The Abel type transformations into Gw

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THE ABEL-TYPE TRANSFORMATIONS INTO

G

MULATU LEMMA and GEORGE TESSEMA

(Received 16 January 2001)

Abstract.The Abel-type matrixAα,twas introduced and studied as a mapping intoby Lemma (1999). The purpose of this paper is to study these transformations as mappings intoGw. The necessary and sufficient conditions forAα,tto beGw-Gw are established. The strength ofAα,tin theGw-Gw setting is investigated. Also, it is shown thatAα,t is translative in theGw-Gw senses for certain sequences.

2000 Mathematics Subject Classification. 40A05, 40D25.

1. Introduction. The Abel-type power series method [1], denoted byAα,α >−1, is

the following sequence-to-function transformation: if

∞

k=0

k+α

k

ukxkis convergent, for 0< x <1,

lim

x→1(1−x) α+1∞

k=0

k+α

k

ukxk=L,

(1.1)

then we sayuisAα-summable toL. The matrix analogue ofAαis theAα,tmatrix [2]

whosenkth entry is given by

ank=

k+α

k

tnk

1−tn

α+1

, (1.2)

where 0< tn<1 for allnand limtn=1. Thus, the sequenceuis transformed into

the sequenceAα,tuwhosenth term is given by

Aα,tu

n=

1−tn

α+1∞

k=0

k+α

k

uktnk. (1.3)

The matrixAα,t is called the Abel-type matrix [2]. Throughout,α >−1 and t will

denote such a sequence: 0< tn<1 for alln, and limtn=1.

2. Basic notations and definitions. LetA=(ank)be an infinite matrix defining a

sequence-to-sequence summability transformation given by

(Ax)n= ∞

k=0

where(Ax)n denotes thenth term of the image sequenceAx. The sequenceAx is

called the A-transform of the sequencex. If X and Z are sets of complex number sequences, then the matrixAis called anX-Zmatrix if the imageAuofuunder the transformationAis inZwheneveruis inX.

Suppose thatyis a complex sequence; then throughout we use the following basic notations and definitions:

=   y:

∞

k=0

_{yk is convergent} ,

d(A)=   y:

∞

k=0

ankykis convergent for eachn≥0

  ,

(A)= {y:Ay∈},

Gw=

y:yk=O

rk_{for somer∈(0, w), 0< w <1,}

c(A)= {y:yis summable byA},

Gw(A)=

y:Ay∈Gw

,

∆xk=xk−xk+1.

(2.2)

Definition_2.1. _{The summability matrixAis said to beGw-translative for a}

se-quenceuinGw(A)provided that each of the sequencesTuandSuis inGw(A), where Tu= {u1, u2, u3, . . .}andSu= {0, u0, u1, . . .}.

Definition_2.2. _{The matrixAis said to beGw-stronger than the matrixBprovided}

Gw(B)⊆Gw(A).

3. The main results

Theorem_3.1. _{The matrixAα,tis aGw-Gw matrix if and only if(1}−_t)α+1∈_Gw.

Proof. _{Suppose thatx}∈_{Gw, then we show thatY} ∈_{Gw, whereY is theAα,t}

-transform of the sequencex. Sincex∈G, it follows that|xk| ≤M1rkfor somer∈ (0, w)andM1>0. Now we have

|Yn| =

1−tn

α+1

∞

k=0

k+α

k

xktkn

,

|Yn| ≤1−tnα+1 ∞

k=0

k+α

k

_{xk t}k n

≤M11−tnα+1 ∞

k=0

k+α

k

rktnk

≤M1

1−tn

α+1

1−r tn

−(α+1)

≤M2

1−tn

α+1

, for someM2>0.

Hence if(1−t)α+1∈Gw, then it follows thatY ∈Gw. Conversely, if(1−t)α+1is not

inGw, then the first column ofAα,tis not inGw becausean,0=tn(1−tn)α+1. Thus, Aα,tis not aGw-Gw matrix.

Remark3.2. In theG_w-G_wsetting,A_α,tbeing aG_w-G_w matrix does not imply that

(1−t)∈Gw. Also,(1−t)∈Gw does not imply thatAα,tis aGw-Gw matrix.

This can be demonstrated as follows.

(1) Lettn=1−(1/3)n,α=1, andw=1/4. So, we have(1−tn)α+1=(1/9)n and

hence(1−t)α+1∈Gw. This implies thatAα,tis aGw-Gw matrix byTheorem 3.1. But

observe that(1−t)is notGw. Hence,Aα,tbeing aGw-Gwmatrix does not imply that (1−t)∈Gw.

(2) Lettn=1−(1/4)n,α= −1/2, andw=1/3. Then we have(1−t)∈Gw. But note

that(1−tn)α+1=(1/2)nand hence(1−t)α+1is not inGw. This implies thatAα,tis

not aGw-Gw matrix byTheorem 3.1. Hence,(1−t)∈Gw does not imply thatAα,tis

aGw-Gw matrix.

Corollary3.3. (1)If−1< α≤0, thenA_α,tis aG_w-G_wmatrix implies that(1−t)∈

Gw.

(2)Ifα >0, then(1−t)∈Gw implies thatAα,tis aGw-Gwmatrix.

Proof. _{(1) Since}−_{1< α}≤_{0 implies that (1}−_tn)≤₍₁−_tn)α+1_{, it follows that}

(1−t)∈Gw byTheorem 3.1.

(2) Ifα >0, then we have(1−tn)α+1< (1−tn)and hence byTheorem 3.1,Aα,ta Gw-Gw matrix whenever(1−t)∈Gw.

Corollary_3.4. _{The matrix Aα,t is aG-Gw matrix if and only ifAα,t is aGw-Gw}

matrix.

Proof. _{SinceGw is a subset ofG, Aα,tbeing aG-Gwmatrix yieldsAα,tis aGw-Gw}

matrix. Conversely, ifAα,t is a Gw-Gw matrix, then by Theorem 3.1, we have (1− t)α+1_∈G

w. Now using the same technique used in the proof ofTheorem 3.1, we can

easily show thatAα,tis aG-Gwmatrix. Thus, the corollary follows.

The next results indicate that the Aα,t matrix is a strong method in the Gw-Gw

setting. TheAα,tmatrix isGw-stronger than the identity matrix.

Theorem3.5. Suppose that−1< α≤0andA_α,tis aG_w-G_w matrix; thenG_w(A_α,t)

contains the class of all sequencesxwhose partial sums are bounded.

Proof. _{The theorem follows using a similar argument as in the proof of [2,}

Theo-rem 8].

Remark _3.6. _{Although Theorem 3.5is stated for} −_{1< α}≤_{0, it is also true for}

allα >−1 for some sequences, which we will demonstrate as follows. Letx be the unbounded sequence defined by

xk=(−1)k

k+α+1

LetY be theAα,t-transform ofx. Then we have

Yn=

1−tn

α+1

1+tn

α+2<

1−tn

α+1_{. (3.3)}

Thus, ifAα,tis aGw-Gwmatrix, then byTheorem 3.1,(1−t)α+1∈Gw, sox∈Gw(Aα,t).

Corollary_3.7. _{Suppose that}−_{1< α}≤_{0andAα,tis aGw-Gwmatrix; thenGw(Aα,t)}

contains the class of all sequencesxsuch that∞_k=0xkis conditionally convergent.

Our next results deal with theGw-translativity of theAα,tmatrix. We will show that

theAα,tmatrix isGw-translative for some sequences inGw(Aα,t).

Theorem 3.8. Every G_w-G_wA_α,t matrix is G_w-translative for each sequence x∈

Gw(Aα,t)for which{xk/k} ∈Gw,k=1,2,3, . . . .

Proof. Letx∈G_w(A_α,t). Then we will show that

(1) Tx∈Gw(Aα,t)and

(2) Sx∈Gw(Aα,t).

We first show that (1) holds. Note that

Aα,tTx

n =

1−tn

α+1

∞

k=0

k+α

k

xk+1tkn

=

1−tn

α+1

tn

∞

k=0

k+α

k

xk+1tnk+1

=

1−tn

α+1

tn

∞

k=1

k−1+α k−1

xktkn

=

1−tnα+1 tn

∞

k=1

k+α

k

xktkn k k+α

=

1−tn

α+1

tn

∞

k=1

k+α

k

xktkn

1−_k+α_α

≤An+Bn,

(3.4)

where

An=

1−tnα+1 tn

∞

k=1

k+α

k

xktkn

,

Bn=|

α|1−tn

α+1

tn

∞

k=1

k+α

k

xk k+αt

k n

.

(3.5)

The use of the triangle inequality is legitimate as the radii of convergence of the two power series are at least 1. Now if we show bothAandB are inGw, then (1) holds.

But the conditions thatA∈Gw andB∈Gw follow easily from the given hypothesis

Next we will show that (2) holds. Observe that

Aα,tSx

n =

1−tn

α+1

∞

k=1

k+α

k

xk−1tnk

=1−tn

α+1 ∞ k=0

k+α+1 k+1

xktnk+1

=1−tn

α+1

∞

k=0

k+α

k

xktnk+1

_k+α+1

k+1

=1−tn

α+1 ∞ k=0 k+α

k

xktnk+1

1+_kα₊₁

≤En+Fn,

(3.6)

where

En=

1−tn

α+1 ∞ k=0 k+α k

xktkn

,

Fn=

1−tn

α+1_|α|

∞

k=0

k+α

k

xk k+1t

k+ n . (3.7)

Now the given hypothesis thatx∈Gw(Aα,t)and{xk/k} ∈Gwimplies that bothEand F are inGw. Consequently, (2) holds and hence the theorem follows.

Theorem 3.9. Suppose that −1< α≤0; then every Gw-G_w matrix A_α,t is G

w-translative for eachAα-summable sequencexinGw(Aα,t).

Proof. _{Since the case α}=_{0 can be easily proved using the technique used in}

the proof of [4, Theorem 4.1], here we only consider the case −1< α <0. Letx∈ c(Aα)∩Gw(Aα,t). Then we will show that

(1) Tx∈Gw(Aα,t)and

(2) Sx∈Gw(Aα,t).

We first show that (1) holds. Note that

Aα,tTx

n =

1−tn

α+1

∞

k=0

k+α

k

xk+1tkn

=

1−tn

α+1 tn ∞ k=0 k+α k

xk+1tnk+1

=

1−tn

α+1 tn ∞

k=1

k−1+α k−1

xktkn

=

1−tn

α+1 tn ∞ k=1 k+α

k

xktkn k k+α

=

1−tn

α+1 tn ∞ k=1 k+α

k

xktkn

1−_k+α

α

≤An+Bn,

where

An=

1−tnα+1 tn ∞ k=1 k+α

k

xktnk

,

Bn= −

α1−tn

α+1

tn ∞

k=1

k+α

k

xk k+αt

k n . (3.9)

The use of the triangle inequality is legitimate as the radii of convergence of the two power series are at least 1. Now if we show that bothAandBare inGw, then (1) holds.

The conditionA∈Gwfollows from the hypothesis thatx∈Gw(Aα,t), andB∈Gwwill

be shown as follows. Observe that

Bn= −α

1−tnα+1 tn

x1tn+

∞

k=2

k+α

k

xk k+αt

k n

≤ −α x1 1−tn

α+1 +−α

1−tn

α+1

tn ∞

k=2

k+α

k

xk k+αt

k n

≤Cn+Dn,

(3.10)

where

Cn= −α x1 1−tn

α+1_,

Dn= −α

1−tnα+1 tn ∞ k=2 k+α k xk k+αt

k n . (3.11)

ByTheorem 3.1, the hypothesis thatAα,tisGw-Gw implies thatC∈Gw, hence there

remains only to showD∈Gwto prove that (1) holds. Now using the same techniques

used in the proof of [3, Theorem 2], we can show that

Dn≤M1M2 α

1−tn−M1M2 α

1−tnα+1, (3.12)

whereM1andM2are some positive real numbers. Note thatAα,tbeing aGw-Gwmatrix

implies that(1−t)α+1∈Gw by Theorem 3.1, and −1< α <0 yields (1−t)∈Gw.

Consequently, we haveD∈Gw and hence (1) holds. Next we show that (2) holds. We

have

Aα,tSx

n =

1−tn

α+1 ∞ k=1 k+α

k

xk−1tnk

=1−tn

α+1

∞

k=0

k+α+1 k+1

xktnk+1

=1−tn

α+1

∞

k=0

k+α

k

xktnk+1

_k+α+1

k+1

=1−tnα+1

∞ k=0 k+α

k

xktnk+1

1+_k+α

1

≤En+Fn,

where

En=

1−tn

α+1

∞

k=0

k+α

k

xktnk

,

Fn= −1−tnα+1α

∞

k=0

k+α

k

xk k+1t

k+ n

.

(3.14)

The hypothesis thatx∈Gw(Aα,t)implies thatE∈Gw and by proceeding as in the

proof of (1) above, we can easily show thatF ∈Gw. Thus, (2) holds and hence our

assertion follows.

Theorem _3.10. _{Suppose that α >0and (1}−_t)∈_{Gw; then every Aα,t matrix is}

Gw-translative for eachAα-summable sequencexinGw(Aα,t).

Proof. The theorem follows easily by using similar argument used in the proof of

Theorem 3.9.

Our next result is a Tauberian theorem forAα,tmatrix in theGw-Gw setting.

Theorem_3.11. _{LetAα,tbe aGw-Gw matrix. Ifxis a sequence such thatAα,txand}

∆xare inGw, thenxis inGw.

Proof. _{The theorem easily follows by an argument similar to the proof of [4,}

Theo-rem 2.1].

References

[1] D. Borwein,On a scale of Abel-type summability methods, Proc. Cambridge Philos. Soc.53 (1957), 318–322.MR 19,134f. Zbl 082.27602.

[2] M. Lemma,The Abel-type transformations into, Int. J. Math. Math. Sci.22(1999), no. 4, 775–784.MR 2001b:40002. Zbl 961.40005.

[3] , The-translativity of Abel-type matrix, Int. J. Math. Math. Sci.23(2000), no. 3, 189–195.MR 2001i:40005. Zbl 952.40002.

[4] S. Selvaraj,Some theorems on the matrix transformations intoG, Houston J. Math.20 (1994), no. 2, 237–245.MR 95d:40007. Zbl 802.40003.

Mulatu Lemma: Savannah State University, Savannah, GA31404, USA E-mail address:[email protected]