View of On some sequence to function transformations

(1)

MATHEI\,I.4.TICÀ

-

^1{E\¡^UE^D,ANAI,Y SÐ NUì{]trRIQUE ET DÐ THÉORIE DÞ 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)[email protected])

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

- ^Tome9, No 2. 1980.

(2)

234 ^R.N. MOHAPATRA _anctG. 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 ^¿r11I. 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.-"

lØ

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

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

_IB 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,donly

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 _betotøil,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

^euerymeasurable, 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"feren'tentsin ct Bcmack sþace

fi

^hìeß^theþroþerty that ^there

is a

,untber

H

^suclt,

rr.rllà+l,ff _<ø

,::,,::,,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-zsg11-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 _{[email protected])}

f.or

the (/, q)

nrean oT

the

series )ø,,e,; rve have (4 1)

where (4.2)

[email protected]):q,(x)t

^{[email protected]),}

,

^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"

oÐ

^oo.u.

f,ln,,*"É ^nr.,l

,-01 À--o

^I

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

[email protected]))

^dx

d xn dr

whenever lxl <r

^since

^thc

^racliuscon\¡ergence

ol

}q,,x,,

ts

r clrangc

ol order of

surnrnation

in

(4.2), we obtain

@û

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) [email protected])

: (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^SEQUENCETO ^FUNCTIONTRANSFORMATTONS 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)

[email protected])l

< M (forX <x<r

^and

for ail

z).

It

^folows

that

(4.9) lirn sup

[email protected])

I ^<

^r.lu/.

But since, as

x

^--+r

- ^,

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

,"

,å

^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'

^{[email protected])}

| ç liruinf

[email protected])l

(

tim

sup

_lgoçç)l

_{< M,} by

(a.e).

-

^ó._Proofof.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

_{[email protected])}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,[email protected])),}

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

-

^(å

+

^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. : _- _(å _to*) _{[email protected])} : ^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

^{[email protected])}

^:0

I

jt¿

s,e,q,{(df dx)(x'[email protected])))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):

@'[email protected])):

⁽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,*

@" [email protected]))

dx <

^Hbctls)

(,

for

sorne atjsolute

positive

constant

H. rn particular,

(5.8)

implies that

(5.e)

!t

^¡

\ll ^n,,,q,,+

^{@^[email protected]))}

^I _" ⁿ

ólr:0 a* _I ^d

tl* ^x"lq x

>0(x {[email protected]|_n);

<0(*>1t,[email protected]+1-ln).

for

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

by

l,emrna 3, lve have

@

Ð

d

If

rve rvrite

f

@)

for

@"[email protected])), ^{n,e hat,e,}

I

rLl\u-l-a1-11

(5.l0)

x' clxqq

ds

for

every bounded, real

functior

[email protected]).

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 [email protected])

^exists

fo¡ _lxl <r

^as^a

power series expansion ^an<1therefo¡e

[email protected]) is

diffe¡entiable

in

_[0,

r).

Now

we

have

J'@) :

_{[email protected])}

l-

J"@) ^as

in

^(5.4).

But

rvhenever

)lAe,,l _<

co, rve ha\.e,

dx: \ ^*

@"lq(x))ctx

-,u,,1,,, fr {"/,t{ù)a* :

:[email protected]þ.

-l-

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

:

([email protected]

+ 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

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 [email protected]))

e(x 4"

E¡

J ^d x"[email protected]

th

as befo¡e

f

[email protected])l dx

<

^cn.We have only

to

show

that

J

lJJr)l

^etx

<

^x:

Now

@"

[email protected]))

^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"[email protected])

= 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)^rveTind

that

11 ^ON^SOME^SEQUENCE^TO^FUNCTIONTRANSFORM,{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

:[email protected]) _-.f(0)) _-/(co) * f(n)) :[email protected]):2%oc-n,

since/(0)

:/(*) :0,

Nory

)u l.,l Çnþn:

'2Ð,,(le,,ln'c ^-,fn)

=

^X,,(le,,l

ln") < *

Nccessity. x,,

tx S n). ^'lhen

f'(x)[email protected])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

_lAI

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^eⁱn, ^D.,^On^melhod,of sunnnal.ion bøsed on þoaer ^series.Proc. Royal Soc. Edinburgh,

c4,342-349 (1957).

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

23-30 ^(r9s9).

,f}l, ^Ott^methods^ofsuntntability 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}^lyþcszmotnability ntethoils. Proc. Catnbridge Phil. Soc., 53, 318-322

( le57) .

[6] Bosarquet, ^{L. S.,}^Notc^onConuergance an'd' Summabilily Faators, (II)' Proc. Irondon

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

[7] ^D^as, ^{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¡ondonMath, 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., ^¿rndPatnaik, J. N., Ot tlt'eAbsollúeLS'tr'ntmabilill'of a7;ottrier Series. J.I,olrclon Math. Soc., 4rì' 452-456 (1968).

IlSl ^OrIicz, 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 ^Noteol ^aTheorem By l3osanquet. J. ^I,onclonMath. 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'@)[email protected]} :(i

(5

+

It

f'(x)dx:

(x)dx x dx

Ï

x dx

f' f'

0

View of On some sequence to function transformations

Full text

-

BT LA TIIDOIIIE DE I,'APPROXIMATION

pp. 233-243

ON soME SBQUBNCE TO F'UNCTTON

TRANSFORMATIONS

l.

tinal

2 0 and

+ 0 for i[Tinitely Ílany valeus

ict x aíd z

for

. I,et r

the radius of

of the

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

1¡it power

'ioi

< r is give'

1)

:

är,,r" (lzl <

Given an inïinite

) a,,

partial sum

wc

that

(J,q)

tof-

the

þor*r,,r'

for lzl{

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

to function tlansforrnation J'@) : -

exists for

( x 1t'. Further, if

'l' (x

r-),

the

2a,is

to

(J,q) to

is

to be

(/,

i/, qltl

e BV

far a

<

[1]

that the

(J,Ð

if

il

- co

x --+./-.

[1,2, 3]

the

Proof.

for

nth partial srlm is (-1)'"

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

: c-ÌD t-

l)',x,,fnl :

e BI/

tiii:i,* eu:0

: O),

o,:

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

Zan

it

that if this

for I :

then

Ð

'rethods of su'rmability. Das l7l

lJ, þl

lJ,glmethods.

*'ell-k'own partic.lar

ON soME ^SBQUBNCE ^TO ^F'UNCTTON

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

1¡it ^power

är,,r" ^(lzl ^<

) ^a,,

^partial ^sum

^the

( x 1t'. ^Further, if

'l' ^(x

i/, ^qltl

- ^co

^x ^--+./-.

^{[1,2, 3]}

_'rethods of su'rmability. _Das l7l

+ ^1)-'

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

^{is the}

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

⁺ @. rt is k'ow'

A and u, A ¿ ^B

^.rea'

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

'or ^the staterne't that

iÂï ^./ ^oj

^that

^{q) is} thc

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