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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 16 (2015),
No. 2, pp. 1243–1252 DOI: 10.18514/MMN.2015.1288
ON TAUBERIAN REMAINDER THEOREMS FOR CESÀROSUMMABILITY METHOD OF
NONINTEGER ORDER
Ü. TOTUR AND M. A. OKUR
Received 18 June, 2014
Abstract. In this paper, we prove some Tauberian remainder
theorems for Cesàro summabilitymethod of noninteger order ˛ >
�1.
2010 Mathematics Subject Classification: 40E05; 40G05
Keywords: Tauberian remainder theorem, �-bounded series, .C;˛/
summability, Cesàro means
1. INTRODUCTION
Let A˛n be defined by the generating function .1�x/�˛�1
DP1nD0A
˛nxn,
.jxj< 1/, where ˛ > �1: For a real sequence u D .un/, the
Cesàro means of thesequence .un/ of noninteger order ˛ are defined
by
� .˛/n .u/D1
A˛n
nXjD0
A˛�1n�juj :
We say that a sequence .un/ is .C;˛/ summable to a finite number
s, where ˛ > �1if
limn!1
� .˛/n .u/D s; (1.1)
and we write un! s .C;˛/. We denote the backward difference of
.un/, by �un Dun�un�1, with �u0 D u0: We define �n.u/D n�un .nD
0;1;2; ::/ and indicate�.˛/n .u/ as .C;˛/ mean of .�n.u//.
Note that if taking ˛ D k where k is a nonnegative integer, then
we obtain the.C;k/ summability method and for ˛D 0, the .C;0/
summability is ordinary conver-gence.
The .C;˛/ summability method is regular, more generally, if a
sequence .un/ is.C;˛/ summable to s, where ˛ > �1 and ˇ � ˛ for
˛;ˇ, then .un/ is also .C;ˇ/summable to s. However, the converse is
not always true. The converse of thisstatement is valid under some
conditions called Tauberian conditions. Any theoremwhich states
that convergence of a sequence follows from a summability method
andsome Tauberian condition(s) is said to be a Tauberian theorem.
Recently, a number
c 2015 Miskolc University Press
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1244 Ü. TOTUR AND M. A. OKUR
of authors such as Estrada and Vindas [4, 5], Natarajan [15],
Çanak et al. [2], Erdemand Çanak [3], Çanak and Erdem [1] have
investigated Tauberian theorems for severalsummability methods.
For a sequence .un/ and for each integer m� 1,
.n�/mun D n�..n�/m�1un/; (1.2)
where .n�/0un D un and .n�/1un D n�un.For ˛ > �1, the
identity
� .˛/n .u/D n��.˛/n .u/ (1.3)
was proved by Kogbetliantz [9]. Note that � .0/n .u/D �n.u/.The
identity
� .˛/n .u/��.˛C1/n .u/D
1
˛C1� .˛C1/n .u/ (1.4)
is used in the various steps of proofs (see [10]).Çanak et al.
[2] represent the identity
n��˛C1n D .˛C1/.�˛n � �
˛C1n /; (1.5)
for ˛ > �1.Erdem and Çanak [3] prove that for ˛ > �1 and
any integer k � 1
.n�/k�.˛Ck/n .u/D
kXjD1
.�1/jC1A.j /
k.˛/n�� .˛Cj /n .u/; (1.6)
where A.j /k.˛/D a
.j�1/
k.˛/Ca
.j /
k.˛/, a.0/
k.˛/D 0 and
a.j /
k.˛/D˘kiDjC1.˛Ci/
Xj C1� t1; t2; : : : ; tj�1 � k
r < s) tr � ts
.˛Ct1/.˛Ct2/ : : : .˛Ctj�1/:
2. TAUBERIAN REMAINDER THEOREMS
Let �D .�n/ be a nondecreasing sequence of positive numbers such
that �n!1.A sequence .un/ is called bounded with the rapidity .�n/
(in short �-bounded) if
�n.un� s/DO.1/;
with limn!1
un D s. Let
m� D fuD .un/j limn!1
un D s and �n.un� s/DO.1/g: (2.1)
A sequence .un/ is called �-bounded by the .C;˛/ method of
summability if
�n.�.˛/n .u/� s/DO.1/; (2.2)
with limn!1
� .˛/n .u/D s. Shortly, we write u 2 ..C;˛/;m�/:
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ON TAUBERIAN REMAINDER THEOREMS FOR CESÀRO SUMMABILITY 1245
G. Kangro [7] introduced the concepts of Tauberian remainder
theorems usingsummability with given rapidity �. G. Kangro [8] and
Tammeraid [16, 17] provedsome Tauberian remainder theorems for
several summability method, such as Riesz,Cesàro, Hölder and
Euler-Knopp methods. Recently, various authors have represen-ted
some Tauberian remainder theorems (see [12, 13]). In [18],
Tammeraid provedsome Tauberian remainder theorems in which the
.C;˛/ summability method is used.Tauberian remainder theorems have
also been studied by many authors via the Four-ier integral method.
[6, 11]
Meronen and Tammeraid [14] proved the following Tauberian
remainder theor-ems:
Theorem 1. Let the condition
�n�.1/n .u/DO.1/
be satisfied. If u 2 ..C;1/;m�/; then u 2m�:
Theorem 2. Let the conditions
�n�n.u/DO.1/;
�nn��.1/n .u/DO.1/
be satisfied. If u 2 ..C;1/;m�/, then u 2m�:
The main purpose of this paper is to prove several Tauberian
remainder theoremsfor Cesàro summability method of noninteger
order ˛ > �1. Our main theoremsimprove Theorem 1 and Theorem 2
given by Meronen and Tammeraid [14].
3. A LEMMA
We require the following lemma to be used in the proofs of main
theorems.
Lemma 1. Let ˛ > �1. For any integer k � 2,
.n�/k�.˛Ck/n .u/D B1;1�˛�
.˛/n .u/�B1;1�
.˛/n .u/CB1;1�
.˛C1/n .u/
C
kXjD2
�Bj;j�1�
.˛Cj�2/n .u/�2Bj;j� 1
2� .˛Cj�1/n .u/CBj;j�
.˛Cj /n .u/
�;
where Bm;l D .˛Cm/.˛C l/.�1/mC1A.m/
k.˛/ and A.j /
k.˛/D a
.j�1/
k.˛/Ca
.j /
k.˛/,
a.0/
k.˛/D 0 and
a.j /
k.˛/D˘kiDjC1.˛Ci/
Xj C1� t1; t2; : : : ; tj�1 � k
r < s) tr � ts
.˛Ct1/.˛Ct2/ : : : .˛Ctj�1/:
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1246 Ü. TOTUR AND M. A. OKUR
Proof. From identity (1.6), we have
.n�/k�.˛Ck/n .u/D
kXjD1
.�1/jC1A.j /
k.˛/n�� .˛Cj /n .u/
D A.1/
k.˛/n�� .˛C1/n .u/C
kXjD2
.�1/jC1A.j /
k.˛/n�� .˛Cj /n .u/:
It follows from identity (1.5) that
.n�/k�.˛Ck/n .u/D .˛C1/A
.1/
k.˛/.� .˛/n .u/� �
.˛C1/n .u//
C
kXjD2
.˛Cj /.�1/jC1A.j /
k.˛/.� .˛Cj�1/n .u/� �
.˛Cj /n .u//:
By identity (1.4), we can write the above equation as
.n�/k�.˛Ck/n .u/D .˛C1/A
.1/
k.˛/� .˛/n .u/� .˛C1/
2A.1/
k.˛/
�� .˛/n .u/��
.˛C1/n .u/
�C
kXjD2
.˛Cj /.�1/jC1A.j /
k.˛/
�.˛Cj �1/.� .˛Cj�2/n .u/��
.˛Cj�1/n .u//
�.˛Cj /.� .˛Cj�1/n .u/��.˛Cj /n .u//
�:
Therefore,
.n�/k�.˛Ck/n .u/
D .˛C1/A.1/
k.˛/� .˛/n .u/� .˛C1/
2A.1/
k.˛/� .˛/n .u/C .˛C1/
2A.1/
k.˛/� .˛C1/n .u/
C
kXjD2
�.˛Cj /.˛Cj �1/.�1/jC1A
.j /
k.˛/� .˛Cj�2/n .u/
� .˛Cj /.˛Cj �1/.�1/jC1A.j /
k.˛/� .˛Cj�1/n .u/
� .˛Cj /2.�1/jC1A.j /
k.˛/� .˛Cj�1/n .u/C .˛Cj /
2.�1/jC1A.j /
k.˛/� .˛Cj /n .u/
�:
Hence, we have
.n�/k�.˛Ck/n .u/
D .˛C1/A.1/
k.˛/� .˛/n .u/� .˛C1/
2A.1/
k.˛/� .˛/n .u/C .˛C1/
2A.1/
k.˛/� .˛C1/n .u/
C
kXjD2
�.˛Cj /.˛Cj �1/.�1/jC1A
.j /
k.˛/� .˛Cj�2/n .u/
� .˛Cj /.2˛C2j �1/.�1/jC1A.j /
k.˛/� .˛Cj�1/n .u/
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ON TAUBERIAN REMAINDER THEOREMS FOR CESÀRO SUMMABILITY 1247
C .˛Cj /2.�1/jC1A.j /
k.˛/� .˛Cj /n .u/
�:
Taking .˛Cm/.˛C l/.�1/mC1A.m/k.˛/D Bm;l , we obtain
.n�/k�.˛Ck/n .u/D B1;1�˛�
.˛/n .u/�B1;1�
.˛/n .u/CB1;1�
.˛C1/n .u/
C
kXjD2
�Bj;j�1�
.˛Cj�2/n .u/�2Bj;j� 1
2� .˛Cj�1/n .u/CBj;j�
.˛Cj /n .u/
�:
Thus, we conclude that Lemma 1 is true for each integer k � 2.
�
4. MAIN RESULTS
In the main theorems, we prove some Tauberian remainder theorems
to recover �-bounded by the .C;˛/ summability of a sequence out of
�-bounded by the .C;˛Cj /summability for j D 1;2 and any integer j
D k, and some suitable conditions. Inspecial cases of main
theorems, we obtain some classical type Tauberian remaindertheorems
for the .C;1/ summability method.
Theorem 3. Let the conditions
�nn��.˛C1/n .u/DO.1/; (4.1)
and�n�
.˛/n .u/DO.1/ (4.2)
be satisfied for ˛ > �1. If u 2 ..C;˛C1/;m�/, then u 2
..C;˛/;m�/.
Proof. From identity (1.5), we have
�nn��.˛C1/n .u/D �n.˛C1/.�
.˛/n .u/� �
.˛C1/n .u//
D �n.˛C1/�.˛/n .u/��n.˛C1/�
.˛C1/n .u/:
From identity (1.4), we obtain
�nn��.˛C1/n .u/D �n.˛C1/�
.˛/n .u/��n.˛C1/
2.� .˛/n .u/��.˛C1/n .u//:
Rewritten the above equation, we have
�n.˛C1/2.� .˛/n .u/� s/D �n.˛C1/
2.� .˛C1/n .u/� s/
C�n.˛C1/�.˛/n .u/��nn��
.˛C1/n .u/:
Using (4.1) and (4.2), we get
�n.˛C1/2.� .˛/n .u/� s/DO.1/CO.1/CO.1/DO.1/:
Therefore, �n.�.˛/n .u/� s/DO.1/. That means u 2 ..C;˛/;m�/.
�
Notice that taking ˛ D 0, we obtain Theorem 2.
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1248 Ü. TOTUR AND M. A. OKUR
Proposition 1. Let the conditions
�n.n�/2�.˛C2/n .u/DO.1/; (4.3)
�n�.˛/n .u/DO.1/; (4.4)
and�n.�
.˛C2/n .u/� s/DO.1/ (4.5)
be satisfied for ˛ > �1. If u 2 ..C;˛C1/;m�/, then u 2
..C;˛/;m�/.
Proof. Taking k D 2 in Lemma 1, we have
�n.n�/2�.˛C2/n .u/D �n.˛C2/.˛C1/
�� .˛/n .u/� �
.˛C1/n .u/
���n.˛C2/
2�� .˛C1/n .u/� �
.˛C2/n .u/
�D �n.˛C2/.˛C1/�
.˛/n .u/��n.˛C2/.˛C1/�
.˛C1/n .u/
��n.˛C2/2� .˛C1/n .u/C�n.˛C2/
2� .˛C2/n .u/:
From identity (1.4), we get
�n.n�/2�.˛C2/n .u/
D �n.˛C2/.˛C1/�.˛/n .u/��n.˛C2/.˛C1/
�.˛C1/.� .˛/n .u/ � �
.˛C1/n .u//
���n.˛C2/
2�.˛C1/.� .˛/n .u/��
.˛C1/n .u//
�C�n.˛C2/
2�.˛C2/.� .˛C1/n .u/��
.˛C2/n .u//
�D �n.˛C2/.˛C1/�
.˛/n .u/��n.˛C1/
2.˛C2/� .˛/n .u/
C�n.˛C1/2.˛C2/� .˛C1/n .u/��n.˛C2/
2.˛C1/� .˛/n .u/
C�n.˛C2/2.˛C1/� .˛C1/n .u/C�n.˛C2/
3� .˛C1/n .u/��n.˛C2/3� .˛C2/n .u/
D �n.˛C2/.˛C1/�.˛/n .u/��n.˛C1/.˛C2/.2˛C3/˛
.˛/n .u/
C�n..˛C2/3C .˛C2/2.˛C1/C .˛C1/2.˛C2//� .˛C1/n .u/
��n.˛C2/3� .˛C2/n .u/:
Rewritten the above equation, we have
�n.˛C1/.˛C2/.2˛C3/.�.˛/n .u/� s/
D��nn��.˛C2/n .u/C�n.˛C2/.˛C1/�
.˛/n .u/C�n..˛C2/
3C .˛C2/2.˛C1/
C .˛C1/2.˛C2/� s/� .˛C1/n .u/��n..˛C2/3� s/� .˛C2/n .u/:
Using (4.3), (4.4) and (4.5), we get
�n.˛C1/.˛C2/.2˛C3/.�.˛/n .u/� s/DO.1/CO.1/CO.1/DO.1/:
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ON TAUBERIAN REMAINDER THEOREMS FOR CESÀRO SUMMABILITY 1249
Therefore, �n.�.˛/n .u/� s/DO.1/. That means u 2 ..C;˛/;m�/.
�
Now, we represent a Tauberian remainder theorem which
generalizes Theorem 3and Proposition 1.
Theorem 4. Let the conditions
�n.n�/k�.˛Ck/n .u/DO.1/; (4.6)
�n�.˛/n .u/DO.1/; (4.7)
and�n.�
.˛Cj /n .u/� s/DO.1/ for 2� j � k (4.8)
be satisfied for ˛ > �1. If u 2 ..C;˛C1/;m�/, then u 2
..C;˛/;m�/.
Proof. From Lemma 1 we have
�n.n�/k�.˛Ck/n .u/D B1;1�˛�n�
.˛/n .u/�B1;1�n�
.˛/n .u/CB1;1�n�
.˛C1/n .u/
C�n
kXjD2
�Bj;j�1�
.˛Cj�2/n .u/�2Bj;j� 1
2� .˛Cj�1/n .u/CBj;j�
.˛Cj /n .u/
�;
Rewritten the above equation, we have
B1;1�n.�.˛/n .u/� s/
D B1;1�˛�n�.˛/n .u/��n.n�/k�
.˛Ck/n .u/CB1;1�n.�
.˛C1/n .u/� s/
C�n
kXjD2
Bj;j�1.�.˛Cj�2/n .u/� s/��n
kXjD2
2Bj;j� 12.� .˛Cj�1/n .u/� s/
C�n
kXjD2
Bj;j .�.˛Cj /n .u/� s/:
Using (4.6), (4.7) and (4.8), we get
B1;1�n.�.˛/n .u/� s/DO.1/CO.1/CO.1/CO.1/CO.1/CO.1/DO.1/:
Therefore, �n.�.˛/n .u/� s/DO.1/. That means u 2 ..C;˛/;m�/.
�
Theorem 5. Let the condition
�n�.˛CjC1/n .u/DO.1/ for 0� j � k�1; (4.9)
be satisfied for ˛ > �1. If u 2 ..C;˛Ck/;m�/, then u 2
..C;˛/;m�/.
Proof. Suppose that u 2 ..C;˛C k/;m�/. Taking j D k� 1 in (4.9),
it followsfrom the idendity
� .˛Ck/n .u/D .˛Ck/.�.˛Ck�1/n .u/��
.˛Ck/n .u//
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1250 Ü. TOTUR AND M. A. OKUR
that we obtain
�n.˛Ck/.�.˛Ck�1/n .u/� s/D �n�
.˛Ck/n .u/C�n.˛Ck/.�
˛Ckn .u/� s/
DO.1/CO.1/DO.1/
then we obtain �n.�.˛Ck�1/n � s/DO.1/. Hence, that means
u 2 ..C;˛Ck�1/;m�/:
From identity (1.4), we have
� .˛Ck�1/n .u/D .˛Ck�1/.�.˛Ck�2/n .u/��
.˛Ck�1/n .u//:
Taking j D k�2 in (4.9), we obtain
�n.˛Ck�1/.˛.˛Ck�2/n .u/� s/
D �n�.˛Ck�1/n .u/C�n.˛Ck�1/.�
.˛Ck�1/n .u/� s/DO.1/CO.1/DO.1/
Therefore we haveu 2 ..C;˛Ck�2/;m�/:
Continuing in this way, we obtain that
u 2 ..C;˛C1/;m�/:
Taking j D 0 in (4.9), we obtain �n�.˛C1/n DO.1/: From identity
(1.4), we have
�n.˛C1/.�.˛/n .u/� s/D �n�
.˛C1/n .u/C�n.˛C1/.�
.˛C1/n .u/� s/
DO.1/CO.1/DO.1/
This completes the proof. �
Theorem 6. Let the condition
�n.n�/j �.˛Cj /n .u/DO.1/ for 0� j � k; (4.10)
be satisfied for ˛ > �1. If u 2 ..C;˛Ck/;m�/, then u 2
..C;˛/;m�/.
Proof. By identity (1.6) for k D 1, it follows
�nn��.˛C1/n .u/D �n.˛C1/.�
.˛/n .u/� �
.˛C1/n .u//
D �n.˛C1/�.˛/n .u/��n.˛C1/�
.˛C1/n .u/:
Taking j D 0 and j D 1 in (4.10), we obtain
�n�.˛C1/n .u/DO.1/
From identity (1.6) for k D 2, we get
�n.n�/2�.˛C2/n .u/D �n.˛C2/.˛C1/
�� .˛/n .u/� �
.˛C1/n .u/
���n.˛C2/
2�� .˛C1/n .u/� �
.˛C2/n .u/
�:
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ON TAUBERIAN REMAINDER THEOREMS FOR CESÀRO SUMMABILITY 1251
Taking j D 0 and j D 2 in (4.10), we obtain
�n�.˛C2/n .u/DO.1/
Continuing in this way, by Lemma 1, we obtain
�n.n�/k�.˛Ck/n .u/D .˛C1/A
.1/
k.˛/�n.�
.˛/n .u/� �
.˛C1/n .u//
C�n
kXjD2
.˛Cj /.�1/jC1A.j /
k.˛/.� .˛Cj�1/n .u/� �
.˛Cj /n .u//:
Taking j D 0 and j D k in (4.10), we obtain
�n�.˛Ck/n .u/DO.1/:
The conditions in Theorem 5 hold, the proof is completed. �
ACKNOWLEDGEMENT
The authors are grateful to the anonymous referee who has made
invaluable sug-gestions that helped to improve this paper.
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Authors’ addresses
Ü. ToturAdnan Menderes University, Department of Mathematics,
09010, Aydin, TurkeyE-mail address: [email protected]
M. A. OkurAdnan Menderes University, Department of Mathematics,
09010, Aydin, TurkeyE-mail address: [email protected]
http://dx.doi.org/10.3846/13926292.2013.758674
1. Introduction2. Tauberian remainder theorems3. A lemma4. Main
resultsAcknowledgementReferences