TAUBERIAN CONDITIONS FOR A GENERAL
LIMITABLE METHOD
İBRAHİM C.ANAK AND ÜMİT TOTUR
Received 16 July 2006; Revised 21 August 2006; Accepted 21 August 2006
Let (un ) be a sequence of real numbers, L an additive limitable method with some property, and ᐁ and ᐂ different spaces of sequences related to each other. We prove that if the
classical control modulo of the oscillatory behavior of (un ) in ᐁ is a Tauberian condition
for L, then the general control modulo of the oscillatory behavior of integer order m of
(un ) in ᐁ or ᐂ is also a Tauberian condition for L.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
In this paper, un = O(1) and un = o(1) denote O(1) as n → ∞ and o(1) as n → ∞, respectively. Let ᏺ, Ꮾ, ᏿, and ᏹ denote the space of sequences converging to 0, bounded, slowly
oscillating, and moderately oscillating, respectively.
The classical control modulo of the oscillatory behavior of (un ) is denoted by ωn(0) (u) =
nΔun and the general control modulo of the oscillatory behavior of order m of (un ) is
defined by ωn(m) (u) = ωn(m−1) (u) − σn(1) (ω(m−1) (u)), where
⎧
⎪
⎨un − un−1 ,
n ≥ 1,
Δun = ⎪
⎩u0 ,
σn(1) (u) =
n = 0,
n
1 uk .
n + 1 k =0
(1.1)
Tauber [10] proved that if (un ) is Abel limitable and
ωn(0) (u) ∈ ᏺ,
(1.2)
then (un ) is convergent. The condition (1.2) on the sequence (un ) is called a Tauberian condition for Abel limitable method and the resulting theorem is called a Tauberian
theorem.
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 82342, Pages 1–6
DOI 10.1155/IJMMS/2006/82342
2
Tauberian conditions for a general limitable method
Tauber [10] further proved that the condition
σn(1) ω(0) (u)
∈ᏺ
(1.3)
is also a Tauberian condition. It was shown by Littlewood [6] that the condition (1.2)
could be replaced by
ωn(0) (u) ∈ Ꮾ.
(1.4)
Hardy and Littlewood [5] improved Littlewood’s theorem replacing (1.4) by onesided
boundedness of (ωn(0) (u)).
Stanojević [9] reformulated the definition of slow oscillation given by Schmidt [8] in a
more suitable form and then proved that the conditions (1.2) and (1.3) could be replaced
by
ωn(0) (u) ∈ ᏿,
(1)
σn
ω(0) (u)
∈ ᏿,
(1.5)
(1.6)
respectively.
A generalization of slow oscillation, moderate oscillation, was introduced by Stanojević and it was proved by Dik [4] that (1.5) could be replaced by
ωn(0) (u) ∈ ᏹ,
(1.7)
and (1.6) could not be replaced by
σn(1) ω(0) (u)
∈ ᏹ.
(1.8)
Recently, C.anak and Totur [3] have shown that for any nonnegative integer m ≥ 1,
ωn(m) (u) ∈ ᏹ
(1.9)
is a Tauberian condition for Abel limitable method.
Meyer-König and Tietz [7] proved that if (1.2) is a Tauberian conditions for an additive
and regular limitability method, then (1.3) is a Tauberian condition for L. C.anak et al. [1]
extended and generalized Meyer-König and Tietz’s [7] result and obtained the following
theorems for an additive and (C,1) regular method L.
Theorem 1.1. If (ωn(0) (u)) ∈ ᏿ is a Tauberian condition for an additive and (C,1) regular
limitable method L, then (ωn(1) (u)) ∈ ᏿ is a Tauberian condition for L.
Theorem 1.2. If (ωn(0) (u)) ∈ Ꮾ is a Tauberian condition for an additive and (C,1) regular
limitable method L, then (ωn(1) (u)) ∈ Ꮾ is a Tauberian condition for L.
Let ᐁ and ᐂ be distinct spaces of sequences related to each other. In this paper, we
prove that if the classical control modulo of the oscillatory behavior of (un ) in ᐁ is a
Tauberian condition for an additive and (C,1) limitable method L, then the general control modulo of the oscillatory behavior of integer order m of (un ) in ᐁ or ᐂ is also a
Tauberian condition for L.
İ. C.anak and Ü. Totur 3
2. Notations and definitions
Throughout this paper, let u = (un ) be a sequence of real numbers. For each integer m ≥ 0
and for all nonnegative integers n denote σn(m) (u) by
⎧
⎪
⎪
⎨ 1
n
σn(m) (u) = ⎪ n + 1 k=0
⎪
⎩
un ,
σk(m−1) (u) = u0 +
n
Vk(m−1) (Δu)
k
k =1
,
m ≥ 1,
(2.1)
m = 0,
where
⎧
(m−1)
(1)
⎪
⎪
(Δu) ,
⎪
⎨σn V
n
Vn(m) (Δu) = ⎪ 1 ⎪
kΔuk ,
⎪
⎩n+1
m ≥ 1,
m = 0.
(2.2)
k =0
The identity
un − σn(1) (u) = Vn(0) (Δu)
(2.3)
is well known and will be extensively used. We define inductively for each integer m ≥ 1
and for all nonnegative integers n,
(nΔ)m un = nΔ (nΔ)m−1 un ,
where (nΔ)0 un = un .
(2.4)
It is proved in [2] that for each integer m ≥ 1,
ωn(m) (u) = (nΔ)m Vn(m−1) (Δu).
(2.5)
Definition
2.1. A sequence u = (un ) is Abel limitable to s if the limit limx→1− (1 −
∞
x) n=0 un xn = s.
Definition 2.2. A sequence u = (un ) is L limitable to s if L − limn un = s.
A limitation method L is called additive if L − limn un = s and L − limn vn = t imply
that L − limn (un + vn ) = s + t. A limitation method L is called regular if the L − limit of
every convergent sequence is equal to its limit. L is called (C,1) regular if L − limn un = s
implies L − limn σn(1) (u) = s. It is clear that every regular limitable method is (C,1) regular.
Definition 2.3. A sequence u = (un ) is one-sidedly bounded if for some C ≥ 0 and for
each nonnegative integer n, un ≥ −C.
Definition 2.4. A sequence u = (un ) is slowly oscillating [9] if
k
= 0,
Δu
lim+ lim max j
λ→1 n n+1≤k≤[λn] j =n+1
where [λn] denotes the integer part of λn.
(2.6)
4
Tauberian conditions for a general limitable method
A sequence u = (un ) ∈ ᏿ if and only if (Vn(0) (Δu)) ∈ ᏿ and (Vn(0) (Δu)) ∈ Ꮾ (see [4]).
The next definition is a generalization of slow oscillation.
Definition 2.5. A sequence u = (un ) is moderately oscillating [9] if for λ > 1,
k
< ∞.
lim max Δu
j
n n+1≤k≤[λn] (2.7)
j =n+1
A sequence (un ) ∈ ᏹ if and only if (Vn(0) (Δu)) ∈ Ꮾ (see [4]).
3. Results and proofs
Theorem 3.1. If (ωn(0) (u)) ∈ ᏹ is a Tauberian condition for L, then for any integer m ≥ 1,
(ωn(m) (u)) ∈ ᏹ is also a Tauberian condition for L.
Proof. Assume that (ωn(0) (u)) ∈ ᏹ is a Tauberian condition for L. Let L − limn un = s.
Since L is (C,1) regular, it follows by (2.3) that L − limn Vn(0) (Δu) = 0. It is obvious that
L − limn un = s implies L − limn (nΔ)m−1 Vn(m−1) (Δu) = 0. Since
ωn(m) (u) = nΔ (nΔ)m−1 Vn(m−1) (Δu)
∈ ᏹ,
(3.1)
by assumption, we have
(nΔ)m−1 Vn(m−1) (Δu) = o(1).
(3.2)
By the same reasoning, we deduce that
(nΔ)m−1 Vn(m−1) (Δu) = nΔ (nΔ)m−2 Vn(m−1) (Δu) = o(1)
(3.3)
and L − limn (nΔ)m−2 Vn(m−1) (Δu) = 0. Again by assumption, we have
(nΔ)m−2 Vn(m−1) (Δu) = o(1).
(3.4)
(nΔ)m−1 Vn(m−1) (Δu) = (nΔ)m−2 Vn(m−2) (Δu) − (nΔ)m−2 Vn(m−1) (Δu),
(3.5)
From the identity
(3.2), and (3.4), we have
(nΔ)m−2 Vn(m−2) (Δu) = o(1).
(3.6)
Continuing in this vein, we have
nΔVn(1) (Δu) = o(1).
(3.7)
Since L − limn Vn(1) (Δu) = 0, it follows by (3.7) that
Vn(1) (Δu) = o(1).
(3.8)
İ. C.anak and Ü. Totur 5
From (3.7) and (3.8), we have Vn(0) (Δu) = o(1). L − limn σn(1) (u) = s and Vn(0) (Δu) =
nΔσn(1) (u) = o(1) imply that limn σn(1) (u) = s. Hence, by (2.3), (un ) converges to s.
Theorem 3.2. If (ωn(0) (u)) ∈ Ꮾ is a Tauberian condition for L, then for any integer m ≥ 1,
(ωn(m) (u)) ∈ Ꮾ is also a Tauberian condition for L.
Proof. Assume that ωn(0) (u) = O(1) is a Tauberian condition for L. Let L − limn un = s.
Since L − limn (nΔ)m−1 Vn(m−1) (Δu) = 0 and ωn(m) (u) = nΔ((nΔ)m−1 Vn(m−1) (Δu)) = O(1),
(nΔ)m−1 Vn(m−1) (Δu) = o(1) by assumption. The rest of the proof is as in the proof of
Theorem 3.1.
Theorem 3.3. If for some C ≥ 0, ωn(0) (u) ≥ −C is a Tauberian condition for L, then for any
integer m ≥ 1, ωn(m) (u) ≥ −C is also a Tauberian condition for L.
Proof. Assume that ωn(0) (u) ≥ −C for some C ≥ 0 is a Tauberian condition for L. Let
L − limn un = s. Since L − limn (nΔ)m−1 Vn(m−1) (Δu) = 0 and ωn(m) (u) = nΔ((nΔ)m−1 Vn(m−1)
(Δu)) ≥ −C, (nΔ)m−1 Vn(m−1) (Δu) = o(1) by assumption. The rest of the proof is as in the
proof of Theorem 3.1.
We now prove that if (ωn(0) (u)) ∈ ᏹ (or ∈ Ꮾ) is a Tauberian condition for L, then for
any integer m ≥ 1, (ωn(m) (u)) ∈ Ꮾ (or ∈ ᏹ) is a Tauberian condition for L, respectively.
Theorem 3.4. If (ωn(0) (u)) ∈ ᏹ is a Tauberian condition for L, then for any integer m ≥ 1,
(ωn(m) (u)) ∈ Ꮾ is also a Tauberian condition for L.
Proof. It is sufficient to note that ωn(m) (u) = (nΔ)m Vn(m−1) (Δu) = Vn(0) (Δω(m−1) (u))
(m−1)
= O(1) implies (ωn
(u)) ∈ ᏹ. Proof now follows from Theorem 3.1.
Theorem 3.5. If (ωn(0) (u)) ∈ Ꮾ is a Tauberian condition for L, then for any integer m ≥ 1,
(ωn(m) (u)) ∈ ᏹ is also a Tauberian condition for L.
Proof. It is sufficient to note that (ωn(m) (u)) ∈ ᏹ implies Vn(0) (Δω(m) (u)) = ωn(m+1) (u) =
O(1). Proof now follows from Theorem 3.4.
Remark 3.6. Because of the inclusion ᏺ ⊂ ᏿ ⊂ ᏹ, the condition “belonging to ᏹ” can
be replaced by “belonging to ᏿” or “belonging to ᏺ.”
In Theorems 3.1, 3.2, and 3.3, taking m = 1 and replacing ᏹ by ᏿, we have [1, Theorems 4.1, 4.2, and 4.4] by C.anak et al.
Acknowledgment
This research was supported by Adnan Menderes University under Grant FEF-06011.
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İbrahım C.anak: Department of Mathematics, Adnan Menderes University, 09010 Aydin, Turkey
E-mail address: icanak@adu.edu.tr
Ümıt Totur: Department of Mathematics, Adnan Menderes University, 09010 Aydin, Turkey
E-mail address: utotur@adu.edu.tr