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

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�/:

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/:

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/

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.

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/:

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//

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/

�:

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.

REFERENCES

[1] Çanak, İ. and Erdem, Y., “On Tauberian theorems for .A/.C;˛/ summability method,” App. Math.and Comp., vol. 218, no. 6, pp. 2829–2836, 2011, doi: 10.1016/j.amc.2011.08.026.

[2] Çanak, İ. and Erdem, Y. and Totur, Ü., “Some Tauberian theorems for .A/.C;˛/ sum-mability method,” Math. Comput. Modell., vol. 52, no. 5-6, pp. 738–743, 2010, doi:10.1016/j.mcm.2010.05.001.

[3] Erdem, Y. and Çanak, İ., “A Tauberian theorem for .A/.C;˛/ summability,” Comput. Math. Appl.,vol. 60, no. 11, pp. 2920–2925, 2010, doi: 10.1016/j.camwa.2010.09.047.

[4] Estrada, R. and Vindas J., “On Tauber’s second Tauberian theorem,” Tohoku Math. J., vol. 64,no. 4, pp. 539–560, 2012.

[5] Estrada, R. and Vindas J., “Distributional versions of Littlewood’s Tauberian theorem,”Czechoslovak Math. J., vol. 63, no. 2, pp. 403–420, 2013, doi: 10.1007/s10587-013-0025-1.

[6] Ganelius, T. H., “Tauberian remainder theorems,” Lecture Notes in Mathematics, vol. 232, 1971.[7] Kangro, G., “Summability factors of Bohr-Hardy type for a given rate. I, II,” Eesti NSV Tead.

Akad. Toimetised Füüs.-Mat., vol. 18, pp. 137–146, 387–395, 1969.[8] Kangro, G., “Summability factors for series that are �-bounded by the Riesz and Cesàro methods,”

Tartu Riikl. Ül. Toimetised Vih., vol. 277, pp. 136–154, 1971.[9] Kogbetliantz, E., “Sur le séries absolument sommables par la méthode des moyennes

arihtmétiques,” Bulletin sc. Math., vol. 49, pp. 234–251, 1925.[10] Kogbetliantz, E., “Sommation des séries et intégrals divergents par les moyennes arithmétiques et

typiques,” Memorial Sci. Math., vol. 51, pp. 1–84, 1931.[11] J. Koreavar, Tauberian theory. Berlin: Springer, 2004, vol. 329.[12] Meronen, O. and Tammeraid, I., “Generalized linear methods and gap Tauberian remainder

theorems,” Math. Model. Anal., vol. 13, no. 2, pp. 223–232, 2008, doi: 10.3846/1392-6292.2008.13.223-232.

[13] Meronen, O. and Tammeraid, I., “Several theorems on �-summable series,” Math. Model. Anal.,vol. 15, no. 1, pp. 97–102, 2010, doi: 10.3846/1392-6292.2010.15.97-102.

http://dx.doi.org/10.1016/j.amc.2011.08.026http://dx.doi.org/10.1016/j.mcm.2010.05.001http://dx.doi.org/10.1016/j.camwa.2010.09.047http://dx.doi.org/10.1007/s10587-013-0025-1http://dx.doi.org/10.3846/1392-6292.2008.13.223-232http://dx.doi.org/10.3846/1392-6292.2008.13.223-232http://dx.doi.org/10.3846/1392-6292.2010.15.97-102

1252 Ü. TOTUR AND M. A. OKUR

[14] Meronen, O. and Tammeraid, I., “General control modulo and Tauberian remainder theor-ems for (C,1) summability,” Math. Model. Anal., vol. 18, no. 1, pp. 97–102, 2013, doi:10.3846/13926292.2013.758674.

[15] Natarajan, P. N., “A Tauberian theorem for weighted means,” Bull. Pure Appl. Math., vol. 6, no. 1,pp. 45–48, 2012.

[16] Tammeraid, I., “Tauberian theorems with a remainder term for the Cesàro and Hölder summabilitymethods,” Tartu Riikl. Ül. Toimetised No., vol. 277, pp. 161–170, 1971.

[17] Tammeraid, I., “Tauberian theorems with a remainder term for the Euler-Knopp summabilitymethod,” Tartu Riikl. Ül. Toimetised No., vol. 277, pp. 171–182, 1971.

[18] Tammeraid, I., “Two Tauberian remainder theorems for the Cesàro method of summability,” Proc.Estonian Acad. Sci. Phys. Math., vol. 49, no. 4, pp. 225–232, 2000.

Authors’ addresses

Ü. ToturAdnan Menderes University, Department of Mathematics, 09010, Aydin, TurkeyE-mail address: utotur@adu.edu.tr

M. A. OkurAdnan Menderes University, Department of Mathematics, 09010, Aydin, TurkeyE-mail address: mali.okur@adu.edu.tr

http://dx.doi.org/10.3846/13926292.2013.758674

1. Introduction2. Tauberian remainder theorems3. A lemma4. Main resultsAcknowledgementReferences