Bulletin of the Institute of Mathematics
Academia Sinica (New Series)
Vol. 5 (2010), No. 1, pp. 75-82
SOME THEOREMS ON WEIGHTED
MEAN SUMMABILITY
BY
MEHMET A. SARIGÖL
Abstract
In this paper we have proved some theorems on weighted
mean summability method by using analytical and summability
techniques, which also extends the well known result of Hardy on
the Cesàro summability.
1. Introduction
P
Let
xv be a given infinite series of complex number with s = (sn )
as the sequence its n-th partial sum and let (pn ) be a sequence of positive
numbers where for n = 0, 1, 2, . . ..
Pn = p0 + p1 + · · · + pn → ∞ as n → ∞,
P−1 = p−1 = 0.
The sequence-to-sequence transformation
Tn =
n
1 X
pv s v ,
Pn v=0
n = 0, 1, 2, . . .
(1)
defines the sequence (Tn ) of (N , pn ) means of the sequence (sn ), generated by
P
the sequence of coefficients (pn ) [2]. The series
xk is said to be summable
(N , pn ) to a number λ if
Tn → λ as n → ∞.
Received December 9, 2008 and in revised form January 6, 2009.
AMS Subject Classification: 40D25, 40G15, 40G99.
Key words and phrases: Cesàro, weighted mean, summability field.
75
76
MEHMET A. SARIGÖL
[March
It is well known that (N , pn ) summability method is regular if and only if
Pn → ∞ as n → ∞. In the special case where pv = 1 for v = 0, 1, 2, . . ., it
reduces to the Cesàro summability (C, 1).
In [1] Hardy proved the following theorem on Cesàro summability.
P
Theorem 1.1. The series
xv is summable (C, 1) to a finite number
P
λ if and only if
bn converges to λ, where
bn =
∞
X
xv
,
v+1
v=n
n = 0, 1, . . . .
We need the following lemma to prove the main theorems.
Lemma 1.2. An infinite matrix B = (bnv ) regular if and only if (see
[2])
(a) sup
n
∞
X
|bnv | < ∞,
(b) lim bnv = 0 (v = 0, 1, . . .),
n
(c) lim
n
v=0
∞
X
bnv = 1.
v=0
2. The Main Results
In this section we have proved the following theorems on weighted mean
summability method using analytical and summability techniques, which
also extend the well known result of Hardy on the Cesàro summability.
Theorem 2.1. Let (pn ) and (qn ) be two sequences of positive numbers
satisfying the following conditions:
Pn → ∞,
Then the series
P
Qn → ∞ as n → ∞.
xv is summable (N , pn ) to λ whenever
a finite number λ if and only if
n X
pv Qv+1 pv−1 Qv−1 = O(Pn )
qv+1 −
qv
v=1
(2)
P
bn converges to
(3)
2010]
SOME THEOREMS ON WEIGHTED MEAN SUMMABILITY
77
where
bn = qn
∞
X
xv
,
Q
v=n v
n = 0, 1, . . . .
(4)
Theorem 2.2. Let (pn ) and (qn ) be two sequences of positive numbers
satisfying the following conditions:
(a) Pn = (pn Qn ) and (b) Pn → ∞,
Qn → ∞ as n → ∞.
(5)
P
P
Then the series
bn converges to λ whenever the series
xv is summable
(N , pn ) to a finite number λ if and only if
∞
X
Pv qv+1
qv+2 pv Qv − pv+1 Qv+2 = O(1/Qn ),
Q
v=n v+1
(6)
where bn is defined by (4).
It is noticed that, if take pn = qn = 1 for v = 0, 1, . . . in Theorem 2.1
and Theorem 2.2, then we obtain Theorem 1.1.
Proof of Theorem 2.1. Let Bn =
n
P
bv → λ. It follows from (4) that
v=0
xn = Qn
b
∞
n
qn
−
X xv
bn+1 bn
=
, n = 0, 1, . . . and
qn+1
qn v=n Qv
and so
sm =
m
X
v=0
Qv
b
v
qv
−
bv+1 bm+1
= Bm − Qm
.
qv+1
qm+1
∞
Since the series
which implies
∞ x
X xv
P
bn
v
is convergent, we have
=
→ 0 as n → ∞,
qn v=n Qv
v=n Qv
sm
Bm
bm+1
=
−
→ 0 as n → ∞,
Qm
Qm qm+1
by virtue of (2). Hence, for n ≥ 0,
bn = qn
∞
m
X
X
xv
sv − sn−1
= lim qn
,
m
Q
Qv
v=n v
v=n
(s−1 = 0)
78
MEHMET A. SARIGÖL
= lim qn
m
= −
[March
m−1
X 1
sm
1
sn−1
+
−
sv −
Qm v=n Qv
Qv+1
Qn
∞
X
1
1
qn sn−1
+ qn
cv sv , where cv =
−
.
Qn
Q
Q
v
v+1
v=n
On the other hand, using Abel’s summation by parts gives us
Bn
n
X
n−1
∞
s
X
X
bv
xv
bn
n−1
=
qv =
Qv
+ Qn
= sn−1 + Qn −
+
cv sv
qv
Qv
qn
Qn
v=n
v=0
v=0
∞
X
∞
X
Bn
Bn
Bn+1
= Qn
cv sv ⇒
=
cv sv ⇒ cn sn =
−
, (n = 0, 1, . . .).
Qn v=n
Qn Qn+1
v=n
By the last equality, we write
Tn
n
n
1 X
1 X 1 Bv
Bv+1 =
pv s v =
pv
−
Pn v=0
Pn v=0 cv Qv
Qv+1
n−1
X 1 pv
1
p0
pv−1
pn
=
B0 +
−
Bv −
Bn+1
Pn c0 q0
Qv cv
cv−1
cn Qn+1
v=1
=
∞
X
cnv Bv ,
v=0
where the matrix C = (cnv ) is defined by
cnv

1
p0

·
,



P
c
n
0 Q0





 1 pv − pv−1 1 ,
Pn cv
cv−1 Qv
=


1
pn


−
,


Pn cn Qn+1




0,
v=0
1≤v≤n
v =n+1
v ≥ n.
Now, it is clear that cnv → 0 as n → ∞ and
∞
X
v=0
cnv =
1
Pn
n
X 1
p0
+
c0 Q0 v=1 Qv
pv
pv−1
−
cv
cv−1
−
pn
cn Qn+1
= 1.
By Lemma 1.2, Tn → λ as n → ∞ if and only if the matrix C is regular, or
2010]
SOME THEOREMS ON WEIGHTED MEAN SUMMABILITY
79
equivalently
n
p0
1 X 1 pv pv−1 pn
|cnv | =
+
−
+
= O(1) as n → ∞.
Pn c0 Q0 Pn
Qv cv cv−1
Pn cn Qn+1
v=0
v=1
(7)
∞
X
Because of that the boundedness of the middle term implies the other term,
(7) is equivalent to (3), whence the result.
Proof of Theorem 2.2. Suppose that if Tn → λ as n → ∞, then
converges to λ. Then, by (1), since
s0 = T0 ,
sn =
1
(Pn Tn − Pn−1 Tn−1 ),
pn
P
bn
n = 1, 2, . . . ,
we have
sm
→ 0 as m → ∞,
Qm
by virtue of (5a) and (5b). Hence, for n ≥ 0,
∞
m
X
X
xv
sv − sn−1
= lim qn
, (s−1 = 0)
m
Q
Qv
v=n v
v=n
m−1
X 1
1
sn−1
sm
= lim qn
+
−
sv −
m
Qm v=n Qv
Qv+1
Qn
bn = qn
∞
X
1
1
qn sn−1
+ qn
cv sv , where cv =
−
.
= −
Qn
Q
Q
v
v+1
v=n
On the other hand, it follows from Abel’s summation by parts that
Bn
n
X
n−1
X
bv
xv
bn
=
qv =
Qv
+ Qn
= sn−1 + Qn
qv
Qv
qn
v=0
v=0
∞
X
m
X
m
X
∞
sn−1 X
−
+
cv sv
Qn
v=n
1
(Pv Tv − Pv−1 Tv−1 )
m
m
pv
v=n
v=n
v=n
m−1
X cv
Pm
Pn−1
cv+1
= Qn lim cm
Tm − cn
Tn−1 +
Pv
−
Tv .
(8)
m
pm
pn
pv
pv+1
v=n
= Qn
cv sv = Qn lim
cv sv = Qn lim
cv
80
MEHMET A. SARIGÖL
[March
Now define
∞
X
N = (xk ) :
xk is summable (N , pn )
k=0
and
X
n
B = (xk ) :
bk is convergent .
k=0
These are BK-spaces (i.e., Banach spaces with continuous coordinates) with
respect to the norms
kxkN
n
1 X
= sup |Tn | = sup pk sk Pn
n
n
(9)
k=0
and
kxkB
X
X
∞
X
n
n
xv ,
= sup bk = sup qk
Qv n
n
k=0
k=0
(10)
v=k
respectively. By the Banach-Steinhaus theorem, there exists a constant M >
0 such that
kxkB ≤ M kxkN
(11)
for all x ∈ N . Applying (9) and (10) to the special sequence


n=k
 1,
xn = −1,
n = k + 1,
k = 0, 1, 2, . . .

 0,
otherwise
we have
kxkN =
pk
and kxkB = Qk ck .
Pk
It follows from (11) that for k = 0, 1, 2, . . .
Qk ck ≤ M
1 pk
Pk
⇔ ck ·
=O
,
Pk
pk
Qk
(12)
which implies
cm
Pm
→ 0 as m → ∞,
pm
(13)
2010]
SOME THEOREMS ON WEIGHTED MEAN SUMMABILITY
81
by virtue of (5b). Thus, considering (8) we write
Bn = −
∞
∞
c
X
X
Qn cn Pn−1
cv+1 v
Tn−1 + Qn
Pv
−
Tv =
bnv Tv , (14)
pn
p
p
v
v+1
v=n
v=0
where the matrix B = (bnv ) is defined by
bnv =







0,
0 ≤ v < n − 1,
Qn cn Pn−1
,
pn





 Qn Pv cv − cv+1 ,
pv
pv+1
−
v = n − 1,
(15)
v ≥ n.
By hypothesis, the matrix B is regular. Now it is clear that lim bnv = 0 and
n
∞
P
bnv = 1. Therefore it follows from Lemma 1.2 that B is regular if and
v=0
only if
∞
X
∞
X
cv
Qn cn Pn−1
cv+1 |bnv | =
+ Qn
Pv −
= O(1) as n → ∞.
pn
pv
pv+1 v=n
v=0
(16)
By considering (13) we obtain
∞
c
X
cv+1 cn
v
Qn Pn−1
≤ Qn
Pv −
pn
pv
pv+1
v=n
and so (16) is equivalent to
∞
X
cv
cv+1
Qn
Pv −
| = O(1) as n → ∞.
pv
pv+1
v=n
Thus, the condition (6) is necessary.
Conversely, if the condition (6) is satisfied, then the series
∞
X
v=0
Pv
cv
cv+1
−
pv
pv+1
c
n+1
converges. Hence it follows that the sequence Pn
converges, which
pn+1
82
MEHMET A. SARIGÖL
[March
implies that
cn
→ 0 as n → ∞
pn
Pn cn
→ 0 as n → ∞
pn
and so (14) is valid. Therefore the result is seen by the regularity of the
matrix B, and completes the proof.
by virtue of (5b). By considering (6) again, we have
References
1. G. H. Hardy, A theorem concerning summable series, Proc. Cambridge Philos.
Soc., 20(1920-21), 304-307.
2. G. H. Hardy, Divergent Series, Oxford Univ. Press., Oxford, (1949).
Department of Mathematics, Pamukkale University, Denizli 20017, Turkey.
E-mail: msarigol@pamukkale.edu.tr