# A Tauberian Theorem for Double Cesàro Summability Method

Authors: Bidu Bhusan Jena; Susanta Kumar Paikray; Umakanta Misra
Journal: International Journal of Mathematics and Mathematical Sciences
 (2016)

---

## Abstract

We have generalized Littlewood Tauberian theorems for(
C
,
k
,
r
) summability of double sequences by using oscillating behavior and de la Vallée-Poussin mean. Further, the generalization of (
C
,
r
) summability from (
C
,
k
,
r
) summability is given as corollaries which were earlier established by the authors.


---

## Body

## 1. Introduction

Letu
=
(
u
m
n
) be a double real sequence. We define Δ
n
u
m
n
=
u
m
n
-
u
m
,
n
-
1, Δ
m
u
m
n
=
u
m
n
-
u
m
-
1
,
n, and Δ
m
,
n
u
m
n
=
u
m
n
-
u
m
-
1
,
n
-
u
m
,
n
-
1
+
u
m
-
1
,
n
-
1.A double sequence(
u
m
n
) is said to be bounded if there exists a finite real number C
>
0 such that u
m
n
≤
C
,
∀
m
,
n
∈
Z
+. Let us write(1)
σ
m
n
11
u
=
1
m
+
1
n
+
1
∑
p
=
0
m
∑
q
=
0
n
u
p
q
=
∑
p
=
1
m
∑
q
=
1
n
V
p
q
11
p
q
+
u
00 (see [1]),(2)
σ
m
n
10
u
=
1
m
+
1
∑
p
=
0
m
u
p
n
,
σ
m
n
01
u
=
1
n
+
1
∑
q
=
0
n
u
m
q
.Then, we say that a double sequence u
=
(
u
m
n
) is (
C
,
1,1
) summable to s if σ
m
n
(
11
)
(
u
) converges to s, as m
,
n
→
∞. Similarly, we say that it is (
C
,
1,0
) summable to s if σ
m
n
(
10
)
(
u
) converges to s as m
,
n
→
∞ and (
C
,
0,1
) summable to s if σ
m
n
(
10
)
(
u
) converges to s as m
,
n
→
∞. For all nonnegative integers k and r, we define σ
m
n
(
k
r
)
(
u
) as follows: (3)
σ
m
n
k
r
u
=
1
m
+
1
n
+
1
∑
p
=
0
m
∑
q
=
0
n
σ
p
q
k
-
1
,
r
-
1
u
f
o
r
k
,
r
≥
1
,
u
m
n
f
o
r
k
,
r
=
0
.A double sequenceu
=
(
u
m
n
) is said to be (
C
,
k
,
r
) summable to s if σ
m
n
(
k
r
)
(
u
) converges to s. If k
=
1 and r
=
1, then (
C
,
k
,
r
) summability reduces to (
C
,
1,1
) summability. Again, if k
≠
0 and r
=
0, then (
C
,
k
,
r
) summability reduces to (
C
,
k
,
0
) summability. Further, if k
=
0 and r
≠
0, then (
C
,
k
,
r
) summability reduces to (
C
,
0
,
r
) summability. Here, Cesàro summability of u
=
(
u
m
n
) refers to (
C
,
k
,
r
) summability of u
=
(
u
m
n
). It may also be noted that the convergence of a double sequence u
=
(
u
m
n
) implies the Cesàro summability of u
=
(
u
m
n
), but the converse is not generally true.For example, consider a functionf
(
x
,
y
)
=
e
2
x
sin
⁡
(
3
y
): the sequence (
u
m
n
), which is the sequence of coefficients in the Taylor series expansion of the function f
(
x
,
y
) about origin, is Cesàro summable but not convergent.For the proof of the converse part, certain conditions are presented in terms of oscillatory behavior of double sequenceu
=
(
u
m
n
).Let us define(
u
m
n
) as(4)
u
m
n
=
V
m
n
11
+
∑
p
=
1
m
∑
q
=
1
n
V
p
q
11
p
q
+
u
00
,
m
,
n
=
1,2
,
…
,where(5)
u
m
n
-
σ
m
n
11
u
=
V
m
n
11
Δ
u
=
1
m
+
1
n
+
1
∑
p
=
0
m
∑
q
=
0
n
p
q
Δ
p
,
q
u
p
q(see [2]).Moreover, in analogy to Kronecker identity for a single sequence, we can write(6)
V
m
n
1,0
Δ
u
=
1
m
+
1
∑
p
=
0
m
p
Δ
p
,
u
p
,
n
,
V
m
n
0,1
Δ
u
=
1
n
+
1
∑
q
=
0
n
q
Δ
q
,
u
m
,
qas the (
C
,
1,0
) mean of the sequence m
Δ
m
u
m
n and the (
C
,
0,1
) mean of the sequence “n
Δ
n
u
m
n,” respectively.Further, as the sequenceV
m
n
(
11
)
(
Δ
m
n
u
m
n
) is the (
C
,
1,1
) mean of the sequence m
n
(
Δ
m
n
u
m
n
), the sequence m
n
(
Δ
m
n
u
m
n
) is (
C
,
1,1
) summable to s whenever V
m
n
(
11
)
(
Δ
m
n
u
m
n
) converges to s as m
,
n
→
∞. For all nonnegative integers k and r, let us define V
m
n
(
k
r
)
(
Δ
u
) as follows: (7)
V
m
n
k
r
Δ
u
=
1
m
+
1
n
+
1
∑
p
=
0
m
∑
q
=
0
n
V
p
q
k
-
1
,
r
-
1
Δ
u
f
o
r
k
,
r
≥
1
,
m
n
Δ
m
,
n
u
m
n
f
o
r
k
,
r
=
0
.The sequencem
n
(
Δ
m
n
u
m
n
) is said to be (
C
,
k
,
r
) summable to s if V
m
n
(
k
r
)
(
Δ
m
n
u
m
n
) converges to s as m
,
n
→
∞. In particular, if k
=
1 and r
=
1, then (
C
,
k
,
r
) summability reduces to (
C
,
1,1
) summability. Again, if k
≠
0 and r
=
0, then (
C
,
k
,
r
) summability reduces to (
C
,
k
,
0
) summability. Further, if k
=
0 and r
≠
0, then (
C
,
k
,
r
) summability reduces to (
C
,
0
,
r
) summability.Then, the de la Vallée-Poussin mean of double real sequence(
u
m
n
) is defined by(8)
τ
m
n
u
=
1
λ
m
-
m
λ
n
-
n
∑
i
=
m
+
1
λ
m
∑
j
=
n
+
1
λ
n
u
i
j
,for sufficiently large nonnegative integers m
,
n and λ
>
1, and(9)
τ
m
n
u
=
1
m
-
λ
m
n
-
λ
n
∑
i
=
λ
m
+
1
m
∑
j
=
λ
n
+
1
n
u
i
j
,for sufficiently large nonnegative integers m
,
n and 0
<
λ
<
1.A single sequenceu
=
(
u
n
) is slowly oscillating [3] if (10)
lim
λ
→
1
+
⁡
lim
sup
n
⁡
max
n
≤
k
≤
λ
n
⁡
u
k
-
u
n
=
0
.A double sequence u
=
(
u
m
n
) is slowly oscillating [4] if (11)
lim
λ
→
1
+
⁡
lim
sup
m
,
n
⁡
max
m
+
1
,
n
+
1
≤
i
,
j
≤
λ
m
,
λ
n
⁡
∑
x
=
m
+
1
i
∑
y
=
n
+
1
j
Δ
x
,
y
u
x
y
=
0
.In an earlier paper by Jena et. al. [5], a proof of the generalized Littlewood Tauberian theorem by Cesàro summability method has been established. For a proof of Littlewood Tauberian theorem differently, the paper of Çanak and Totur [6] and Çanak [7–9] can be referred to. Also, a similar result was introduced earlier by Çanak [10] under the consideration of improper integral. Recently, Totur [4] has introduced Littlewood Tauberian theorem by (
C
,
1,1
) mean for double real sequence.In the proposed paper, with certain novelty, we have generalized it for(
C
,
k
,
r
) summability of a double real sequence defined in (4).


## 2. Known Results

Theorem 1 (see [4]).
If the sequence(
u
m
n
) is (
C
,
1,1
) summable to s and (
u
m
n
) is slowly oscillating (in the sense of (
1,1
)), then l
i
m
m
,
n
→
∞
u
m
n
=
s.Corollary 2 (see [5]).
If the sequence(
u
n
) is (
C
,
1
) summable to s and (
u
n
) is slowly oscillating, then l
i
m
n
→
∞
u
n
=
s.Theorem 3 (see [4]).
If the sequenceu
=
(
u
m
n
) is (
C
,
1,1
) summable to s and V
m
n
(
11
)
(
Δ
m
n
u
m
n
) is slowly oscillating, then l
i
m
m
,
n
→
∞
u
m
n
=
s.Corollary 4 (see [5]).
If the sequence(
u
n
) is (
C
,
1
) summable to s and V
m
n
(
1,0
)
(
Δ
u
) is slowly oscillating, then l
i
m
n
→
∞
u
n
=
s.


## 3. Main Result

Theorem 5.
If(
u
m
n
) is (
C
,
k
,
r
) summable to s and (
u
m
n
) is slowly oscillating, then l
i
m
m
,
n
→
∞
u
m
n
=
s.To prove the above theorem, we need the help of the following lemmas.Lemma 6.
A double sequenceu
=
(
u
m
n
) is slowly oscillating if and only if V
m
n
(
11
) is slowly oscillating and bounded.Proof.
Letu
=
(
u
m
n
) be slowly oscillating. Initially, let us show that V
m
n
(
11
)
=
O
(
1
).
We have by definition of slow oscillation, forλ
>
1, (12)
lim
λ
→
1
+
⁡
lim
sup
m
,
n
⁡
max
m
+
1
,
n
+
1
≤
i
,
j
≤
λ
m
,
λ
n
⁡
∑
x
=
m
+
1
i
∑
y
=
n
+
1
j
Δ
x
,
y
u
x
y
=
0
,and let us rewrite the finite sum ∑
i
=
1
m
‍
∑
j
=
1
n
‍
i
j
Δ
u
i
j as the series ∑
x
=
0
∞
‍
∑
y
=
0
∞
‍
∑
m
/
2
x
+
1
,
n
/
2
y
+
1
≤
i
,
j
<
m
/
2
x
,
n
/
2
y
‍
i
j
Δ
u
i
j.
Clearly,(13)
∑
i
=
1
m
∑
j
=
1
n
i
j
Δ
u
i
j
≤
∑
x
=
0
∞
∑
y
=
0
∞
∑
m
/
2
x
+
1
,
n
/
2
y
+
1
≤
i
,
j
<
m
/
2
x
,
n
/
2
y
i
j
Δ
u
i
j
≤
∑
x
=
0
∞
∑
y
=
0
∞
m
n
2
x
+
y
max
m
/
2
x
+
1
+
1
,
n
/
2
y
+
1
+
1
≤
i
,
j
≤
λ
m
/
2
i
+
1
,
λ
n
/
2
j
+
1
⁡
∑
x
=
m
/
2
x
+
1
+
1
i
∑
y
=
n
/
2
y
+
1
+
1
j
Δ
x
,
y
u
x
y
≤
m
n
C
∑
x
=
0
∞
∑
y
=
0
∞
1
2
x
+
y
=
m
n
C
∗
,where C
∗
>
0.
Consequently, we have(14)
V
m
n
11
Δ
u
=
1
m
+
1
n
+
1
∑
p
=
0
m
∑
q
=
0
n
p
q
Δ
p
,
q
u
p
q
=
O
1
,
a
s
m
,
n
⟶
∞
.Since σ
m
n
(
11
)
(
u
)
=
∑
p
=
1
m
‍
∑
q
=
1
n
‍
V
p
q
(
11
)
/
p
q
+
u
00 is slowly oscillating, V
m
n
(
11
) is slowly oscillating.
To prove the converse part, considerV
m
n
(
11
) to be bounded and slowly oscillating. Now, the boundedness of V
m
n
(
11
) implies that σ
m
n
(
11
)
(
u
) is slowly oscillating. Further, V
m
n
(
11
) is slowly oscillating, so, by Kronecker identity (5), (
u
m
n
) is slowly oscillating.
This completes the proof of Lemma6.Next, we represent the difference(
u
m
n
-
σ
m
n
(
11
)
(
u
)
) under two different cases in the following lemma.Lemma 7 (see [1]).
Letu
=
(
u
m
n
) be a sequence of real numbers with m
,
n sufficiently large; then one has the following: (i)
Forλ
>
1,(15)
u
m
n
-
σ
m
n
11
u
=
λ
m
+
1
λ
n
+
1
λ
m
-
m
λ
n
-
n
σ
λ
m
,
λ
n
11
u
-
σ
λ
m
,
n
11
u
-
σ
m
,
λ
n
11
u
+
σ
m
n
11
u
+
λ
m
+
1
λ
n
-
m
σ
λ
m
,
n
11
u
-
σ
m
,
n
11
u
+
λ
n
+
1
λ
m
-
m
σ
m
,
λ
n
11
u
-
σ
m
n
11
u
-
1
λ
m
-
m
λ
n
-
n
∑
i
=
m
+
1
λ
m
∑
j
=
n
+
1
λ
n
u
i
j
-
u
m
n
.
(ii)
For0
<
λ
<
1,(16)
u
m
n
-
σ
m
n
1
u
=
λ
m
+
1
λ
n
+
1
m
-
λ
m
n
-
λ
n
σ
m
n
11
u
-
σ
λ
m
,
n
11
u
-
σ
m
,
λ
n
11
u
+
σ
λ
m
,
λ
n
11
u
+
λ
m
+
1
m
-
λ
m
σ
m
,
n
11
u
-
σ
λ
m
,
n
11
u
+
λ
n
+
1
n
-
λ
n
σ
m
n
11
u
-
σ
m
,
λ
n
11
u
-
1
m
-
λ
m
n
-
λ
n
∑
i
=
λ
m
+
1
m
∑
j
=
λ
n
+
1
n
u
m
n
-
u
i
j
.Proof of Theorem5.
Letu
=
(
u
m
n
) be slowly oscillating; then, σ
m
n
(
k
,
r
)
(
u
) is slowly oscillating (by Lemma 6). Further, since u
=
(
u
m
n
) is (
C
,
k
,
r
) summable to s, by Theorem 1,(17)
lim
m
,
n
→
∞
⁡
σ
m
n
k
,
r
u
=
s
.
Next, from the definition,(18)
σ
m
n
k
,
r
u
=
σ
m
n
1,1
u
σ
m
n
k
-
1
,
r
-
1
u
.Clearly, (17) and (18) imply that u
=
(
u
m
n
) is (
C
,
k
-
1
,
r
-
1
) summable to s. Again, (
σ
m
n
(
k
-
1
,
r
-
1
)
(
u
)
) is also slowly oscillating (by Lemma 6).
Thus, by Theorem1, we have(19)
lim
m
,
n
→
∞
⁡
σ
m
n
k
-
1
,
r
-
1
u
=
s
.Continuing in this way, we get l
i
m
m
,
n
→
∞
(
u
m
n
)
=
s.Remark 8.
Ifk
=
0 and r
≠
0, then (
C
,
k
,
r
) summability reduces to (
C
,
0
,
r
) summability. Again, for k
≠
0 and r
=
0, (
C
,
k
,
r
) summability reduces to (
C
,
k
,
0
) summability and, consequently, the following corollary is generated from the main result.Corollary 9 (see [5]).
Ifu
=
(
u
n
) is (
C
,
0
,
r
) or (
C
,
k
,
0
) summable to s and u
=
(
u
n
) is slowly oscillating, then l
i
m
n
→
∞
u
n
=
s.Theorem 10.
If(
u
m
n
) is (
C
,
k
,
r
) summable to s and V
m
n
(
11
)
(
Δ
u
) is slowly oscillating, then l
i
m
m
,
n
→
∞
u
m
n
=
s.Proof.
AsV
m
n
(
11
)
(
Δ
u
) is slowly oscillating, setting u
=
(
u
m
n
) in place of V
m
n
(
11
)
(
Δ
u
), σ
m
n
(
k
,
r
)
(
V
m
n
(
11
)
(
Δ
u
)
) is slowly oscillating by Lemma 6. Again, as V
m
n
(
11
)
(
Δ
u
) is (
C
,
k
,
r
) summable to s, by Theorem 3, we have(20)
lim
m
,
n
→
∞
⁡
σ
m
n
k
,
r
V
m
n
11
Δ
u
=
s
.By definition, (21)
σ
m
n
k
,
r
V
m
n
11
Δ
u
=
σ
m
n
1,1
V
m
n
11
Δ
u
σ
m
n
k
-
1
,
r
-
1
V
m
n
11
Δ
u
.From (20) and (21), we have that V
m
n
(
11
)
(
Δ
u
) is (
C
,
k
-
1
,
r
-
1
) summable to s. Again, by Lemma 6, since (
σ
m
n
(
k
-
1
,
r
-
1
)
(
V
m
n
(
11
)
(
Δ
u
)
)
) is slowly oscillating, we have l
i
m
m
,
n
→
∞
σ
m
n
(
k
-
1
,
r
-
1
)
(
V
m
n
(
11
)
(
Δ
u
)
)
=
s (by Theorem 3). Continuing in this way, we get l
i
m
m
,
n
→
∞
(
V
m
n
(
11
)
(
Δ
u
)
)
=
s.Remark 11.
Ifk
=
0 and r
≠
0, then (
C
,
k
,
r
) summability reduces to (
C
,
0
,
r
) summability. Again, for k
≠
0 and r
=
0, (
C
,
k
,
r
) summability reduces to (
C
,
k
,
0
) summability and consequently the following corollaries are generated from the main result.Corollary 12 (see [5]).
Ifu
=
(
u
n
) is (
C
,
0
,
r
) summable to s and V
m
n
(
1,0
)
(
Δ
u
) is slowly oscillating, then l
i
m
n
→
∞
u
n
=
s.Corollary 13 (see [5]).
Ifu
=
(
u
n
) is (
C
,
k
,
0
) summable to s and V
m
n
(
0,1
)
(
Δ
u
) is slowly oscillating, then l
i
m
n
→
∞
u
n
=
s.



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*Source: 2431010-2016-05-12.xml*