Some absolutely effective product methods

VOL. 15 NO. 4 (1992) 641-652

SOME ABSOLUTELY EFFECTIVE PRODUCT METHODS

H.P. DIKSHIT

Dept

ofMath

&

Computer Science Rani Durgavati University,

Jabalpur482001 India

J.A. FRIOY

Dept.

ofMath

&

ComputerScience Kent State University Kent, Ohio,44242 U.S.A.

(Received June 20, 1991 and in revised form March 24, 1992)

ABSTRACT. Itisprovedthat theproduct method

A(C, 1),

where(C, 1)isthe Ceshroarithmetic

eannatrix.istotallyeffectiveundercertainconditionsconcerning thematrixA. Thisgeneral

resultisappliedtostudy absolute NSrlund summability ofFourier seriesand otherrelatedseries.

KEh" WORDS AND PIIRASES. Absolute sunmability, NSrlund suminability.

.\MS SUB.IECTCLASSIFICATION NUMBER..10F05 1. INTRODIi(’,TION AND SOMERESULTS.

Let A (a,,) bean infinite matrix witha,kxs real numbers. A givenseriesu

ut.

or

k=0

les,’,luenceofitspartial sums

{U}

issaid to be

IAI

summableif

,.

a.u (1.1)

is(h’finedforalln.atd v

,,

converges absolutely.

,

isthe series-to:series.:l-transfornatio ofu andwo<h’not ib

I>y

A. TimmLrix

A

is 1]bsolutlyconsrvtivin c timabsolute

,’otav<,rge<’(,ofu

inl>lies

tirol of ..lu. W will I>eninly co,cQrmd wiLl timcasQ i, whid ,,t is

,l)solttelv conservativeand

k=O

fl)reverytixed n >0. it maybe worthwhiletomention thatthe conditio:

k=O

lbr ,v,ry lixed n 0. which is stronger than (1.2), isanessentialrequiretnent for thedefinitiot

of,lu tbrevery

,,

_{tbr which tie sequence}

{u,}

isconvergent (seeSzhsz

[10],

Lemm ,ud 2 of

’lapter 111).

Let

A, B

begivenmatricesfor which Bu and the A-trsformationof

Bu,

viz.,

ABu

are defined. Then the

]AI

summability ofBuorequivMentlythe absolute convergence ofABudefines

642 H.P. DIKSHIT AD J.A. FRIDY

thantheabsolute summability defined by the productmatrix

A.B

(ifitexists).

It may observed that the condition

(1.2’)

and a]ortiori (1.2) are automatically satisfied whenever

A

is alowertriangularmatrix

(that

is,a. 0for k

>

n).

Letp

{p,,

be givensequenceofreal numberssuch that

P,,

’pk

0 and

P-t

0.

k---O

(N.p) matrix isthelowertriangularmatrix

A

with

c0o 1, O,,o 0, n

_>

1, c,, _p.

(1.3)

(a)

for

each

fixed

n, there is apositive integer

r(n)

such that

{cnk}

isnondecreasing

for

<_

k

<_

r(n) andnonincreasing

for

k

>_

r(n),

or

and then(1.1) definestheseries-to-series(N,p)-transformation. The(N, p)-transformationwith

p. forn 0,1,2 defines the series-to-series

(C,

1)-trazxsformation,given by

bo

aoand

b, kay,

S

S_

t, n

>_

1,

(1.4)

n(n

+

l)=t

where

S

sk/(n

+

1) and

{,

isthe sequence ofpartialsums.

kO

For

anysequence

{,},

wewrite

,

=/3,-

O,+t.

{O,},BV

meansthat

la.I

<

let

f(t)

bean (L)-integrableperiodic function withperiod2r.

We

ssume,without lossof generality that theconstant termof the Fourier series

F(t)

of

f(t)

is zero. Let

F(t)

A,,(t)

and

l(t)denotes

itsconjugateseries:

B.(t).

Also, l"’(t)

{,,B.(t)},

and l"’(t) isthederived

nl

seriesofF(I). Foragiven point .,’. wewrite

(t)

0(l)

{f(x

+

t)+ f(x

-t)},

(t)

{f(x

+

i)-

f(x

and takingr= 0or 1,wesay that

A

is

F,-effeetive

providedthat

,()BV(O,

)implies that b’(.r) is

A

sum,nabh,. Wc say tl,at

A

is (i)

I,1-effective

irIj

I,()l-’m

<

inplies tl,at

b’(x) is

A

summable.

(ii)b’-effective

ir

()eBv(o,l

implies tlmt F’(x)is

A

summabh,.

(iii)

IF-effective

ir

{-’e,()lv(0,

)inplies that

F’(x)is

IAI

ummable end absolutely total

effectiveifit iseffectiveineach of thesensdefined almve.

Unless stated otherwise, 0

[/],

i.e., the greetet integernt greeter then

/.

and

denotes psitive constant notneeeserily theseineeteeeheurrence.

For a general mtrix

A.

’rripathy

[11]

(s als

Kuttner

and Tripathy

[8])

and,

Kuttner

and Sahney

[7]

have obtained sufficient conditions s that

A

is

F-effeetive.

The restrictions

itnposedon ,I it

[11]

arcqtit(’ general.Ittt it is,tsually (lillicult to v(,rifvthem forspecial 1"iterest. Theatureof the corresponding conditionsusedin

[7]

issuch that theycan beeasily v.ritied, and, theretbre the following result due to

Kuttner

and Sahney h the advantage o1"

Iavigsonedirect al)plications.

(b)

for

each

fized

n, there ts a posttive tnteger s(n) such that

{cr,,,/k}

is

nondecveasin9

for

<

k

< s(n)

and nonmcreasng

for

k

>

s(n).

Suppose

also that in case

(a).

for ko

>

r(n)+ko

r(n))_2ko and tn case (b),

for

ko

>_

o.t,

0(1

(1.5)

Z

Z

s(n))_2ko

(.6)

ThenA is

IFol-effective.

Starting withan absolutelyconservativemethod A. weobtain inthe present paperasub

licient condition that connects the proof of absolutely total effectiveness of ..I(C’, 1) with the proof of

It"l-effectiveness

ofthe Amatrix. As weshall see. such aresult has someinteresting

applications. Wewillfirst provethe following.

THEOREM 1.

IJ’A

is absolutelyconservative,

(I.2)

holds and

ikt)l-

O(l)

7)

n=l

Jb,.t,(O.

r].

then A(C,1) ,s_{absolutely total}

@ctive.

Weshallobtain the followingcorollariesto’rhrcm I.

COROLLAI[Y [. _IfA satisfies the hypotheses of Theorem

A

and (1.2). then A(C, 1) is

,,bsolutclytotaleffective.

COROLLARY2. If{p,} is ay sequencesuchthat

{(n

+

i)p,/I,}

{R,}tBV

and

i.e..

{,’

isabounded sequence,titan (V, p)(C, l) isabsolutely total effective.

AswesiailseeinLiraImtsectionof Lhc present paper,aspecialceofCorollary providesa result vlicl iciudes. nteralia. siarper results than thoseprovedelseiaere.

(Cf.

[3],

Threms

ud2). Ontim

ottaer

imnd.Corollary2includesinteraliatleresultscontained in

[,l]

(Tiworems

and2).

In Section 4, weobtain anotimr timorem whichprovidesa somewi]atmoredirect suiScicnt

,’o,,,litio,, for

I/,1

i I/r/l-,qfoctivo,essofA(C, ). This tlmoren, wlicla isless gonm’al titan

’l’heorem inthe sense t,iat itdoes not providetimabsolutely total effectivenessofA(C, 1), is

readily applicable, and wededucefi’om itconsiderably slorter proofsof someearlier results(see

1].

’L’lmorms nd 2).

2.

SOME

PRELIMINARY

RESULTS.

Weshall nd thefollowinglemm for theproofofTheorem and itscorollies.

LEMMA

1. Thenecessaryandsufficient conditionthat

A

beabsolutelyconservative isthat for all k 0,

644 H.P. DIKSHIT AND J.A. FRIDY

LEMMA

2. Let

{a,}

beagivensequence;thenforanynumberz wehave -(1 .r akz

az

TM

a,,z"

+

a,+

wheremand n areintegers such thatn m 0.

The proof of

Lemma

2 isastraight forward calculation.

LEMMA

3. IfAsatisfiesthehypothesesofThrem

A,

then

(1.7)

holds.

PROOF. Observingthat (sin

kt)/k

istheimaginarypart ofexp(ikt)/k,weshMlprove

(1.7)

by following closely the proof of Threm

A,

which

contorts

the proof for

]b,(t)l

O(1)

with

b,,(t)

the A-transformation of

(sinkt)/k.

Thus. considering thosevalues ofn

(if

any)

for whichr(n)

5 20(or

s(n)

5

20), wehave that

{a,/k}

is nonnegative and nonincreing for k 20. Therefore, byanapplication of Abel’s

Lemma

totheinner sumand using the fact that the partialsums exp(ik/) are

O(l/t),

wehave

cxp(ikt)

,(n)<O k=20 n=l

(or (,)<O)

th

0(1)

timatiotincdfron

Lmma

and timhypothssthat

A

issolutlyonsrvtiv.

Consideringthe(’;e(I)), wefollow exaCtly theargumentin

[7],

I

I=o().

a(n)2e

In e (a), the part of the sum for k

r(n)

-O may be dealt with in

[7],

p. 43, and it remainstoshowthat whenr(n) -0

>

20, then

.,

.

i=o().

(.)

’1 rifythis woapply I,cmma 2and get (of.

[7],p.

,I13)

(n)-o

k=28

(

a,.0

(n,,’(,)-0+

1)

/;t-’

I(.,/)1

+

+

(.)

0

+

whcrc. Ibr conw,nicnc(,, we writct,(n. k) for a,.. Tlte rcstof the proofof (’2.’2) follows (lirectly

from

[7].

Thisconplctesthe proofofthe present lcmma.

LEMMA

1. If{p, satisfiesthehypothesesofCorollary2, then(N,p)isabsolutely_regular and a

Jbrtov

absolutclvconservative.

Lemma

4may be

prov

by followingtheprfof

mma

10in

[4].

LEMMA

5. If

{}tB,

then

{S}B

implies that

C

Z,o

P*]

O(P,[)

d

{S}tB

is

equivMenttothefollowing:

=o(1)

IP.I

lPl

The first part of the lemma follows directly when weobserve that

k

+

ancarlierresult (see

[5],

Lemma3; seealso

[2]).

LEMMA

6. If

P

O([P,,[),

then uniformly for all r,

s(r

>_

s

>_

O)

L

Pk

exp(-ikt)l

s

Kt-’IP,

uniformly in0

<

<

r.

Lemma

6 may be proved byapplyingAbel’s transformationand using the result that the partialsums exp(-ikt)are

O(1/t).

I,EMMA 7. If

{p,,}

satisfiesthe hypothesesofCorollary2, then

(1.7)

holds for the

(N,p)-transformation defined by

(1.3).

PROOF. Considerthecondition:

_-

_-

p

P_,

whichisobtained from

(1.7)

by replacing 0 by0. Veshallfirst prove

(2.3).

,’Now_observingthat

Pk

P-t

P.

P,,-t

weh&ve

exp(ikt)

0(1)

(2.3)

k

(n

+

t)P._,

[P(n,

)

+

(,,

Nt

+

N,

say. Applying Abel’sLemmatotheinner sum,wehave

n-O

max

P,

exp(-ikt)

AR,.[.

N

<

Ift

,=o

nlP.-t[

o<.,<.-o_ =,,,

(2.4)

0

<

n

<

n

n-I

_<

K,t-’

IAR.I

IP.I,

emO

byvirtueof

Lemma

6. Thus,

N

g=

nlP._lllAl.=o

=0(1)

g,lA&l

IP,

I

(n

+

1)IP.I

byvirtueofLemma5 and thehypothesis that

{ }eBV.

Taking 0 ifa

>

&

breaking the range for k into

w

parts vi.. k

<

0 and k 0and observingthatby Lemma

,

Pg

O(IP),

wehere

N

I,lPol

nlP.-,I

+

I,

nlP._,l

pexp(-ikt)l.

Since p

P(R/(k

+

1)}

and the partialsumofexp(-ik)

O(1/),

anapplication of the

(2.5)

646 H.P. DIKSHIT AND J.A. FRIDY

Abel’stransformationshows that foran5" mwith0

<

m

<

n,wehave

.k=,,,

P,exp(-ikt)l

< Kt-t

,,-t

\k+lll+K

tn

Thus. using the resultof

Lemma

5andasuitablechangeinthe orderof summations,we have

t-’

.

O(tl,

(2.)

because R,, e/3V.

Combining (2.5) with (2.6), we prove (2.3). Since the above proof remains valid if0 is

replaced by 20,wededuce (t.7) from theproofof(2.3).

3. I’ROOF OFTIIEOI{EM

AND ITS

C.OROLLARIES.

(1).

I"t

I-effectiveness:

Wehaveassumed without loss ofgeneralitythat the constant termin

/"(.r) is zerosothat

NOW.

,(r)

(t)dt

O.

(l)cos ,,t dr.

A,,(.r)

r

l)enoling by

A.(.r)

the series-to-series(C, l)-transformationofF(.r), wehave

A.(.)

(t)

,- cos ,.td.

r

k(’k

+

)

Itttcgrating by parts wesee that

..l.(.r)

isthe real part of

where

6:(t)

/,’(k

+

l’i

’’exp(i’/)"

r’--I

Thus,if

b,(z)

denotesthe

A(C,

1)-transformationof

F(x),

thenwehave

+-

,

[Re6(u)ldu)d,(t).

Nowit iseilyseenthat

6(t)

O(1/kt),

therefore,

by virtueofthe condition

(1.2).

Nextwe noticethat

’[Re’(u)ldu

k(k

+

1) sinr. Thus,

I,.,,I

[Re6k(u)]du

0(1),

(3.3)

(3.4)

byvirtueof the condition

(1.2). In

viewof

(3.4)

and

(3.5),

wehave

2

[tR(n,t)-

ot,,,

ReS,(u)du]dda(t)

(3.6)

b,,(x)

-rr

where

R(n,

t)istherealpart of

Sinceby hyl)othcsis

fo

Id,(t)l <_

K,inordertoprove that

A(C, 1)

is

IF,

l-,:frertive,it issufficient toslmw that the followingestimateshold forre(0,

S’

Id(n,t)l

O(1)

(3.7)

S

.,1,

c,.

ReS(u)dul

0(1).

(3.8)

n----I k---I

We lirst proceed toprove (3.7). Breaking therangeofsummation for the inner suminto

< k < 20 and 20

<

k, we usefor the former range the fact that

&,(t)

O(1), while for the

latterrlulge wereplace b(t)by thefollowingexpression whicl isequal to it:

(t

exp(it))-’k+l {.1

-oxp(i,’t)-exp(i[k

+

,]t)}.

I’lus, wewrite

n----I

k=l k=30

O(1),

by virtueof thehypothesis

(1.7)

and that Ais absolutelyconservativesothat

(2.1)

holds.

We

thus prove

(3.7). Next,

wehave

0-sinrt

s,

-<

.:,1

+

’:’""

-(:

+

t)

.:, O-I

k= kO

O(1),

by reasoning paralleltothat usedinthe preceding paragraph. Thus

(3.8),

holds andwecomplete the proof of

IFtl-elfectiveness

of

A(C, 1).

648 H.P. DIKSHIT AND J.A. FRIDY

it follows from(:3..1) that

,(t)

sin nt dr.

t3.(.r

r

"

where 1(n, t)isthe imaginary part ofJ(n, t)d

.(z)

istheA(C,

l)-transformation

of Butbylypothesis

f

t-[(,(t)ldt

K;

therefor,inorderto provethe

ll-elfectiveness

of

A(G, 1)

it issuificienttoshow that

I/(.,t)l

o().

This fbllows directly from (3.7). and we complete the proof of the

{{-.ffectiveness

of .,l((’.t).

(lIl).

[l’l-effectiveness:

For thesequenceF’(x),

cos

.

d,(),

since ,() 0. Thus. timtermsof thecorrespondingseriesareobtained as .,,(.r)

sin(

2 sin(,

5

)

d,().

I)enoting by

u(.r)

tim series-to-series

(C.

l)-transformationof ,,(.r).we have

u.(a

.)

sin(){cos()t,n6(1)

sin(g)Rc$(t)}d,(t).

Thus. if

b(x)

denotes the

A(C.

1)-trsformatioa

of

F’(x)

thenin viewof

(3.4),

wehave

_4

"

!

lt)I(n,t)

sin(t)R(n,t)IdO(t).

b(x)

r

sin(2t){cos(.

But by hypothesis

(t)eBY(O,r);

therefore, in orderto prove the

[Fl-effectiveness

of

A(C, 1)

it is sufficienttoshowthat

(3.7)

holds. Wethus completetheproofof the

[Fl-effectiveness

of

(IV).

IFl-effectiveness:

Fortheseries

F’(x)

Z

u,(x),

wehave

and. therefore,

2

[

d

u,(z)

--

(t)tt

Re{

(t)}dt.

(3.9)

Following the proof of the

[F

l-effectiveness

part,wededuce

IF]-effectiveness

ofA(C,

1)

from

(3.7)

and

(3.8)

bycomparing

(3.9)

with

(3.2)

and thenappealingtothe hypothesisthat

((t)/t)eBV(O, ).

Since

(t)eBV(O,

)implies that

,(t)eBV(O,

r),

the

IFol-effectiveness

of

A(C,

)

isincluded

in its

F-effectiveness.

Thuscombining

(I)

(IV),

wecompletetheproofofThrem 1.

To

proveCorollary wenotethat if

A

satisfiesthehypothesesof Threm

A,

then

Lcmma

3 ensures thecondition

(1.7)

ofThrem 1. This together with

(1.2)

implies theconclusionof Theoreml, which isalso the conclusion ofCorollary 1. Since for the

(N,

p)transformation,

(1.2)

holds automaticallyweuseLcmma4and Lemma7tosthat thehypothesesofThrem are

satisfiedand in conclusion weobtaitt Corollary2. 4.

ADDITIONAL RESULTS.

TIIEOREM ’2.

If

A

is absolutely conservative andthe condition

9(t)Bl/(O,r)

impliesthe

IAI

summability

of

.

to,, where

dr,

(4.1)

1)t

w,,

#(t) cos(n

+

then the A(C, 1) is

IFI-

and

IF[I-effective.

Comparing the hypotheses of Theorem with those of Theorem ’2, we observe that the

condition (1.7) of Theorem is directly associated with the

I.’ll

summability of the series

k__cxp(ikt)/k. Similarly, the corresponding condition of Theorem 2 is closely associated

with I/;’o]-elt’ectiveness of /1, which is usually proved by establishing the stronger result that

:E

Ic,,(t)l

O(1)

where

E

c,,(l)

isthe A-transformation of

:E(sin

kt)/k.

Weshall need thefollowingadditionallemmasfor theproofofTheorem’2.

I.EMMA

8.

For

any fixed.\

>

0, weset

,,(t)

{sin(k

+

,)t}/(k

+

).

k-’-I

If

-c,,(l)

is either (i) the A-transformation of

u(t}

such that

A

satisties the hypotheses of ’Tlworem :\. or (it) the (:V.p)-transtbrmation of u(i} such that

{p,,}

satisfies the hypothesesof

(’orollary ’2. then

’’l’,,(t}l

O(1 for

tel0,

PROOF. In

viewof

Lemma

4andthehypothesesofTheorem

A

we noticethat both

A

and

(N,

p) areabsolutelyconservativeanditfollowsthatinordertoprove

Lemma

8 it isenoughto

prove it with

u(t)

replaced by

v(t)

{sin(k

+

)t}[k.

But wemay prove thisby following closelytheproofof

Lemma

8for thecase

,

0 whichis contained in

[7], [2]

(proofof Theorem

C).

Thiscompletesthe proof of thelemma.

LEMMA

9. Ifg(t)eBV(O,r),then

{V,,}eBV,

where

rV,,=

[sin1/2(n+X)t]

at.

(4.2)

n

’-t-

9(t)

mn

t

PROOF. Let

A,,(z) denotethe Fourier series at x, ofa 2r-periodic function such that

9(t)

1/2{f(:t

+

t)

+

f(z

t)}.

Thenwe seethat

Vn

’s(z)/(n +

1),

where

isthe sequence of partial sumsof

A(z).

Thus, in viewof

(1.4)

the result ofthe lemmais

essentially that

(C, 1)

is

IFol-effective.

The lemmanow follows from

Bosanquet

[1]

whereithas been shown that

(C, a)

is

IFol-effettive

for

>

0.

5.

PROOF OF THEOREM

2.

(I).

IFtl-effectiveness:

Taking

E

a,,

F(:t)in

(1.4)

wehave

+

1)

fo"

s

i-

()

{.i.(

+

)/.

1/2}a

rO

_[-(n+ 1)

(t){sin(n

+

1)t/sin-t}dt.

Using

(3.1)

and integrationby parts

(cf.

[4],

p.

480),

wehave

"

2--

sin

It

sin

1/2t

"’,(t){

}{sin(n+l)t}’

(5.1)

+r(

+tfo

cos

sin1/2t

650 H.P. DIKSHIT AND J.A. FRIDY

W +

say. Thus.inorderto provethat A(C, is

}FI

I-effective

it isenough toshowthat both

(V.

I’,_,) and

(14,

W,,_,)are

[.4[

smnmable. That theformerof theseis

[A[

summablefollows from

Lemma

9 and the hypothesis that

A

is absolutely conservative when we observe that

(t)cBV(O,). Next

we seethat

I,V.

w._

"

1(/)

-g

_sin

t

cos(n

+

)

at

and. therefore.

]A[

summability of

(I,V

l,V._t) follows from the hypotheses of Theorem

)eBV(O,r)

This completes the proofofthe

"

when we appeal to the fact that

((t)/sin

it

]Ft

I-effectiveness

ofA(C. 1).

(II).

]F]-effectiveness

Ting a.

F’(x)

in

(1.4),

wehave

(cf.

[4],

p.

480)

-r(n

+

1

O(t)

;sin

t

dt

"

@(t){sin(n+l)t}d

(5.2)

2

Sin

t

sin

t

1

I"

@(t)

tcsl

{sin(n+l)t}

dt"

+r(n

+

1)

sin

t

t)

sin

t

Comparing

(5.2)with (5.1)and

observingthat byhypothesis

{(t)/sin

t}eBV(O,r),

weprove

that

A(C,

1)

is

]Fl-effective

by following the preceding proof. This completes the proof of Theorem2.

6.

REMARKS.

Weobservethat the hypothesis concerning the

IAI

summability of

w.

wappliedinthe d

@(t)/sin

t.

Thus, in viewof

(3.1)

d

(r)

0 proof of Threm 2 forg(t)

(t)

/

sin

t

wemay

te

g(x) 0. Therefore,onintegrationbyptswesfrom

(4.1)

that

w.,

{sin(n

+

)t}/(n

+

)dg(t).

Ifwenow denoteby

EY,

the A-transform of w, and assume

(1.2]

and that

g(t)eBV(O,r),

then

U,

-"c,

{sin(k

+

)t}/(

+

)d(t)

-’.()d()

andasucientconditionfor the

]a

summability of

w.

isthat

Z

[c.(t)

0(1)

where

Z

isthe a-trsformation of

sin(k

+

)t

/

(k

+

).

Thisissatisfied,however, byvirtueof

Lemma

8, wheneither

A

satisfies thehypothesesofCorolly or

A

is

(N, p)

and satisfiesthe hypotheses ofCorolly2. Thuswededuce from Threm 2,

]F[-

d

F-effective

parts ofCorollary and Corollary 2.

It

is interestingto observe that the foregoing arguments provide remarkably shorter proofsof someearlier results

{[4],

Theorems and

2).

Following

[7],

weobserve that if

{p.}

isnonnegative, nonincreing and

{S}eB.

then the

(N,

p)satisfiesthe

hypotheses

ofTheorem

A,

and Corollary providesthefollowing resultwhich

COROLLARY3. If

{p,,}

isnonnegative, nonincreasing, and

{S}elt,

then the(N,p)(C,1}

isabsolutelv total effectiw..

Corollary also shows that if, in Corollary ’2, we impose the additional restriction that {p,,} is nonnegative and nonincreasing, then thehypothesis

{R,}eBV

maybedeleted. It may

be ,uentioned here. however, that thereexist sequences

{p,}

which satisfy tile hypotheses of ’o,’ollarv"2 but whichare not even monotonic. Theabsolutely total effectiveness oftile(’es/tro

tnatrix (C,//) for b > vas initiated by

Bosanquet

[1]

and was completed subsequently. As

a specialcaseof Corollaries and 2, weobtain theabsolutely total effectivenessof the

(C,

)

matrixfor8

>

1.

The authors should liketothank the referee forsomehelpfulsuggestions.

REFERENCES

l.

BOSANQUET,

L.S. The absolute Cesarosummability

of

a Fourierseries, Proc. London Math. Soc., 41(1936), 517-528.

2. DIKSHIT,

It.P.

Absolute summability

of

some series related to a Fourier series, 3.

Aus-tralianMath. Soc.,

13(1971),

7-14.

3. DIKSHIT,

H.P.

Absolute total

effective

(N,p,)(C, 1)

method, Pacific

J.

Math.,

41(1972).

89-98.

4. DIKSHIT,

H.P.

Absolute summability

of

aFourierseriesanditsderivedseriesbyaproduct

method,J. AustralianMath. Soc.,

14(1972),

470-IS1.

5.

DIKSHIT, II.P.

and

KUMAR,

A. Absolute

total-effective

(N,

p,) means1,Proc. Cambridge

Phil.

Soc., 77(1975),

103-108.

6. FRIDY, J.A.

A

noteonabsolute summability,Proc. Amer. Math. Soc., 20(1969),285-286.

7.

KUTTNER,

B.andSAHNEY,

B.N. On

the absolutematrizsummability

of

Fourierseries,

Pacific

J.

Math.,

.t3(1972),

-t07-419.

8.

KUTTNER,

B.andTRIPATtlY,N.

An

inclusion theorem

for

llausdoffsummabilitymethod

associatedwith

functional

integrals,

Quart. J.

Math(Oxford),

(2),22(1971),

299-308.

9.

MEARS, F.M.

Absolute regularity and the NSrlund means, Annals of Math,

38(1937),

594-601.

10.

SZ/SZ,

O. Introductionto the

Theor!l

of

Divergent Series,

New

York, 1952.