Another Theorem on Weighted Means

(1)

P N Natarajan

Another Theorem on Weighted Means

Abstract. In this short paper, which is a continuation of [2], we prove another interesting result concerning weighted means.

2000 Mathematics Subject Classification: 40,46.

Key words and phrases: Regular matrix, weighted means.

1. Introduction. For the sake of completeness, we recall the following. For a given infinite matrix A = (ank), n, k = 0, 1, 2, . . . and a given sequence x = {x^k}, k = 0, 1, 2, . . . , by the A-transform of x = {x^k}, we mean the sequence Ax = {(Ax)ⁿ},

(Ax)n= X∞ k=0

ankxk, n = 0, 1, 2, . . . , where it is assumed that the series on the right converges. If lim

n→∞(Ax)n = `, we say that x = {x^k} is A-summable or summable A to `. If lim_n→∞(Ax)n= `, whenever

klim→∞xk = `, we say that A is regular. The following result is well-known (see [3], Theorem II.1, pp. 11-12).

Theorem 1.1 A = (ank)is regular if and only if (i) sup

n

X∞ k=0

|a^nk| < ∞;

(ii) lim

n→∞ank= 0, k = 0, 1, 2, . . . ; and

(iii) lim

n→∞

X∞ k=0

ank= 1.

(2)

An infinite series X∞ k=0

xk is said to be A-summable to ` if {sn} is A-summable

to `, where sn= Xn k=0

xk, n = 0, 1, 2, . . . .

2. Weighted Means.

Definition 2.1 ([3], p.16) The weighted mean method or ( ¯N , pn) method is defi- ned by the infinite matrix A = (ank), where

ank= (_p

k

Pn, k¬ n;

0, k > n,

Pn= Xn k=0

pk, n = 0, 1, 2, . . . , Pn 6= 0, n = 0, 1, 2, . . . .

Theorem 2.2 ([3], p.16) The weighted mean method ( ¯N , pn)is regular if and only if

(i) Xn k=0

|p^k| = O(Pⁿ), n → ∞;

and

(ii) Pn → ∞, n → ∞.

Remark 2.3

|Pⁿ| ¬ Xn k=0

|p^k|

¬

n+mX

k=0

|p^k|

¬ L|P^n+m|, for some L > 0, m = 0, 1, 2, . . . ; n = 0, 1, 2, . . . .

3. Main Result. Following Móricz and Rhoades [1], we prove the main result of the paper, which supplements [2].

Theorem 3.1 Let ( ¯N , pn), ( ¯N , qn)be two regular weighted mean methods and

(1) Pn = O(pnQn), n → ∞,

(3)

i.e., pn^PQⁿn

¬ M for some M > 0, n = 0, 1, 2, . . . . Let

X∞ n=0

xn be ( ¯N , pn) summable to `. Then X∞ n=0

bn converges to ` if and only if

sup

n

"

|Qⁿ| X∞ k=n

Pk

Qk+1

qk+1

pkQk − qk+2

pk+1Qk+2

#

<∞,

where bn= qn

X∞ k=n

xk

Qk, n = 0, 1, 2, . . .. Proof Let

sn = Xn k=0

xk and

tn =p0s0+ p1s1+ · · · + pⁿsn

Pn

, n = 0, 1, 2, . . . . Then

s0= t0 and sn = 1

pn

(Pntn− Pⁿ−1tn−1), n = 1, 2, . . . .

Let X∞ n=0

xn be ( ¯N , pn) summable to ` so that lim

n→∞tn= `. Now, sn

Qn

= 1

pnQn

(Pntn− Pn−1t_n−1)

= 1

pnQn

[Pn(tn− `) − Pⁿ−1(tn−1− `) + `(Pⁿ− Pⁿ−1)]

= 1

pnQn

[Pn(tn− `) − Pⁿ−1(tn−1− `) + `pⁿ]

= Pn

pnQn

(tn− `) − Pn−1

pnQn

(tn−1− `) + ` Qn

so that sn

Qn

¬ M [|tⁿ− `| + L|tⁿ−1− `|] + |`|

|Qⁿ|,

since |Pn−1| ¬ L|Pⁿ|, using Remark 2.3

→ 0, n → ∞, since lim_n

→∞tn= ` and lim

n→∞Qn= ∞,

( ¯N , qn) being regular, using Theorem 2.2.

As already worked out in [2],

bn= −qnsn−1

Qn

+ qn

X∞ k=n

cksk,

(4)

where

ck = 1 Qk − 1

Qk+1, k = 0, 1, 2, . . . Now,

Bn= sn−1+ Qn −sn−1

Qn

+ X∞ k=n

cksk

!

(see [2])

= sn−1− sⁿ−1+ Qn

X∞ k=n

cksk

= Qn

X∞ k=n

cksk

= Qn lim

m→∞

Xm k=n

cksk

= Qn lim

m→∞

Xm k=n

ck· 1

pk {P^ktk− Pk−1t_k−1}

= Qn lim

m→∞

"

cmPmtm

pm −cnP_n−1t_n−1 pn

+

mX−1 k=n

Pktk

ck

pk − ck+1

pk+1

#

(2) .

Let

A1= (

{x^k} : X∞ k=0

xk is ( ¯N , pn) summable )

;

A2= (

{x^k} : X∞ k=0

bk converges )

.

Note that A1, A2are BK spaces with respect to the norms defined by

||x||^A1= sup

n0|tⁿ|, x = {x^k} ∈ A1; and

||x||^A2= sup

n0|Bⁿ|, x = {x^k} ∈ A², respectively. In view of Banach-Steinhaus theorem,

(3) ||x||^A2 ¬ U||x||^A1, for some U > 0.

For every fixed k = 0, 1, 2, . . . , define the sequence x = {xⁿ}, where

xn=







1, if n = k;

−1, if n = k + 1;

0, otherwise.

(5)

For this sequence x,

||x||^A1= pk

Pk

and ||x||^A² = |Q^kck|.

Using (3), we have, for k = 0, 1, 2, . . . ,

|Q^kck| ¬ U|p^k|

|P^k|, so that

ckPk

pk

¬ U

|Q^k|

→ 0, k → ∞, since lim_k

→∞Qk= ∞, in view of Theorem 2.2.

Consequently

(4) lim

k→∞

ckPk

pk

= 0.

Using (4) in (2), we have,

Bn= −cnPn−1tn−1

pn

Qn+ Qn

X∞ k=n

Pktk

ck

pk −ck+1

pk+1

= X∞ k=0

anktk,

where (ank) is defined by

ank=









0, if 0 ¬ k < n − 1;

−^Qⁿ^cⁿpn^Pⁿ⁻¹, k = n− 1;

QnPk

ck

pk −^cp^k+1_k+1

, k n.

We first note that lim

n→∞ank= 0, k = 0, 1, 2, . . . and X∞ k=0

ank= 1, n = 0, 1, 2, . . . so

that lim

n→∞

X∞ k=0

ank= 1. Thus, appealing to Theorem 1.1, X∞ n=0

bn converges to ` if and only if

(5) sup

n

"

|Qⁿ|

cnPn−1

pn

+

X∞ k=n

P^k

ck

pk −ck+1

pk+1

!#

<∞.

(6)

However,

QncnP_n−1 pn

¬ L

QncnPn

pn

since |Pⁿ−1| ¬ L|Pⁿ|, using Remark 2.3

= L|Qⁿ| Pn

X∞ k=n

ck

pk −ck+1

pk+1

, using (4)

¬ L²|Qⁿ|

X∞ k=n

Pk

ck

pk − ck+1

pk+1

since|Pⁿ| ¬ L|P^k|, k n, using Remark 2.3

¬ L²|Qⁿ|X

kn

P^k

ck

pk −ck+1

pk+1

. (6)

Using (6), it is now clear that (5) is equivalent to

sup

n0|Qⁿ|



X

kn

P^k

ck

pk −ck+1

pk+1



 < ∞.

Now,

ck

pk −ck+1

pk+1 = 1 pk

1 Qk − 1

Qk+1

− 1

pk+1

1

Qk+1 − 1 Qk+2

= qk+1

pkQkQk+1 − qk+2

pk+1Qk+1Qk+2.

Thus X∞ n=0

bn converges to ` if and only if

sup

n0|Qⁿ|



X

kn

P^k

qk+1

pkQkQk+1 − qk+2

pk+1Qk+1Qk+2



 < ∞,

i.e., sup

n0|Qⁿ|



X

kn

Pk

Qk+1

qk+1

pkQk − qk+2

pk+1Qk+2



 < ∞,

which completes the proof of the theorem. _■

References

[1] F. Móricz and B.E. Rhoades, An equivalent reformulation of summability by weighted mean methods, revisited, Linear Algebra Appl.349 (2002), 187–192.

[2] P.N. Natarajan, A generalization of a theorem of Móricz and Rhoades on weighted means (Communicated for publication).

(7)

[3] A. Peyerimhoff, Lectures on summability, Lecture Notes in Mathematics,107, Springer, 1969.

P N Natarajan

Old No. 2/3, New No. 3/3, Second Main Road, R.A. Puram, Chennai 600 028, India E-mail: pinnangudinatarajan@gmail.com

(Received: 4.10.2010)