl(d-i- I + n)):

:

(nl@

+ 1+

n))" (o.

+ 1).r'(n +

o

f 1)-*-t :Q(1x-e-t).

IIence frolrr Theoreln 2 rve obtain

D

@ 1.,,1 Çuþu-

o (D

lr,,llr."-")

Ai) : o (D n

'le

,,1): o(l),

by

thc ,'roi.n"r,r.

tYeccssitjt. We

lirst

observe th.at we do

not

impose

any additional

res-

triction by

assuming

that

(3.4) holcls

for

every boundcd coruplcx Tuuction 9(ø). We

next

set

q(x): (l -

*)n

(r: l=) in

(3.4).

'lhen the integral

is girren b¡,'

l

i,' - ¡' * ut - x")a,'x'))d.xl :

1,

i

(t - x)o+i*"n.|:

: Itl(l I

a

I

i)I-þø

| l)lt(rø +

2

+

ø.)

| =

(1

+ i +

ø")lndt.

IIence

2,,1e,,1 A,l,

(r -

*)n

h ttt -

x)æ'rtr+¡r,* i.e.'),,(le"l ltr.)

<

c<t

'I'hus

the llroof of

the theorem,is c'otnplete',r lq@))

e(x 4"

J d x"lq@

th

as befo¡e

f

l.Tr@)l dx

<

cn. We have only

to

show

that

J

lJJr)l

etx

<

x:

Now

@"

lq@))

d.x

< h,l

e,, qnþo < lt

J'@) dx a\

s,e,Qn ¿ n:0

i

tlø

æ

This

completes

the

proof

of

Theorern 2.

6. In this section u'e appl1,

Theorem

l and 'lheorem 2 to

ol¡tairr results for summability method

lA,l. In this

case

/:l,q(x):(i-")-a-r =

: Ð

@ ¿i" *".

n:Q

<co

(6)

242

IJ

R. N. MOIIAPATRA and G DÀS

10

7. fn this

section r,ve

apply

Theorern

I and

Theorem

2 to

absolute

Borel

sr.mmability.

I' this

case

q(r) :¿Í :L @þl),

and.

/ : oo.

Orlr

result is the

following:

:rHrloREùr

4. e,e

(c, IBI )

if

and, onl,y

if

(7.1)

X,,lAe,l

<

co.

qnd.

(7.2)

),,{lu,l

lni}.*.

Proof. I-,et

f(x):

x"lq@)

= c-"x".

IIence f'@) : -e-r

xtt

!

n,y,,-ts-"

:

¿-ryn-t(12

-

ø)

.

So

(7.3)

Í'(*) : >0(x <n);

<

0 (x

>

n).

Sufficiency.

fn

vieu'

of

(7.3) rve Tind

that

11 ON SOME SEQUENCE TO FUNCTION TRANSFORM,{TIONS 243

pROBI,EII.

Does

there

exist a

series

lvhich is

convergent

but is

not summable lL | ?

We feel that the

answer

will

be

in the affirmative.

Concerning the convergence factors

for

series

sunmable I,

rve conjecturc

the folloli'ing:

cooJEcTuRÊ. €n

e (t, ILD if and only if

X,,l

Ae,,l <

co

and ),,{l

e,,l(talog

n)} <æ.

RDITDRENCES

:f@) -.f(0)) -/(co) * f(n)) :2f@):2%oc-n,

since/(0)

:/(*) :0,

Nory

)u l.,l Çnþn:

'2Ð,,(le,,ln'c -,f n)

=

X,,(le,,l

ln") < *

Nccessity. x,,

tx S n). 'lhen

f'(x)g@)dx

Choose

p(r) :1 ("<n), a1d q(x):-1

Strbstituting

the

r-alue

in

(3.4) we see

the

necessit¡'part o1 (7.2).

B. If

one sets q(ø)

:1og(1 l(1-x)) and r:1then (/,q)

rnethod

reduces

to the

logarithrnic method

(I,)

(see

[a]).

Consiclering

a

series )1a,, strch

tlrat

F(x):Ðn&nx."

is

ø-(t-t-r¡,

it

is easy to sec

that It(x) is oI

bouded rrariation

over

(0, 1) and

thus making L¡t.oe lAl.

Holvever

this

series is

rrot

sumrnable (C,

å) for any

lt,

) 0

(see

[9], p.

109).

Since

lA I

C lI,I

(see

[12], p.

453)

not all

series

lf l

are Cesàro

summable

and

a

fort,iori convergent.

This

raíses

thc

follorving problem :

[f ] B o r rv e i n, D., On melhod, of sunnnal.ion bøsed on þoaer series. Proc. Royal Soc. Edinburgh,

c4,342-349 (1957).

12) - , On tnclltod,s of summat|on bøsed, on integrøl funat'ions. Proc. Cambticlge Phit. Soc., 55,

23-30 (r9s9).

,f}l, Ott methods of suntntability bøsed on integrøl ftr.nctiort.s. Proc. Cambridge Phil. Soc., 56, 123- 131 (1960).

l4l -, A togarithmia m'etolrotl of summability. J. Londol NIath' Soc.,'J'J,212-220 (1958).

[S] -, On a scale of Abel lyþc szmotnability ntethoils. Proc. Catnbridge Phil. Soc., 53, 318-322

( le57) .

[6] Bosarquet, L. S., Notc on Conuergance an'd' Summabilily Faators, (II)' Proc. Irondon

llatir. Soc.,5ll' 295-304 (1948).

[7] D a s, G., On some methods of summability. Quarterly J. Math., l7' 244-356 (1966 ).

[Sl -, Iq.clttsion, T]rcoyetns for an Absolul,e fuIethod of Su.mmabi,lity, Jour. I¡ondon Math, Soc.,

8, 467 472 (1973ì|.

[9] IIarcly, G. H., Diuergenl' Sevies. (Oxfortl, 1949).

tl¡l Ilurrr.itz, H. LL.,'lolal llegr,r,lnrily of Ge.nerøl'1'ransfonnalíor¿s. Ilull. Arner. llath.

Soc,, zrc' 833-837 (1940).

Illl Mc Fadden, Ir., Absolute Nörluncl Surnnrability. Duke I\{ath. Jout., 9, 168 -208 (1s42).

Il2l Molr¿nty, R., ¿rnd Patnaik, J. N., Ot tlt'eAbsollúeLS'tr'ntmabilill'of a7;ottrier Series. J.I,olrclon Math. Soc., 4rì' 452-456 (1968).

IlSl O rIicz, rl[. W., Beitrrige zuy '].'lrcorie det Orlhogonalenlwichh'øtgcrl 11. Stutlia Math, I' 241--25s (1929).

[f4] Prasad, lJ. N.,'fhe Äbsolute Surnrnability (r\) of a Ëourier Series. Pro¿. Ldin.\latlt,

Soc., 2, 129-134 (1929)'

[r5] Tatchell, J.B.,,A Note ol a Theorem By l3osanquet. J. I,onclon Math. soc.,2Ð, 203-211 (1954).

,[16] Writtaker, J. ][., The Absolnte Surnmability of Iroutier Series. Pzo¿. IÌd'itt. NÍath', Sc¡c., -

:¡, l-5 (1930).

American Un'iuersily of lleirut, Beirut,

orlæbanon Uniaersily oJ California, Santa tsarbara,California, U.S.A.

Reeived 17. IV. 1980

i )t',',*,',

ct x

:2,'n

e_-"

": i r'@)e@d. :(i

(5

+

It

f'(x)dx:

(x)dx x dx

Ï

x dx

f' f'

0

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