for review only - rdo.psu.ac.th · for review only orlicz arithmetic convergence defined by matrix...

For Review OnlyOrlicz Arithmetic convergence defined by Matrix

Transformation and Lacunary sequence

Journal: Songklanakarin Journal of Science and Technology

Manuscript ID SJST-2018-0018.R4

Manuscript Type: Original Article

Date Submitted by the Author: 12-Oct-2018

Complete List of Authors: Raj, Kuldip; Choudhary, Anu

Keyword: Orlicz function, Infinite matrix, arithmetic convergence

For Proof Read only

Songklanakarin Journal of Science and Technology SJST-2018-0018.R4 Raj

For Review Only

Orlicz Arithmetic Convergence Defined by Matrix

Transformation and Lacunary Sequence

Kuldip Raj1*and Anu Choudhary1

1School of Mathematics, Shri Mata Vaishno Devi University,

Katra-182320, J & K, India

* Corresponding author, Email address:[email protected]

Abstract

In this paper we introduce and study some spaces of Orlicz arithmetic convergence sequences

with respect to matrix transformation and lacunary sequence. We make an effort to examine

some algebraic and topological properties of these sequence spaces. Some inclusion relation

between these sequence spaces have been established.

Keywords: Orlicz function, infinite matrix, lacunary sequence, arithmetic convergence,

matrix transformation.

1. Introduction

Ruckle (2012) introduced the idea of arithmetic convergence. A sequence is said to kxx

be arithmetically convergent, if for each there is an integer such that we have0 l

for every integer , where denotes the greatest common divisor of two || , lkk xx k lk,

integers and . The sequence space of all arithmetic convergent sequences is denoted by k l

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𝒜C. Subsequently, the arithmetic convergence has been discussed in (Yaying & Hazarika,

2017a, 2017b) etc.

Let , ℝ and be the sets of natural, real and complex numbers respectively. We writeℕ ℂ

ℝ or𝑤 = {𝑥 = (𝑥 k ) kx: ℂ}

the space of all real or complex sequences. An increasing non-negative integer sequence

with and as is known as lacunary sequence. The )( rk 00 k ,1 rr kk r

intervals determined by will be denoted by We write and 𝜃 ].,( 1 rrr kkI 1 rrr kkh rq

denotes the ratio . The space of lacunary strongly convergent sequence was defined by 1r

r

hh

Freedman, Sember and Raphael (1978) as follows:

somefor0||1lim:)({

rIk

kr

rk Lxh

xxN 𝐿 }.

The space is a BK-space with the normN

rIk

krr

xh

x .||1sup||||

The concept of lacunary convergence has been studied by many authors. Some of them are

Çolak, Tripathy, and Et (2006); Kara and İlkhan (2016); Raj and Pandoh (2016); Savaş and

Patterson (2007); Tripathy and Baruah (2010); Tripathy and Dutta (2012); Tripathy and Et

(2005); Tripathy, Hazarika, and Choudhary (2012); Tripathy and Mahanta (2004). Freedman,

Sember, and Raphael (1978) also introduced the concept of lacunary refinement. A lacunary

refinement of a lacunary sequence is a lacunary sequence ( ) satisfying . 𝜃 𝜃′ = 'rk )()( '

rr kk

For more details about sequence spaces see (Et, Lee & Tripathy, 2006; Raj & Sharma, 2016;

Raj & Kilicman, 2015).

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Let be an infinite matrix of real or complex numbers , where . We)( kjaA kja jk, ∈ ℕ

write the transform of as and if converges A )( kxx Ax ))(( xAAx k

1

)(j

jkjk xaxA

for each . 𝑘 ∈ ℕ

A function is said to be an Orlicz function, if it is convex, continuous and ,0,0:M

non-decreasing with for and as . An Orlicz 0,00 xMM 0x xM x

function is said to satisfy 2-condition for all values of , if there exists such that M ∆ u 0R

.0,2 uuRMuM

The idea of Orlicz function was used by Lindenstrauss and Tzafriri (1971) to define the

following sequence space:

0somefor,:1

k

kM

xMwx

known as Orlicz sequence space. The space is a Banach space with the normM

.

1:0inf1k

kxMx

An Orlicz function can always be represented in the following integral formM

M ,)()(0x

dttx

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where is known as the kernel of is right differentiable for t 0, (0) = 0, (t)>0, is 𝜇 ,M 𝜇 𝜇 𝜇

non- decreasing and (t) A sequence of Orlicz functions is called a 𝜇 →∞,𝑡→∞. kMM

Musielak-Orlicz function (Maligranda, 1989; Musielak, 1983). The complementary function

of a Musielak-Orlicz function is denoted by and is defined by 𝑁 = 𝑁𝑘

𝑁𝑘(𝜐) = sup {|𝜐|𝑢 ― 𝑀𝑘(𝑢):𝑢 > 0}, 𝑘 = 1, 2, …

Let be a positive sequence with sup = , . Then for all )( kpp kp 𝐻 𝐶 = max (1,2𝐻 ― 1 )

for all , we havekk ba , ∈ ℂ 𝑘 ∈ ℕ

(1.1) kk ba | kp| )|||(| kk pk

pk baC

Let be a Musielak-Orlicz function, be a bounded sequence of positive kMM )( kpp

real numbers and a sequence of positive real numbers. Let be an infinite )( kuu )( kjaA

matrix of real or complex numbers , where . In the present paper we define the kja jk, ∈ ℕ

Orlicz arithmetic sequences using matrix transformations and lacunary sequence as follows:

[𝒜C , ]= AMpu ,,,

r

k

Ik

p

lkkkk

rrk l

xxAMu

hxx 0andintegersomefor,0

|)(|1lim:)( ,

and

[𝒜C , ]=1 AMpu ,,,

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.

0someforas

0|)(|1thatsuchintegeranexistthere:)( ,

1

q

xxAMu

qlxx

kp

lkkk

q

kkk

For 1kp

[𝒜C , ]= , ,u M A

,| ( ) |1( ) : lim 0, for someinteger and 0 .r

k k lk k kr k Ir

A x xx x u M l

h

The arithmetic convergent sequence space [𝒜C, ] is defined AMpu ,,,

[𝒜C, ]=AMpu ,,,

,| ( ) |( ) : lim 0, for someinteger and 0 .

kpk k l

k k kk

A x xx x u M l

The main aim of the paper is to introduce the sequence spaces [𝒜C , ] and [𝒜C AMpu ,,,

, ]. We shall investigate some algebraic properties, topological properties and 1 AMpu ,,,

inclusion relation between these sequence spaces.

2. Main Results

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Theorem 2.1. Let be a Musielak-Orlicz function, be a bounded )( kMM )( kpp

sequence of positive real numbers and be any sequence of positive real numbers. )( kuu

Then the space [𝒜C , ] is linear over the complex field . AMpu ,,, C

Proof. Let and [𝒜C , ]. Then for an integer )( kxx )( kyy AMpu ,,, l

r

k

Ik

p

lkkkk

rr

xxAMu

h0

|)(|1lim ,

and

.0|)(|1lim ,

r

k

Ik

p

lkkkk

rr

yyAMu

h

Let and be two scalars in , then there exist integers and such that and C A B A||

. Therefore B||

r

k

Ik

p

lklkkkkk

r

yxyAxAMu

h

|)()()(|1 ,,

A

r r

kk

Ik Ik

p

lkkkk

r

p

lkkkk

r

yyAMu

hB

xxAMu

h

|)(|1|)(|1 ,,

.)()( ,, lklkkk yxyAxA

Therefore, [𝒜C , ] is a linear space. AMpu ,,,

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Then, we have

[𝒜C, ] [𝒜C , ].AMpu ,,, AMpu ,,,

Proof. Let [𝒜C, ]. Then there is an integer such that )( kxx AMpu ,,, l

forkp

lkkkk

xxAMu

|)(| , , .0

Thus, we have

r

k

Ik

p

lkkkk

r

xxAMu

h

|)(|1 ,

kr

kr

p

lkkk

k

kk

p

lkkk

k

kk

r

xxAMu

xxAMu

h

|)(||)(|1 ,

1

,

1

1

< =)(1 rr

hh

.

Therefore, [𝒜C , ]. )( kxx AMpu ,,,



Then, [𝒜C , ] is a topological space paranormed by AMpu ,,,

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,0somefor,1|)(|1:inf)(

/1

,/

H

Ik

p

lkkkk

r

Hp

r

k

kxxA

Muh

xg

where ).sup,1max( kk pH

Proof.(i) Clearly for [𝒜C , ]. Since we get ,0)( xg )( kxx AMpu ,,, ,0)0( kM

.0)0( g

(ii) ).()( xgxg

(iii) Let [𝒜C , ], then there exist such that )(),( kk yyxx AMpu ,,, 0, 21

r

k

Ik

p

lkkkk

r

xxAMu

h 1

, |)(|1

1

and

.

r

k

Ik

p

lkkkk

r

yyAMu

h 2

, |)(|1

1

Let then by Minkowski’s inequality, we have,21

r

k

Ik

p

lkklkkkk

r

yyAxxAMu

h

|))(())((|1 ,,

21

1

r

k

Ik

p

lkkkk

r

xxAMu

h 1

, |)(|1

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21

2

r

k

Ik

p

lkkkk

r

yyAMu

h 2

, |)(|1

and thus

yxg

1|))(())((|1:inf

/1

,,/21

H

Ik

p

lkklkkkk

r

Hp

r

k

kyyAxxA

Muh

).()( ygxg

(iv) Finally, we show that scalar multiplication is continuous. Let be any complex number,

then

)( xg

1|)(|1:inf

/1

,/

H

Ik

p

lkkkk

r

Hp

r

k

kxxA

Muh

,1|)(|1:)|(|inf

/1

,/

H

Ik

p

lkkkk

r

Hp

r

k

k

sxxA

Muh

s

where Hence [𝒜C , ] is a paranormed space..||

s AMpu ,,,

Theorem 2.4. Let for all and let be bounded. Thenkk qp 0 k )/( kk pq

[𝒜C , ] [𝒜C , ]. AMqu ,,, AMpu ,,,

Proof. Let [𝒜C , ] and )( kxx AMqu ,,,

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kdkq

lkkk

xxAM

|)(| ,

and for all . Then for all . Take for .kkk qp / k ℕ 10 k k ℕ k 0 k ℕ

Define sequences and as follows:)( ky )( kz

For let and and for let and Then clearly,,1kd kk dy 0kz ,1kd 0ky .kk dz

for all , we havek ℕ

., kkkkkkkkk zydzyd

Now it follows that and Therefore,kky

kk dy kkz .

kz

( )

rr

k

Ikk

rIkkk

r

uh

duh

11 kkkk zy

rr Ik

kkrIk

kkr

zuh

duh

.11

Now for each ,k

1111

rr Ikk

rkk

rIkkk

r

uh

zuh

zuh

=

11/11/1

11

rr Ikk

rIkkk

r

uh

zuh

=

rIk

kkr

zuh1

and hence

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r rrIk Ikkk

rIkkk

rkk

r

zuh

duh

zuh

.111

Therefore, [𝒜C , ]. )( kxx AMpu ,,,

Theorem 2.5. (i) If for all , then1inf0 kk pp k ℕ

[𝒜C , ] [𝒜C , ]. AMpu ,,, AMu ,,

(ii) If for all , then Hpp kk sup1 k ℕ

[𝒜C , ] [𝒜C , ]. AMu ,, AMpu ,,,

Proof. (i) Let [𝒜C , ], then )( kxx AMpu ,,,

r

k

Ik

p

lkkkk

rr

xxAMu

h.0

|)(|1lim ,

Since , implies 1inf0 kk pp

r r

k

Ik Ik

p

lkkkk

rr

lkkkk

rr

xxAMu

h

xxAMu

h.

|)(|1lim|)(|1lim ,,

Thus, Hence, [𝒜C , ] [𝒜C ,

rIk

lkkkk

rr

xxAMu

h.0

|)(|1lim ,

AMpu ,,, AMu ,,

].

(ii) Let and [𝒜C , ], then for each we havekk pp sup1 )( kxx AMu ,, ,0

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rIk

lkkkk

rr

xxAMu

h.0

|)(|1lim ,

For given conditions we have

r

k

Ik

p

lkkkk

rr

xxAMu

h

|)(|1lim ,

rIk

lkkkk

rr

xxAMu

h.0

|)(|1lim ,

Therefore, [𝒜C , ].Hence, [𝒜C , ] [𝒜C , ]. )( kxx AMpu ,,, AMu ,, AMpu ,,,

Theorem 2.6. If for all , then,supinf0 Hpp kk k ℕ

[𝒜C , ] [𝒜C , ]. AMpu ,,, AMu ,,

Proof. We omit the details as it is easy to prove.

Theorem 2.7. Let and be two Hpphp kkk supinf0 )(),( kk MMMM

Musielak Orlicz functions satisfying -condition, then we have2

[𝒜C , ] [𝒜C , ]. AMpu ,,, AMMpu ,,,

Proof. Suppose [𝒜C , ], then )( kxx AMpu ,,,

.0|)(|1lim ,

r

k

Ik

p

lkkkk

rr

xxAMu

h

Now choose with and such that for 10 0 )(wM k .0 w

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Let for all , then we have

|)(| ,lkkkk

xxAMz k ℕ

kr

k

kr

k

r

k

zIk

pkkk

rz

Ik

pkkk

rIk

pkkk

r

zMuh

zMuh

zMuh

.111

Therefore,

kr

kr

kk

zIk

zIk

pkkk

r

Hk

pkkk

r

zMuh

MzMuh

1)1(1

(2.1)

kr

k

zIk

pkkk

r

Hk zMu

hM .1)2(

Since are convex and non-decreasing and for we have kM ,1,

kkkk

zzzz

k

kkk

kkkzMMzMzM 2

21)2(

211)(

and as satisfies condition, so we have)( kMM 2

)2(21)2(

21)( k

kk

kkk MzTMzTzM

).2(k

k MzT

Thus,

(2.2) .1)2(,1max1

kr

kr

kk

zIk

zIk

pkkk

r

Hkp

kkkr

zMuh

MTzMuh

From (2.1) and (2.2), we have [𝒜C , ]. This completes the proof. )( kxx AMMpu ,,,

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Theorem 2.8.Let Then for Musielak-Orlicz function .supinf0 Hppph kkk

which satisfies -condition, we have)( kMM 2

[𝒜C , ] [𝒜C , ]. Apu ,, AMpu ,,,

Proof. We omit it, as it is easy to prove.

Theorem 2.9. Let and be two Musielak-Orlicz functions. Then we )( kMM )( kMM

have

[𝒜C , ] [𝒜C , ] [𝒜C , ]. AMpu ,,, AMpu ,,, AMMpu ,,,

Proof. Let [𝒜C , ] [𝒜C , ], then)( kxx AMpu ,,, AMpu ,,,

0|)(|1lim ,

r

k

Ik

p

lkkkk

rr

xxAMu

h

and

.0|)(|1lim ,

r

k

Ik

p

lkkkk

rr

xxAMu

h

Also by using (1.1) we have

kp

lkkkk

xxAMM

|)(| ,

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.Ckk p

lkkk

p

lkkk

xxAMC

xxAM

|)(||)(| ,,

Now multiplying both sides of the above inequality by and applying we getkr

uh

,1 rIk

,

r

k

Ik

p

lkkkkk

r

xxAMMu

h

|)(|1 ,

.k

rr

k p

lkkk

Ikk

Ik

p

lkkkk

r

xxAMu

xxAMu

hC

|)(|

hC|)(| ,

r

,

This completes the proof.



Then, [𝒜C , ] [𝒜C , ],where is the lacunary refinement of a AMpu ,,, AMpu ,,,

lacunary sequence .

Proof. Let each of contains the point of such thatrI )(1, )( r

ccrk ,

.')(,

'2,

'1,1 rrrrrr kkkkk

So as .1)(, rr )( 'rr kk

Suppose be the sequence of interval ordered by increasing right end points.

1*

eI *,trI

Since [𝒜C , ], then for each )( kx ' AMpu ,,, ,0

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re

k

II

p

lkkkk

e

xxAMu

h *

|)(|1 ,*

and in view of . Now for each '1,

',

', trtrtr kkh )( '

rr kk ,0

<

r

k

Ik

p

lkkkk

r

xxAMu

h

|)(|1 ,

re

k

II

p

lkkkk

e

xxAMu

h *

|)(|1 ,*

.

Hence, [𝒜C , ].)( kx AMpu ,,,



Then the space [𝒜C , ] is closed under addition and scalar multiplication.1 AMpu ,,,

The proof can be established using standard technique, so omitted.


sequence of positive real numbers and be any sequence of positive real numbers. If )( kuu

then [𝒜C , ] [𝒜C , ].,1inflim rq1 AMpu ,,, AMpu ,,,

Proof. Suppose and [𝒜C , ]. Then there exists 1inflim rq )( kxx1 AMpu ,,, ,0

such that and for sufficiently large Then

11r

rr k

kq

1

r

r

hk .r

rk

kk

r

uk 1

1kp

lkkk

xxAM

|)(| ,

rk1

r

k

Ik

p

lkkkk

xxAMu

|)(| ,

=

kp

lkkk

r

rxxA

Mkh

|)(|u

h1 ,

Ikk

r r

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.

1

kp

lkkk

xxAM

|)(|u

h1 ,

Ikk

r r

Hence, [𝒜C , ]. )( kxx AMpu ,,,



then [𝒜C , ] [𝒜C , ].,suplim rq AMpu ,,, 1 AMpu ,,,

Proof .Suppose then there exists such that for every ,suplim rq 0N Nqr .r

For [𝒜C , ] and there exists such that for every )( kxx AMpu ,,, ,0 P ,Pr

<)(r

r

k

Ik

p

lkkkk

r

xxAMu

h

|)(|1 , .

For Let be an integer with Then.,)(,0 rQrQ q .1 rr kqk

q

kku

q 1

1kp

lkkk

xxAM

|)(| ,

rk

kk

r

uk 11

1kp

lkkk

xxAM

|)(| ,

=

P

j Ikk

r j

uk 11

1kp

lkkk

xxAM

|)(| ,

r

j

k

Pj Ikk

r

uk 11

1kp

lkkk

xxAM

|)(| ,

P

j Ikk

r j

uk 11

1kp

lkkk

xxAM

|)(| ,

+ Prr

kkk

1

1

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P

jjj

r

hk 11

1 Prr

kkk

1

1

Nkk Pj

Pjr

sup1

1

.1

NQkk

r

P

From here we conclude that [𝒜C , ]. )( kxx1 AMpu ,,,



then [𝒜C , ] = [𝒜C , ].rqinflim ,suplim rq AMpu ,,,1 AMpu ,,,

Proof. The proof is easy and follows from Theorem 2.12 and Theorem 2.13.

Acknowledgement: The authors would like to express their sincere thanks to referees for a

careful reading and several constructive comments that have improved the presentation of

the paper.

References

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lacunary almost statistical convergence. Vietnam Journal of Mathematics, 34, 129-

138.

Et., M., Lee, P. Y., & Tripathy, B. C. (2006). Strongly almost -summable ))(,( rV

sequences defined by Orlicz function, Hokkaido Mathematical Journal, 35, 197-

213.

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Kara, E. E., & İlkhan, M. (2016). Lacunary I-convergent and lacunary I-bounded sequence

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Raj, K., & Kilicman,A. (2015).On certain generalized paranormed spaces. Journal of

Inequalities and Applications, 2015, 37.

Raj, K., & Pandoh, S. (2016). Some generalized double lacunary Zweier convergent

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Ruckle,W. H. (2012). Arithmetical Summability. Journal of Mathematical Analysis and

Applications, 396, 741-748.

Savaş, E., & Patterson,R. F. (2007). Double sequence spaces characterized by lacunary

sequences. Applied Mathematics Letters, 20, 964-970.

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Tripathy, B. C., & Baruah, A. (2010). Lacunary statistically convergent and lacunary

strongly convergent generalized difference sequences of fuzzy real numbers.

Kyungpook Mathematical Journal, 50, 565-574.

Tripathy, B. C., & Dutta, H.(2012). On some lacunary difference sequence spaces defined

by a sequence of Orlicz functions and q-lacunary -statistical convergence. nm

Analele Stiintifice ale Universitatii Ovidius, Seria Mathematica, 20, 417-430.

Tripathy, B. C., & Et, M. (2005). On generalized difference lacunary statistical

convergence. Universitatis Babeş-Bolyai. Studia Mathematica, 50, 119-130.

Tripathy, B. C., Hazarika, B. & Choudhary, B. (2012). Lacunary I-convergent sequences.

Kyungpook Mathematical Journal, 52, 473-482.

Tripathy, B. C., & Mahanta, S. (2004). On a class of generalized lacunary difference

sequence spaces defined by Orlicz function. Acta Mathematicae Applicatae

Sinica. English Series, 20, 231-238.

Yaying, T., & Hazarika,B. (2017).On arithmetical summabilityand multiplier sequences.

National Academy Science Letters, 40, 43-46.

Yaying, T., & Hazarika,B. (2017).On arithmetic continuity.Boletim da Socieda de

Paranaense de Matemática, 35, 139-145.

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