Extended Tauberian Theorem for the weighted mean Method of Summability
We prove a new Tauberian-like theorem. For a real sequence u = (u n ), on the basis of the weighted mean summability of its generator sequence (V (0) n,p (∆u)) and some other conditions, this theorem establishes the property of slow oscillation of the indicated sequence.
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| author | Çanak, І. Totur, Ü. Чанак, І. Тотур, Ю. |
| author_facet | Çanak, І. Totur, Ü. Чанак, І. Тотур, Ю. |
| author_sort | Çanak, І. |
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| description | We prove a new Tauberian-like theorem. For a real sequence u = (u n ), on the basis of the weighted mean summability of its generator sequence (V (0) n,p (∆u)) and some other conditions, this theorem establishes the property of slow oscillation of the indicated sequence. |
| first_indexed | 2026-03-24T02:24:10Z |
| format | Article |
| fulltext |
UDC 517.5
İ. Çanak (Ege Univ., Izmir, Turkey),
Ü. Totur (Adnan Menderes Univ., Aydin, Turkey)
AN EXTENDED TAUBERIAN THEOREM
FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY
УЗАГАЛЬНЕНА ТАУБЕРОВА ТЕОРЕМА
ДЛЯ МЕТОДУ ЗВАЖЕНОГО СЕРЕДНЬОГО ДЛЯ ЗНАХОДЖЕННЯ СУМ
We prove a new Tauberian-like theorem that establishes the slow oscillation of a real sequence u = (un) on the basis of
the weighted mean summability of its generator sequence (V
(0)
n,p (∆u)) and some conditions.
Доведено нову теорему тауберового типу, яка встановлює повiльнi коливання дiйсної послiдовностi u = (un) на
основi збiжностi її генеруючої послiдовностi (V
(0)
n,p (∆u)) у зважених середнiх та певних умов.
1. Introduction. Let u = (un) be a sequence of real numbers. Assume that p = (pn) is a sequence
of nonnegative numbers with p0 > 0 such that
Pn =
n∑
k=0
pk →∞ as n→∞.
The nth weighted mean of (un) is defined by
σ(1)n,p(u) =
1
Pn
n∑
k=0
pkuk.
A sequence (un) is said to be summable by the weighted mean method determined by the
sequence p, in short; (N, p) summable to a finite number s if
lim
n→∞
σ(1)n,p(u) = s. (1)
If the limit
lim
n→∞
un = s (2)
exists, then (1) also exists. However, the converse is not always true. Notice that (1) may imply
(2) under a certain condition which is called a Tauberian condition. Any theorem which states that
convergence of sequences follows from (N, p) summability method and some Tauberian condition is
said to be a Tauberian theorem.
If pn = 1 for all nonnegative n, then (N, p) summability method reduces to Cesàro summability
method.
The difference between un and its nth weighted mean σ
(1)
n,p(u) which is called the weighted
Kronecker identity is given by the identity
un − σ(1)n,p(u) = V (0)
n,p (∆u), (3)
where V (0)
n,p (∆u) :=
1
Pn
∑n
k=1
Pk−1∆uk.
c© İ. ÇANAK, Ü. TOTUR, 2013
928 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
AN EXTENDED TAUBERIAN THEOREM FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY 929
Since
Pn−1
pn
∆σ
(1)
n,p(u) = V
(0)
n,p (∆u), the Kronecker identity can be written as
un = V (0)
n,p (∆u) +
n∑
k=1
pk
Pk−1
V
(0)
k,p (∆u). (4)
Because of the identity (4), the sequence
(
V
(0)
n,p (∆u)
)
is called the generator sequence of (un).
For each integer m ≥ 0, we define σ(m)
n,p (u) and V (m)
n,p (∆u) by
σ(m)
n,p (u) =
1
Pn
n∑
k=0
pkσ
(m−1)
k,p (u), m ≥ 1,
un, m = 0,
and
V (m)
n,p (∆u) =
1
Pn
n∑
k=0
pkV
(m−1)
k,p (∆u), m ≥ 1,
V
(0)
n,p (∆u), m = 0,
respectively.
The weighted classical control modulo of (un) is denoted by ω(0)
n,p(u) =
Pn−1
pn
∆un and the
weighted general control modulo of integer order m ≥ 1 of (un) is defined in [1] by ω(m)
n,p (u) =
= ω
(m−1)
n,p (u)− σ(1)n,p(ωm−1(u)).
If pn = 1 for all nonnegative n, then the weighted classical and general control modulo reduce
to the classical and general control modulo, respectively. The classical and general control modulo
have been used as Tauberian conditions for various summability methods [2 – 5].
For a sequence u = (un), we define(
Pn−1
pn
∆
)
m
un =
(
Pn−1
pn
∆
)
m−1
(
Pn−1
pn
∆un
)
=
Pn−1
pn
∆
((
Pn−1
pn
∆
)
m−1
un
)
,
where
(
Pn−1
pn
∆
)
0
un = un, and
(
Pn−1
pn
∆
)
1
un =
Pn−1
pn
∆un.
Note that by the definition of the weighted general control modulo,
ω(1)
n,p(u) = ω(0)
n,p(u)− σ(1)n,p(ω
(0)(u)) =
Pn−1
pn
∆un − V (0)
n,p (∆u) =
Pn−1
pn
∆V (0)
n,p (∆u)
and
ω(2)
n,p(u) = ω(1)
n,p(u)− σ(1)n,p(ω
(1)(u)) =
=
Pn−1
pn
∆V (0)
n,p (∆u)− Pn−1
pn
∆V (1)
n,p (∆u) =
(
Pn−1
pn
∆
)
2
V (1)
n,p (∆u).
A sequence (un) is said to be slowly oscillating [6] if
lim
λ→1+
lim sup
n→∞
max
n+1≤k≤[λn]
|uk − un| = 0. (5)
Denote by S the class of slowly oscillating sequences.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
930 İ. ÇANAK, Ü. TOTUR
The weighted de la Vallée Poussin means of (un) are defined by
τ>n,[λn],p(u) =
1
P[λn] − Pn
[λn]∑
k=n+1
pkuk
for λ > 1 and sufficiently large n, and
τ<n,[λn],p(u) =
1
Pn − P[λn]
n∑
k=[λn]+1
pkuk
for 0 < λ < 1 and sufficiently large n.
For the definitions of O and o we refer to [7, p. 149].
A number of authors such as Hardy [7], Tietz [8], Tietz and Zeller [9], and Móricz and Rhoades
[10] obtained Tauberian theorems for (N, p) summability method. Tietz [8], Tietz and Zeller [9]
established Tauberian conditions controlling the oscillatory behavior of sequences for (N, p) summa-
bility method. Móricz and Rhoades [10] obtained necessary and sufficient conditions for (N, p)
summable (un) to be convergent. Hardy [7] proved that ω(0)
n,p(u) = O(1) is a Tauberian condition for
the weighted mean summability method. Recently, Çanak and Totur [1] have shown that under some
certain conditions imposed on the sequence p = (pn) the condition
ω(1)
n,p(u) ≥ −C (6)
for some positive constant C is a Tauberian condition for (N, p) summability method.
Instead of recovering convergence of (un) out of the existence of (1) and additional condition
imposed on the sequences (un) and p = (pn), we can obtain more general information on (un) by
replacing (N, p) summability of (un) by (N, p) summability of its generator sequence
(
V
(0)
n,p (∆u)
)
.
In this paper, we shall prove the following extended Tauberian theorem.
Theorem 1. Let
Pn−1
pn
= O(n). (7)
For a real sequence u = (un) let there exist a nonnegative sequenceM = (Mn) with
(∑n
k=1
Mk
k
)
∈
∈ S such that
ω(2)
n,p(u) ≥ −Mn, (8)
lim sup
n→∞
(
P[λn] − Pn
Pn
)
lim sup
n→∞
τ>n,[λn],p(M) = o(1), λ→ 1+, (9)
and
lim sup
n→∞
(
Pn − P[λn]
P[λn]
)
lim sup
n→∞
τ<n,[λn],p(M) = o(1), λ→ 1−. (10)
If
(
V
(0)
n,p (∆u)
)
is (N, p) summable to s, then u = (un) is slowly oscillating.
If we take pn = 1 for all n, we have Theorem 2.1 in [11].
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
AN EXTENDED TAUBERIAN THEOREM FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY 931
2. Auxiliary results. For the proof of Theorem 1 we shall need the following lemmas.
Lemma 1 [1]. Let v = (vn) be a sequence of real numbers.
(i) For λ > 1 and sufficiently large n,
vn − σ(1)n,p(v) =
P[λn]
P[λn] − Pn
(
σ
(1)
[λn],p(v)− σ(1)n,p(v)
)
− 1
P[λn] − Pn
[λn]∑
k=n+1
pk(vk − vn),
where [λn] denotes the integer part of λn.
(ii) For 0 < λ < 1 and sufficiently large n,
vn − σ(1)n,p(v) =
P[λn]
Pn − P[λn]
(
σ(1)n,p(v)− σ(1)[λn],p(v)
)
+
1
Pn − P[λn]
n∑
k=[λn]+1
pk(vn − vk).
The next lemma represents the difference between the weighted de la Vallée Poussin means and
the weighted means of sequence (vn).
Lemma 2. Let v = (vn) be a sequence of real numbers.
(i) For λ > 1 and sufficiently large n,
τ>n,[λn],p(v)− σ(1)n,p(v) =
P[λn]
P[λn] − Pn
(
σ
(1)
[λn],p(v)− σ(1)n,p(v)
)
.
(ii) For 0 < λ < 1 and sufficiently large n,
τ<n,[λn],p(v)− σ(1)n,p(v) =
P[λn]
Pn − P[λn]
(
σ(1)n,p(v)− σ(1)[λn],p(v)
)
.
Proof. (i) From the definition of the weighted de la Vallée Poussin means of (vn) we have, for
λ > 1,
τ>n,[λn],p(v) =
1
P[λn] − Pn
[λn]∑
j=n+1
pjvj =
1
P[λn] − Pn
[λn]∑
j=0
pjvj −
n∑
j=0
pjvj
=
=
1
P[λn] − Pn
(
P[λn]σ
(1)
[λn],p(v)− Pnσ(1)n,p(v)
)
=
=
P[λn]
P[λn] − Pn
σ
(1)
[λn],p(v)− Pn
P[λn] − Pn
σ(1)n,p(v) =
= σ(1)n,p(v) +
P[λn]
P[λn] − Pn
(
σ
(1)
[λn],p(v)− σ(1)n,p(v)
)
,
which proves Lemma 2 (i).
(ii) From the definition of the weighted de la Vallée Poussin means of (vn) we have, for 0 < λ <
< 1,
τ<n,[λn],p(v) =
1
Pn − P[λn]
n∑
j=[λn]+1
pjvj =
1
Pn − P[λn]
n∑
j=0
pjvj −
[λn]∑
j=0
pjvj
=
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
932 İ. ÇANAK, Ü. TOTUR
=
1
Pn − P[λn]
(
Pnσ
(1)
n,p(v)− P[λn]σ
(1)
[λn],p(v)
)
=
=
Pn
Pn − P[λn]
σ(1)n,p(v)−
P[λn]
Pn − P[λn]
σ
(1)
[λn],p(v) =
= σ(1)n,p(v) +
P[λn]
Pn − P[λn]
(
σ(1)n,p(v)− σ(1)[λn],p(v)
)
,
which proves Lemma 2 (ii).
The next lemma states that if (vn) is (N, p) summable to s, then the sequence of the weighted
de la Vallée Poussin means of (vn) converges to s.
Lemma 3. If (vn) is (N, p) summable to s, then
(i) limn→∞ τ
>
n,[λn],p(v) = s,
(ii) limn→∞ τ
<
n,[λn],p(v) = s.
Proof. (i) Since
lim
n→∞
P[λn]
P[λn] − Pn
(
σ
(1)
[λn],p(v)− σ(1)n,p(v)
)
= 0,
we have
lim
n→∞
τ>n,[λn],p(v) = lim
n→∞
σ(1)n,p(v) = s.
Proof of (ii) is similar.
Lemma 4. For a real sequence v = (vn) let there exist a nonnegative sequence M = (Mn)
such that
Pn−1
pn
∆vn ≥ −Mn.
Then
(i) −
(
τ>n,[λn],p(v)− vn
)
≤
P[λn] − Pn
Pn
τ>n,[λn],p(M),
(ii) vn − τ<n,[λn],p(v) ≥ −
Pn − P[λn]
P[λn]
τ<n,[λn],p(M).
Proof. (i) Since ∆vn ≥ −
pn
Pn−1
Mn, we have
−
k∑
j=n+1
∆vj = −(vk − vn) ≤
k∑
j=n+1
pj
Pj−1
Mj . (11)
Multiplying the inequality (11) by pk, summing the resulting inequality from k = n+ 1 to [λn] and
then dividing it by P[λn] − Pn, we get
−(τ>n,[λn],p(v)− vn) = − 1
P[λn] − Pn
[λn]∑
k=n+1
(pkvk − pkvn) ≤
≤ 1
P[λn] − Pn
[λn]∑
k=n+1
pk
k∑
j=n+1
pj
Pj−1
Mj ≤
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
AN EXTENDED TAUBERIAN THEOREM FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY 933
≤
P[λn] − Pn
Pn
1
P[λn] − Pn
[λn]∑
j=n+1
pjMj
=
=
P[λn] − Pn
Pn
τ>n,[λn],p(M).
(ii) Since ∆vn ≥ −
pn
Pn−1
Mn, we have
n∑
j=k+1
∆vj = vn − vk ≥ −
n∑
j=k+1
pj
Pj−1
Mj . (12)
Multiplying the inequality (12) by pk, summing the resulting inequality from k = [λn] + 1 to n and
then dividing it by Pn − P[λn], we get
vn − τ<n,[λn],p(v) =
1
Pn − P[λn]
n∑
k=[λn]+1
(pkvn − pkvk) ≥
≥ − 1
Pn − P[λn]
n∑
k=[λn]+1
pk
n∑
j=k+1
pj
Pj−1
Mj =
= −
Pn − P[λn]
P[λn]
1
Pn − P[λn]
n∑
j=[λn]+1
pjMj
= −
Pn − P[λn]
P[λn]
τ<n,[λn],p(M).
3. Proof of Theorem 1. Since (V
(0)
n,p (∆u)) is (N, p) summable to s, (V
(1)
n,p (∆u)) is convergent
to s, and then
(
V
(2)
n,p (∆u)
)
is convergent to s. Applying the identity (3) to (V
(1)
n,p (∆u)), we obtain
that
V (1)
n,p (∆u)− V (2)
n,p (∆u) = V (0)
n,p (∆V (1)(∆u)) =
Pn−1
pn
∆V (2)
n,p (∆u).
Therefore, we have
Pn−1
pn
∆V
(2)
n,p (∆u) = o(1). Under the assumptions of Theorem 1 we now prove
that
Pn−1
pn
∆V
(1)
n,p (∆u) = o(1). Let vn :=
Pn−1
pn
∆V
(1)
n,p (∆u). Then σ
(1)
n,p(v) =
Pn−1
pn
∆V
(2)
n,p (∆u).
Applying Lemma 1 (i) to vn, and using Lemma 2 (i) and Lemma 4 (i), we have
vn − σ(1)n,p(v) ≤ τ>n,[λn],p(v)− σ(1)n,p(v) +
P[λn] − Pn
Pn
τ>n,[λn],p(M). (13)
Taking lim sup of both sides of (13), we obtain
lim sup
n→∞
(
vn − σ(1)
n,p(v)
)
≤ lim sup
n→∞
(
τ>n,[λn],p(v)− σ(1)
n,p(v)
)
+ lim sup
n→∞
(
P[λn] − Pn
Pn
τ>n,[λn],p(v)
)
. (14)
Noticing that the first term on the right-hand side of (14) vanishes by Lemma 3 (i), we deduce that
lim sup
n→∞
(vn − σ(1)n,p(v)) ≤ lim sup
n→∞
(
P[λn] − Pn
Pn
τ>n,[λn],p(M)
)
≤
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
934 İ. ÇANAK, Ü. TOTUR
≤ lim sup
n→∞
(
P[λn] − Pn
Pn
)
lim sup
n→∞
τ>n,[λn],p(M).
Hence, it follows by (9) that
lim sup
n→∞
(vn − σ(1)n,p(v)) ≤ 0. (15)
Similarly, applying Lemma 1 (ii) to vn, and using Lemma 2 (ii) and Lemma 4 (ii), we have
vn − σ(1)n,p(v) ≥ τ<n,[λn],p(v)− σ(1)n,p(v)−
Pn − P[λn]
P[λn]
τ<n,[λn],p(M). (16)
Taking the lim inf of both sides of (16), we obtain
lim inf
n→∞
(
vn − σ(1)
n,p(v)
)
≥ lim inf
n→∞
(
τ<n,[λn],p(v)− σ(1)
n,p(v)
)
+ lim inf
n→∞
(
−
Pn − P[λn]
P[λn]
τ<n,[λn],p(M)
)
. (17)
Noticing that the first term on the right-hand side of (17) vanishes by Lemma 3 (ii), we deduce
that
lim inf
n→∞
(vn − σ(1)n,p(v)) ≥ lim inf
n→∞
(
−
Pn − P[λn]
P[λn]
τ<n,[λn],p(M)
)
≥
≥ − lim sup
n→∞
(
Pn − P[λn]
P[λn]
)
lim sup
n→∞
τ<n,[λn],p(M).
Hence, it follows by (10) that
lim inf
n→∞
(vn − σ(1)n,p(v)) ≥ 0. (18)
Combining (15) and (18) yields
vn =
Pn−1
pn
∆V (1)
n,p (∆u) = o(1). (19)
Applying the identity (3) to (V
(0)
n,p (∆u)), we obtain that
V (0)
n,p (∆u)− V (1)
n,p (∆u) = V (0)
n,p (∆V (0)(∆u)) =
Pn−1
pn
∆V (1)
n,p (∆u).
Since (V
(0)
n,p (∆u)) is (N, p) summable to s, it follows from (19) that (V
(0)
n,p (∆u)) converges to s.
From the representation un = V
(0)
n,p (∆u) +
∑n
k=1
pkV
(0)
k (∆u)
Pk−1
and the condition (7), it follows that
(un) is slowly oscillating.
As a corollary we have the following classical Tauberian theorem for (N, p) summability method.
Corollary 1. Let (pn) satisfy the condition (7). For a real sequence u = (un) let there exist a
nonnegative sequence M = (Mn) with
(∑n
k=1
Mk
k
)
∈ S such that (8), (9) and (10) are satisfied.
If (un) is (N, p) summable to s, then (un) is convergent to s.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
AN EXTENDED TAUBERIAN THEOREM FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY 935
Proof. Assume that (un) is (N, p) summable to s. It follows by (3) that (V
(0)
n,p (∆u)) is (N, p)
summable to 0. By Theorem 1 (un) is slowly oscillating. Finally, (un) is convergent to s by Theorem
6 in [1].
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Received 16.05.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
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| id | umjimathkievua-article-2478 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:24:10Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ce/78bbb2452cdc8c565a7ae84d69b5a3ce.pdf |
| spelling | umjimathkievua-article-24782020-03-18T19:16:28Z Extended Tauberian Theorem for the weighted mean Method of Summability Узагальнена Тауберова теорема для методу зваженого середнього для знаходження сум Çanak, І. Totur, Ü. Чанак, І. Тотур, Ю. We prove a new Tauberian-like theorem. For a real sequence u = (u n ), on the basis of the weighted mean summability of its generator sequence (V (0) n,p (∆u)) and some other conditions, this theorem establishes the property of slow oscillation of the indicated sequence. Доведено нову теорему тауберового типу, яка встановлює повільні коливання дійсної послідовності u = (u n )) на основі збіжності її генеруючої послідовності (V (0) n,p (∆u)) у зважених середніх та певних умов. Institute of Mathematics, NAS of Ukraine 2013-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2478 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 7 (2013); 928–935 Український математичний журнал; Том 65 № 7 (2013); 928–935 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2478/1722 https://umj.imath.kiev.ua/index.php/umj/article/view/2478/1723 Copyright (c) 2013 Çanak І.; Totur Ü. |
| spellingShingle | Çanak, І. Totur, Ü. Чанак, І. Тотур, Ю. Extended Tauberian Theorem for the weighted mean Method of Summability |
| title | Extended Tauberian Theorem for the weighted mean Method of Summability |
| title_alt | Узагальнена Тауберова теорема для методу зваженого середнього для знаходження сум |
| title_full | Extended Tauberian Theorem for the weighted mean Method of Summability |
| title_fullStr | Extended Tauberian Theorem for the weighted mean Method of Summability |
| title_full_unstemmed | Extended Tauberian Theorem for the weighted mean Method of Summability |
| title_short | Extended Tauberian Theorem for the weighted mean Method of Summability |
| title_sort | extended tauberian theorem for the weighted mean method of summability |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2478 |
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