Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 491768 5 pageshttpdxdoiorg1011552013491768

Research ArticleStatistical Approximation for Periodic Functions ofTwo Variables

Abdullah Alotaibi1 M Mursaleen2 and S A Mohiuddine1

1 Department of Mathematics Faculty of Science King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia2Department of Mathematics Aligarh Muslim University Aligarh 202002 India

Correspondence should be addressed to M Mursaleen mursaleenmgmailcom

Received 4 October 2013 Accepted 17 November 2013

Academic Editor Asadollah Aghajani

Copyright copy 2013 Abdullah Alotaibi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We prove a Korovkin type approximation theorem for a function of two variables by using the notion of statistical summability(119862 1 1) We also study the rate of statistical summability (119862 1 1) of positive linear operators Finally we construct an example toshow that our result is stronger than those previously proved for Pringsheimrsquos convergence and statistical convergence

1 Introduction and Preliminaries

In 1951 Fast [1] and Steinhaus [2] independently introducedan extension of the usual concept of sequential limit which iscalled statistical convergence

The number sequence 119909 is said to be statistically conver-gent to the number 119871 provided that for each 120598 gt 0

lim119899

1

119899

1003816100381610038161003816119896 le 1198991003816100381610038161003816119909119896 minus ℓ

1003816100381610038161003816 ge 1205981003816100381610038161003816 = 0 (1)

where |119896 le 119899 119896 isin 119870| denotes the number of elementsof 119870 not exceeding 119899 In this case we write 119904119905-lim119909

119896= ℓ

The notion of statistical convergence of double se-quences 119909 = (119909

119895119896) has been introduced and studied in [3 4]

independently in the same year 2003Let 119870 sube N times N be a two-dimensional set of positive

integers and let 119870(119899119898) be the numbers of (119894 119895) in 119870 suchthat 119894 le 119899 and 119895 le 119898 Then the two-dimensional analogueof natural density can be defined as follows

The lower asymptotic density of a set 119870 sube NtimesN is definedas

1205752(119870) = lim

119899119898

inf 119870 (119899119898)

119899119898 (2)

In this case the sequence (119870(119899119898)119899119898) has a limit in Pring-sheimrsquos sense then we say that 119870 has a double natural densityand is defined as

119875-lim119899119898

119870 (119899119898)

119899119898= 1205752(119870) (3)

A real double sequence 119909 = (119909119895119896) is said to be statistically

convergent to the number 119897 if for each 120598 gt 0 the set

(119894 119895) 119895 le 119898 119896 le 119899 10038161003816100381610038161003816119909119895119896minus ℓ

10038161003816100381610038161003816ge 120598 (4)

has double natural density zero In this case we write1199041199052-lim119895119896119909119895119896= ℓ

If 119909 is statistically convergent then 119909 need not be con-vergent Also it is not necessarily bounded For examplelet 119909 = (119909

119895119896) be defined as

119909119895119896=

119895119896 if 119895 and 119896 are squares1 otherwise

(5)

It is easy to see that 1199041199052-lim119909

119895119896= 1 since the cardinality of the

set (119895 119896) |119909119895119896minus 1| ge 120598 le radic119895radic119896 for every 120598 gt 0 But 119909 is

neither convergent nor boundedMoricz [5] introduced the notion of statistical summa-

bility (119862 1 1) A double sequence 119909 = (119909119895119896) is said to be sta-

tistically summable (119862 1 1) to the number ℓ if for every 120598 gt0

1205752(119898 119899) isin N times N

1003816100381610038161003816120590119898119899 minus ℓ1003816100381610038161003816 ge 120598 = 0 (6)

2 Journal of Function Spaces and Applications

where

120590119898119899

=1

(119898 + 1) (119899 + 1)

119898

sum119895=0

119899

sum119896=0

119909119895119896 (7)

is the (119862 1 1) mean of 119909 = (119909119895119896) Thus the double se-

quence 119909 is statistically summable (119862 1 1) to 119897 if and onlyif the sequence 120590 = (120590

119898119899) is statistically convergent to ℓ

In this case we write 1199041199052(119862 1 1)-lim

119895119896119909119895119896

= ℓ Note thatif a double sequence is bounded then 119904119905

2-lim119895119896119909119895119896

= ℓ

implies 1199041199052-lim119898119899

120590119898119899

= ℓKorovkin type approximation theorems (cf [6ndash10]) are

useful tools to check whether a given sequence (119871119899)119899ge1

ofpositive linear operators on 119862[0 1] of all continuous func-tions on the real interval [0 1] is an approximation processThat is these theorems exhibit a variety of test functionswhich assure that the approximation property holds on thewhole space if it holds for them Such a property was discov-ered by Korovkin in 1953 for test functions 1 119909 and 1199092 in thespace 119862[0 1] as well as for test functions 1 cos and sin in thespace of all continuous 2120587-periodic functions on the real line

We know that 119862[0 1] is a Banach space with norm

10038171003817100381710038171198911003817100381710038171003817infin = sup

119909isin[01]

1003816100381610038161003816119891 (119909)1003816100381610038161003816 119891 isin 119862 [0 1] (8)

We denote by 1198622120587([0 1]) the space of all 2120587-periodic

functions 119891 isin 119862([0 1]) which is a Banach space with

100381710038171003817100381711989110038171003817100381710038172120587 = sup

119905isin[01]

1003816100381610038161003816119891 (119905)1003816100381610038161003816 (9)

After the paper of Gadjiev and Orhan [11] many papershave appeared in the literature concerning the Korovkin typeapproximation theorems via different statistical summabilitymethods and for different sets of test functions At present weare concerned about applications of such summability meth-ods for double sequences to prove two-dimensional versionofKorovkin theorem For example in [12 13] the authors usedthe notion of statistical 119860-summability of double sequencesin [13ndash16] the authors have used respectively statistical con-vergence and 119860-statistical convergence of double sequencesand in [17 18] the authors used almost summability For somemore related work we refer to [19ndash22]

In this paper we present the Korovkin type approxima-tion theorem for periodic functions via statistical summabil-ity (119862 1 1) and also study the rate of statistical summability(119862 1 1) of a double sequence of positive linear opera-tors defined from 119862

lowast(R2) into 119862lowast(R2)where 119862lowast(R2) is thespace of all 2120587-periodic and real valued continuous functionson R2 equipped with the norm

10038171003817100381710038171198911003817100381710038171003817119862(R2) = sup

(119909119910)isinR2

1003816100381610038161003816119891 (119909 119910)1003816100381610038161003816 (119891 isin 119862 (R

2

)) (10)

2 Main Result

First we state the result due to [15] for 119860-statistical conver-gence of double sequences

Theorem 1 Let (119871119898119899) be a double sequence of positive linear

operators acting from 119862lowast(R2) into 119862lowast(R2) Then for all 119891 isin

119862lowast(R2)

119904119905119860

2- lim119898119899rarrinfin

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) = 0 (11)

if and only if

119904119905119860

2- lim119898119899rarrinfin

1003817100381710038171003817119871119898119899 (119891119894) minus 1198911198941003817100381710038171003817119862lowast(R2) = 0 119894 = 0 1 2 3 4 (12)

where 1198910(119909 119910) = 1 119891

1(119909 119910) = sin119909 119891

2(119909 119910) = sin119910 119891

3(119909

119910) = cos119909 and 1198914(119909 119910) = cos119910

If we replace the matrix 119860 by the identity four-dimen-sional matrix in the above theorem then we immediately getthe following result in Pringsheimrsquos sense

Corollary 2 Let (119871119898119899) be a double sequence of positive linear

operators acting from 119862lowast

(R2) into 119862lowast(R2) Then for all 119891 isin

119862lowast(R2)

119875- lim119898119899rarrinfin

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) = 0 (13)

if and only if

119875- lim119898119899rarrinfin

1003817100381710038171003817119871119898119899 (119891119894) minus 1198911198941003817100381710038171003817119862lowast(R2) = 0 119894 = 0 1 2 3 4 (14)

We prove the following result

Theorem 3 Let (119879119895119896) be a double sequence of positive linear

operators acting from 119862lowast(R2) into 119862lowast(R2) Then for all 119891 isin

119862lowast(R2)

1199041199052(119862 1 1) - lim

119895119896rarrinfin

10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)= 0 (15)

if and only if

1199041199052(119862 1 1) - lim

119895119896rarrinfin

10038171003817100381710038171003817119879119895119896(119891119894) minus 119891119894

10038171003817100381710038171003817119862lowast(R2)= 0

(119894 = 0 1 2 3 4)

(16)

Proof Since each of the functions 1198910 1198911 1198912 1198913 and 119891

4

belongs to 119862lowast(R2) necessity follows immediately from (15)Let condition (16) hold and 119891 isin 119862lowast(R2) Let 119868 and 119869 beclosed subintervals each of length 2120587 of R Fix(119909 119910) isin 119868 times 119869By the continuity of 119891 at (119909 119910) it follows that for given 120576 gt 0

there is a number 120575 gt 0 such that for all (119906 120592) isin R2

1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816 lt 120576 (17)

whenever |119906 minus 119909| |120592 minus 119910| lt 120575 Since 119891 is bounded it followsthat

1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816 le 119872

119891=10038171003817100381710038171198911003817100381710038171003817119862lowast(R2) (18)

for all (119906 120592) isin R2For all (119906 120592) isin (119909 minus 120575 2120587 + 119909 minus 120575] times (119910 minus 120575 2120587 + 119910 minus 120575] it

is well known that

1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816 lt 120576 +

2119872119891

sin2 (1205752)120595 (119906 120592) (19)

Journal of Function Spaces and Applications 3

where 120595(119906 120592) = sin2((119906 minus 119909)2) + sin2((120592 minus 119910)2) Sincethe function 119891 isin 119862lowast(R2) is 2120587-periodic the inequality (19)holds for (119906 120592) isin R2 Then we obtain

10038161003816100381610038161003816119879119895119896(119891 119909 119910) minus 119891 (119909 119910)

10038161003816100381610038161003816

le 119879119895119896(1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)

1003816100381610038161003816 119909 119910)

+1003816100381610038161003816119891 (119909 119910)

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816119879119895119896(120576 +

2119872119891

sin2 (1205752)120595 (119906 120592) 119909 119910)

100381610038161003816100381610038161003816100381610038161003816

+ 119872119891

10038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

le 120576 + ((120576 +119872119891)10038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816)

+119872119891

sin2 (1205752)

times 210038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

+ |sin119909| 10038161003816100381610038161003816119879119895119896 (1198911 119909 119910) minus 1198911 (119909 119910)10038161003816100381610038161003816

+1003816100381610038161003816sin119910

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198912 119909 119910) minus 119891

3(119909 119910)

10038161003816100381610038161003816

+ |cos119909| 10038161003816100381610038161003816119879119895119896 (1198913 119909 119910) minus 1198913 (119909 119910)10038161003816100381610038161003816

+1003816100381610038161003816cos119910

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198914 119909 119910) minus 119891

4(119909 119910)

10038161003816100381610038161003816

lt 120576 + 119870

4

sum119894=0

10038161003816100381610038161003816119879119895119896(119891119894 119909 119910) minus 119891

119894(119909 119910)

10038161003816100381610038161003816

(20)

where 119870 = 120576 + 119872119891+ (2119872

119891sin2(1205752)) Now taking

sup(119909119910)isin119868times119869

we get

10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)lt 120576 + 119870

4

sum119894=0

10038171003817100381710038171003817119879119895119896(119891119894) minus 119891119894

10038171003817100381710038171003817119862lowast(R2) (21)

Now for a given 119903 gt 0 choose 1205761015840 gt 0 such that 1205761015840 lt 119903 Definethe following sets

119863 = (119898 119899) 1003817100381710038171003817119871119898119899 (119891) minus 119891

1003817100381710038171003817119862lowast(R2) ge 119903

119863119894= (119898 119899)

1003817100381710038171003817119871119898119899 (119891119894) minus 1198911198941003817100381710038171003817119862lowast(R2) ge

119903 minus 1205761015840

5119870

(119894 = 0 1 2 3 4)

(22)

where 119871119898119899

= (1(119898 + 1)(119899 + 1))sum119898

119895=0sum119899

119896=0119879119895119896 Then by (21)

119863 sube

4

⋃119894=0

119863119894 (23)

and so

120575(2)

119860(119863) le

4

sum119894=0

120575(2)

119860(119863119894) (24)

Now using (16) we get

1199041199052(119862 1 1) - lim

119895119896rarrinfin

10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)= 0 (25)

Example 4 Now we present an example of double sequencesof positive linear operators showing thatCorollary 2 does notwork but our approximation theoremworksWe consider thedouble sequence of Fejer operators on 119862

lowast(R2)

120590119898119899

(119891 119909 119910) =1

(119899120587)sdot

1

(119899120587)

times int120587

minus120587

int120587

minus120587

119891 (119906 120592) 119865119898(119906) 119865119899(120592) 119889119906 119889120592

(26)

where

119865119898(119906) =

sin2 (119898 (119906 minus 119909) 2)

sin2 ((119906 minus 119909) 2)

1

120587int120587

minus120587

119865119898(119906) 119889119906 = 1 (27)

Observe that

120590119898119899

(1198910 119909 119910) = 119891

0(119909 119910)

120590119898119899

(1198911 119909 119910) =

119898 minus 1

1198981198911(119909 119910)

120590119898119899

(1198912 119909 119910) =

119899 minus 1

1198991198912(119909 119910)

120590119898119899

(1198913 119909 119910) =

119898 minus 1

1198981198913(119909 119910)

120590119898119899

(1198914 119909 119910) =

119899 minus 1

1198991198914(119909 119910)

(28)

Define a double sequence 120572 = (120572119898119899) by 120572

119898119899= (minus1)

119898+119899119898 119899 isin N

We observe that 120572 = (120572119898119899) is neither 119875-convergent nor

statistically convergent but

1199041199052(119862 1 1) -lim120572 = 0 (29)

Let us define the operators 119871119898119899

119862lowast(R2) rarr 119862lowast(R2) by

119871119898119899

(119891 119909 119910) = (1 + 120572119898119899) 120590119898119899

(119891 119909 119910) (30)

Then observe that the double sequence of positive linearoperators (119871

119898119899) defined by (30) satisfies all hypotheses of

Theorem 3 Hence by (28) we have for all 119891 isin 119862lowast(R2)

1199041199052(119862 1 1) - lim

119898119899rarrinfin

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) = 0 (31)

Since (120572119898119899) is neither 119875-convergent nor statistically conver-

gent the sequence (119871119898119899) given by (30) is also neither 119875-

convergent nor statistically convergent to the function 119891 isin

119862lowast(R2) So we conclude that Corollary 2 and Theorem 1do not work for the operators (119871

119898119899) given by (30) while

Theorem 3 still works Hence we conclude that 1199041199052(119862 1 1)-

version is stronger than that of 119875-version as well as statisticalversion

4 Journal of Function Spaces and Applications

3 Rate of Statistical Summability (1198621 1)

Let (120573119898119899) be a positive nonincreasing double sequence We

say that a double sequence 119909 = (119909119898119899) is statistically sum-

mable (119862 1 1) to the number 119871 with the rate 119900(120573119898119899) if for

every 120576 gt 0

119875- lim119898119899rarrinfin

1

120573119898119899

10038161003816100381610038161003816119895 le 119898 119896 le 119899

10038161003816100381610038161003816120590119895119896minus ℓ

10038161003816100381610038161003816ge 120598

10038161003816100381610038161003816= 0 (32)

In this case wewrite 119909119898119899minus119871 = 119904119905

2(119862 1 1)-119900(120573

119898119899) as 119898 119899 rarr

infinNow we recall the notion of modulus of continuity The

modulus of continuity of 119891 isin 119862lowast(R2) denoted by 120596(119891 120575) for120575 gt 0 is defined by

120596 (119891 120575) = sup 1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816 (119906 120592) (119909 119910) isin R

2

radic(119906 minus 119909)2

+ (120592 minus 119910)2

le 120575

(33)

It is well known that

1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816

le 120596 (119891radic(119906 minus 119909)2

+ (120592 minus 119910)2

)

le 120596 (119891 120575)(radic(119906 minus 119909)

2

+ (120592 minus 119910)2

120575+ 1)

(34)

Then we have the following result

Theorem 5 Let (119879119895119896) be a double sequence of positive linear

operators acting from 119862lowast(R2) into 119862lowast(R2) Let (120572119895119896) and (120573

119895119896)

be two positive non-increasing sequences Suppose that

(i) 119879119895119896(1198910) minus 1198910119862lowast(R2)

= 1199041199052(119862 1 1)-119900(120572

119898119899)

(ii) 120596(119891 120582119895119896) = 119904119905

2(119862 1 1) minus 119900(120573

119895119896) where 120582

119898119895119896=

radic 119879119895119896(120593)119862lowast(R2) and

120593 (119906 120592) = sin2 (119906 minus 1199092

) + sin2 (120592 minus 119910

2)

119891119900119903 119890119886119888ℎ (119906 120592) (119909 119910) isin R2

(35)

Then for all 119891 isin 119862lowast(R2)

10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)= 1199041199052(119862 1 1) -119900 (120574

119895119896) (36)

where 120574119895119896= max120572

119895119896 120573119895119896

Proof Let 119891 isin 119862lowast(R2) and (119909 119910) isin minus[120587 120587]times [minus120587 120587] Let 120575 gt

0 we have the following cases

Case I If 120575 lt |119906 minus 119909| le 120587 120575 lt |120592 minus 119910| le 120587 then |119906 minus 119909| le

120587| sin((119906 minus 119909)2)| and |120592 minus 119910| le 120587| sin((120592 minus 119910)2)| Thereforeby (34) we have1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)

1003816100381610038161003816

le 120596 (119891 120575) (1205872sin2 ((119906 minus 119909) 2) + sin2 ((120592 minus 119910) 2)

1205752+ 1)

(37)

Case II If |119906 minus 119909| gt 120587 |120592 minus 119910| le 120587 Let 119896 be an integer suchthat |119906 + 2119896120587 minus 119909| le 120587 then

1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816

=1003816100381610038161003816119891 (119906 + 2119896120587 120592) minus 119891 (119909 119910)

1003816100381610038161003816

le 120596 (119891 120575)

times(1205872sin2 ((119906 + 2119896120587 minus 119909) 2) + sin2 ((120592 minus 119910) 2)

1205752+ 1)

= 120596 (119891 120575) (1205872sin2 ((119906 minus 119909) 2) + sin2 ((120592 minus 119910) 2)

1205752+ 1)

(38)

Similarly in the other two cases when |119906 minus119909| le 120587 |120592 minus119910| gt 120587

and |119906 minus 119909| gt 120587 |120592 minus 119910| gt 120587 we obtain (37)Now using the definition of modulus of continuity and

the linearity and the positivity of the operators 119879119895119896we get

10038161003816100381610038161003816119879119895119896(119891 119909 119910) minus 119891 (119909 119910)

10038161003816100381610038161003816

le 119879119895119896(1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)

1003816100381610038161003816 119909 119910)

+1003816100381610038161003816119891 (119909 119910)

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

le 120596 (119891 120575) 119879119895119896(1198910 119909 119910) + 120587

2120596 (119891 120575)

1205752119879119895119896(120593 119909 119910)

+1003816100381610038161003816119891 (119909 119910)

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

(39)

Taking supremum over (119909 119910) on both sides of the aboveinequality and let

120575 = 120575119895119896= radic

10038171003817100381710038171003817119879119895119896(120593)

10038171003817100381710038171003817119862lowast(R2) (40)

We obtain10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)

le 120596 (119891 120575119895119896)10038171003817100381710038171003817119879119895119896(1198910) minus 1198910

10038171003817100381710038171003817119862lowast(R2)+ (1 + 120587

2

)

times 120596 (119891 120575119895119896) +119872

10038171003817100381710038171003817119879119895119896(1198910) minus 1198910

10038171003817100381710038171003817119862lowast(R2)

(41)

Journal of Function Spaces and Applications 5

where 119872 = 119891119862lowast(R2) Let 119871

119898119899= (1(119898 + 1)(119899 +

1))sum119898

119895=0sum119899

119896=0119879119895119896 Now for a given 120576 gt 0 define the following

sets

119863 = (119898 119899) 1003817100381710038171003817119871119898119899 (119891) minus 119891

1003817100381710038171003817119862lowast(R2) ge 120576

1198631= (119898 119899)

1003817100381710038171003817119871119898119899 (1198910) minus 11989101003817100381710038171003817119862lowast(R2) ge

120576

3

1198632= (119898 119899) 120596 (119891 120575

119898119899) ge

120576

3 (1 + 1205872)

1198633= (119898 119899)

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) ge

120576

3119872

(42)

Then 119863 sub 1198631cup 1198632cup 1198633 Further define

1198634= (119898 119899) 120596 (119891 120575

119898119899) ge radic

120576

3

1198635= (119898 119899)

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) ge radic

120576

3

(43)

We see that 1198631sub 1198634cup 1198635 Therefore 119863 sub ⋃

5

119894=2119863119894 There-

fore since 120574119898119899

= max120572119898119899 120573119898119899 we conclude that for every

(119895 119896) isin N times N

1205752(119863) le

5

sum119894=2

1205752(119863119894) (44)

Using conditions (i) and (ii) we get 119871119898119899(119891) minus 119891

119862lowast(R2) =

1199041199052(119862 1 1)-119900(120574

119898119899)

Acknowledgment

The authors gratefully acknowledge the financial supportfrom King Abdulaziz University Jeddah Saudi Arabia

References

[1] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951

[2] H Steinhaus ldquoSur la convergence ordinaire et la convergenceasymptotiquerdquo Colloquium Mathematicum vol 2 pp 34ndash731951

[3] F Moricz ldquoStatistical convergence of multiple sequencesrdquoArchiv der Mathematik vol 81 no 1 pp 82ndash89 2003

[4] M Mursaleen and O H H Edely ldquoStatistical convergence ofdouble sequencesrdquo Journal of Mathematical Analysis and Appli-cations vol 288 no 1 pp 223ndash231 2003

[5] F Moricz ldquoTauberian theorems for double sequences that arestatistically summable (119862 1 1)rdquo Journal of Mathematical Ana-lysis and Applications vol 286 no 1 pp 340ndash350 2003

[6] F Altomare ldquoKorovkin-type theorems and approximation bypositive linear operatorsrdquo Surveys in ApproximationTheory vol5 pp 92ndash164 2010

[7] P P Korovkin Linear Operators and Approximation TheoryHindustan New Delhi India 1960

[8] O H H Edely S A Mohiuddine and A K Noman ldquoKorovkintype approximation theorems obtained through generalizedstatistical convergencerdquoAppliedMathematics Letters vol 23 no11 pp 1382ndash1387 2010

[9] R F Patterson andE Savas ldquoKorovkin andWeierstrass approxi-mation via lacunary statistical sequencesrdquo Journal ofMathemat-ics and Statistics vol 1 no 2 pp 165ndash167 2005

[10] M Mursaleen and A Alotaibi ldquoStatistical summability andapproximation by de la Vallee-Poussin meanrdquo Applied Mathe-matics Letters vol 24 no 3 pp 320ndash324 2011 ErratumAppliedMathematics Letters vol 25 p 665 2012

[11] ADGadjiev andCOrhan ldquoSome approximation theorems viastatistical convergencerdquoThe Rocky Mountain Journal of Mathe-matics vol 32 no 1 pp 129ndash138 2002

[12] C Belen M Mursaleen and M Yildirim ldquoStatistical 119860-sum-mability of double sequences and a Korovkin type approxima-tion theoremrdquo Bulletin of the Korean Mathematical Society vol49 no 4 pp 851ndash861 2012

[13] K Demirci and S Karakus ldquoKorovkin-type approximationtheorem for double sequences of positive linear operators viastatistical 119860-summabilityrdquo Results in Mathematics vol 63 no1-2 pp 1ndash13 2013

[14] F Dirik and K Demirci ldquoKorovkin type approximation theo-rem for functions of two variables in statistical senserdquo TurkishJournal of Mathematics vol 34 no 1 pp 73ndash83 2010

[15] K Demirci and F Dirik ldquoFour-dimensional matrix transforma-tion and rate of119860-statistical convergence of periodic functionsrdquoMathematical and Computer Modelling vol 52 no 9-10 pp1858ndash1866 2010

[16] S A Mohiuddine and A Alotaibi ldquoStatistical convergence andapproximation theorems for functions of two variablesrdquo Journalof Computational Analysis and Applications vol 15 no 2 pp218ndash223 2013

[17] G A AnastassiouMMursaleen and S AMohiuddine ldquoSomeapproximation theorems for functions of two variables throughalmost convergence of double sequencesrdquo Journal of Compu-tational Analysis and Applications vol 13 no 1 pp 37ndash46 2011

[18] S A Mohiuddine ldquoAn application of almost convergence inapproximation theoremsrdquo Applied Mathematics Letters vol 24no 11 pp 1856ndash1860 2011

[19] S A Mohiuddine A Alotaibi and M Mursaleen ldquoStatisticalsummability (119862 1) and a Korovkin type approximation theo-remrdquo Journal of Inequalities and Applications vol 2012 article172 2012

[20] M Mursaleen and A Alotaibi ldquoKorovkin type approximationtheorem for functions of two variables through statistical 119860-summabilityrdquoAdvances inDifference Equations vol 2012 article65 2012

[21] M Mursaleen and A Alotaibi ldquoKorovkin type approximationtheorem for statistical 119860-summability of double sequencesrdquoJournal of Computational Analysis and Applications vol 15 no6 pp 1036ndash1045 2013

[22] C Belen and S A Mohiuddine ldquoGeneralized weighted statis-tical convergence and applicationrdquo Applied Mathematics andComputation vol 219 no 18 pp 9821ndash9826 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces and Applications

where

120590119898119899

=1

(119898 + 1) (119899 + 1)

119898

sum119895=0

119899

sum119896=0

119909119895119896 (7)

is the (119862 1 1) mean of 119909 = (119909119895119896) Thus the double se-

quence 119909 is statistically summable (119862 1 1) to 119897 if and onlyif the sequence 120590 = (120590

119898119899) is statistically convergent to ℓ

In this case we write 1199041199052(119862 1 1)-lim

119895119896119909119895119896

= ℓ Note thatif a double sequence is bounded then 119904119905

2-lim119895119896119909119895119896

= ℓ

implies 1199041199052-lim119898119899

120590119898119899

= ℓKorovkin type approximation theorems (cf [6ndash10]) are

useful tools to check whether a given sequence (119871119899)119899ge1

ofpositive linear operators on 119862[0 1] of all continuous func-tions on the real interval [0 1] is an approximation processThat is these theorems exhibit a variety of test functionswhich assure that the approximation property holds on thewhole space if it holds for them Such a property was discov-ered by Korovkin in 1953 for test functions 1 119909 and 1199092 in thespace 119862[0 1] as well as for test functions 1 cos and sin in thespace of all continuous 2120587-periodic functions on the real line

We know that 119862[0 1] is a Banach space with norm

10038171003817100381710038171198911003817100381710038171003817infin = sup

119909isin[01]

1003816100381610038161003816119891 (119909)1003816100381610038161003816 119891 isin 119862 [0 1] (8)

We denote by 1198622120587([0 1]) the space of all 2120587-periodic

functions 119891 isin 119862([0 1]) which is a Banach space with

100381710038171003817100381711989110038171003817100381710038172120587 = sup

119905isin[01]

1003816100381610038161003816119891 (119905)1003816100381610038161003816 (9)

After the paper of Gadjiev and Orhan [11] many papershave appeared in the literature concerning the Korovkin typeapproximation theorems via different statistical summabilitymethods and for different sets of test functions At present weare concerned about applications of such summability meth-ods for double sequences to prove two-dimensional versionofKorovkin theorem For example in [12 13] the authors usedthe notion of statistical 119860-summability of double sequencesin [13ndash16] the authors have used respectively statistical con-vergence and 119860-statistical convergence of double sequencesand in [17 18] the authors used almost summability For somemore related work we refer to [19ndash22]

In this paper we present the Korovkin type approxima-tion theorem for periodic functions via statistical summabil-ity (119862 1 1) and also study the rate of statistical summability(119862 1 1) of a double sequence of positive linear opera-tors defined from 119862

lowast(R2) into 119862lowast(R2)where 119862lowast(R2) is thespace of all 2120587-periodic and real valued continuous functionson R2 equipped with the norm

10038171003817100381710038171198911003817100381710038171003817119862(R2) = sup

(119909119910)isinR2

1003816100381610038161003816119891 (119909 119910)1003816100381610038161003816 (119891 isin 119862 (R

2

)) (10)

2 Main Result

First we state the result due to [15] for 119860-statistical conver-gence of double sequences

Theorem 1 Let (119871119898119899) be a double sequence of positive linear

operators acting from 119862lowast(R2) into 119862lowast(R2) Then for all 119891 isin

119862lowast(R2)

119904119905119860

2- lim119898119899rarrinfin

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) = 0 (11)

if and only if

119904119905119860

2- lim119898119899rarrinfin

1003817100381710038171003817119871119898119899 (119891119894) minus 1198911198941003817100381710038171003817119862lowast(R2) = 0 119894 = 0 1 2 3 4 (12)

where 1198910(119909 119910) = 1 119891

1(119909 119910) = sin119909 119891

2(119909 119910) = sin119910 119891

3(119909

119910) = cos119909 and 1198914(119909 119910) = cos119910

If we replace the matrix 119860 by the identity four-dimen-sional matrix in the above theorem then we immediately getthe following result in Pringsheimrsquos sense

Corollary 2 Let (119871119898119899) be a double sequence of positive linear

operators acting from 119862lowast

(R2) into 119862lowast(R2) Then for all 119891 isin

119862lowast(R2)

119875- lim119898119899rarrinfin

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) = 0 (13)

if and only if

119875- lim119898119899rarrinfin

1003817100381710038171003817119871119898119899 (119891119894) minus 1198911198941003817100381710038171003817119862lowast(R2) = 0 119894 = 0 1 2 3 4 (14)

We prove the following result

Theorem 3 Let (119879119895119896) be a double sequence of positive linear

operators acting from 119862lowast(R2) into 119862lowast(R2) Then for all 119891 isin

119862lowast(R2)

1199041199052(119862 1 1) - lim

119895119896rarrinfin

10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)= 0 (15)

if and only if

1199041199052(119862 1 1) - lim

119895119896rarrinfin

10038171003817100381710038171003817119879119895119896(119891119894) minus 119891119894

10038171003817100381710038171003817119862lowast(R2)= 0

(119894 = 0 1 2 3 4)

(16)

Proof Since each of the functions 1198910 1198911 1198912 1198913 and 119891

4

belongs to 119862lowast(R2) necessity follows immediately from (15)Let condition (16) hold and 119891 isin 119862lowast(R2) Let 119868 and 119869 beclosed subintervals each of length 2120587 of R Fix(119909 119910) isin 119868 times 119869By the continuity of 119891 at (119909 119910) it follows that for given 120576 gt 0

there is a number 120575 gt 0 such that for all (119906 120592) isin R2

1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816 lt 120576 (17)

whenever |119906 minus 119909| |120592 minus 119910| lt 120575 Since 119891 is bounded it followsthat

1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816 le 119872

119891=10038171003817100381710038171198911003817100381710038171003817119862lowast(R2) (18)

for all (119906 120592) isin R2For all (119906 120592) isin (119909 minus 120575 2120587 + 119909 minus 120575] times (119910 minus 120575 2120587 + 119910 minus 120575] it

is well known that

1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816 lt 120576 +

2119872119891

sin2 (1205752)120595 (119906 120592) (19)

Journal of Function Spaces and Applications 3

where 120595(119906 120592) = sin2((119906 minus 119909)2) + sin2((120592 minus 119910)2) Sincethe function 119891 isin 119862lowast(R2) is 2120587-periodic the inequality (19)holds for (119906 120592) isin R2 Then we obtain

10038161003816100381610038161003816119879119895119896(119891 119909 119910) minus 119891 (119909 119910)

10038161003816100381610038161003816

le 119879119895119896(1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)

1003816100381610038161003816 119909 119910)

+1003816100381610038161003816119891 (119909 119910)

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816119879119895119896(120576 +

2119872119891

sin2 (1205752)120595 (119906 120592) 119909 119910)

100381610038161003816100381610038161003816100381610038161003816

+ 119872119891

10038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

le 120576 + ((120576 +119872119891)10038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816)

+119872119891

sin2 (1205752)

times 210038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

+ |sin119909| 10038161003816100381610038161003816119879119895119896 (1198911 119909 119910) minus 1198911 (119909 119910)10038161003816100381610038161003816

+1003816100381610038161003816sin119910

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198912 119909 119910) minus 119891

3(119909 119910)

10038161003816100381610038161003816

+ |cos119909| 10038161003816100381610038161003816119879119895119896 (1198913 119909 119910) minus 1198913 (119909 119910)10038161003816100381610038161003816

+1003816100381610038161003816cos119910

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198914 119909 119910) minus 119891

4(119909 119910)

10038161003816100381610038161003816

lt 120576 + 119870

4

sum119894=0

10038161003816100381610038161003816119879119895119896(119891119894 119909 119910) minus 119891

119894(119909 119910)

10038161003816100381610038161003816

(20)

where 119870 = 120576 + 119872119891+ (2119872

119891sin2(1205752)) Now taking

sup(119909119910)isin119868times119869

we get

10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)lt 120576 + 119870

4

sum119894=0

10038171003817100381710038171003817119879119895119896(119891119894) minus 119891119894

10038171003817100381710038171003817119862lowast(R2) (21)

Now for a given 119903 gt 0 choose 1205761015840 gt 0 such that 1205761015840 lt 119903 Definethe following sets

119863 = (119898 119899) 1003817100381710038171003817119871119898119899 (119891) minus 119891

1003817100381710038171003817119862lowast(R2) ge 119903

119863119894= (119898 119899)

1003817100381710038171003817119871119898119899 (119891119894) minus 1198911198941003817100381710038171003817119862lowast(R2) ge

119903 minus 1205761015840

5119870

(119894 = 0 1 2 3 4)

(22)

where 119871119898119899

= (1(119898 + 1)(119899 + 1))sum119898

119895=0sum119899

119896=0119879119895119896 Then by (21)

119863 sube

4

⋃119894=0

119863119894 (23)

and so

120575(2)

119860(119863) le

4

sum119894=0

120575(2)

119860(119863119894) (24)

Now using (16) we get

1199041199052(119862 1 1) - lim

119895119896rarrinfin

10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)= 0 (25)

Example 4 Now we present an example of double sequencesof positive linear operators showing thatCorollary 2 does notwork but our approximation theoremworksWe consider thedouble sequence of Fejer operators on 119862

lowast(R2)

120590119898119899

(119891 119909 119910) =1

(119899120587)sdot

1

(119899120587)

times int120587

minus120587

int120587

minus120587

119891 (119906 120592) 119865119898(119906) 119865119899(120592) 119889119906 119889120592

(26)

where

119865119898(119906) =

sin2 (119898 (119906 minus 119909) 2)

sin2 ((119906 minus 119909) 2)

1

120587int120587

minus120587

119865119898(119906) 119889119906 = 1 (27)

Observe that

120590119898119899

(1198910 119909 119910) = 119891

0(119909 119910)

120590119898119899

(1198911 119909 119910) =

119898 minus 1

1198981198911(119909 119910)

120590119898119899

(1198912 119909 119910) =

119899 minus 1

1198991198912(119909 119910)

120590119898119899

(1198913 119909 119910) =

119898 minus 1

1198981198913(119909 119910)

120590119898119899

(1198914 119909 119910) =

119899 minus 1

1198991198914(119909 119910)

(28)

Define a double sequence 120572 = (120572119898119899) by 120572

119898119899= (minus1)

119898+119899119898 119899 isin N

We observe that 120572 = (120572119898119899) is neither 119875-convergent nor

statistically convergent but

1199041199052(119862 1 1) -lim120572 = 0 (29)

Let us define the operators 119871119898119899

119862lowast(R2) rarr 119862lowast(R2) by

119871119898119899

(119891 119909 119910) = (1 + 120572119898119899) 120590119898119899

(119891 119909 119910) (30)

Then observe that the double sequence of positive linearoperators (119871

119898119899) defined by (30) satisfies all hypotheses of

Theorem 3 Hence by (28) we have for all 119891 isin 119862lowast(R2)

1199041199052(119862 1 1) - lim

119898119899rarrinfin

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) = 0 (31)

Since (120572119898119899) is neither 119875-convergent nor statistically conver-

gent the sequence (119871119898119899) given by (30) is also neither 119875-

convergent nor statistically convergent to the function 119891 isin

119862lowast(R2) So we conclude that Corollary 2 and Theorem 1do not work for the operators (119871

119898119899) given by (30) while

Theorem 3 still works Hence we conclude that 1199041199052(119862 1 1)-

version is stronger than that of 119875-version as well as statisticalversion

4 Journal of Function Spaces and Applications

3 Rate of Statistical Summability (1198621 1)

Let (120573119898119899) be a positive nonincreasing double sequence We

say that a double sequence 119909 = (119909119898119899) is statistically sum-

mable (119862 1 1) to the number 119871 with the rate 119900(120573119898119899) if for

every 120576 gt 0

119875- lim119898119899rarrinfin

1

120573119898119899

10038161003816100381610038161003816119895 le 119898 119896 le 119899

10038161003816100381610038161003816120590119895119896minus ℓ

10038161003816100381610038161003816ge 120598

10038161003816100381610038161003816= 0 (32)

In this case wewrite 119909119898119899minus119871 = 119904119905

2(119862 1 1)-119900(120573

119898119899) as 119898 119899 rarr

infinNow we recall the notion of modulus of continuity The

modulus of continuity of 119891 isin 119862lowast(R2) denoted by 120596(119891 120575) for120575 gt 0 is defined by

120596 (119891 120575) = sup 1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816 (119906 120592) (119909 119910) isin R

2

radic(119906 minus 119909)2

+ (120592 minus 119910)2

le 120575

(33)

It is well known that

1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816

le 120596 (119891radic(119906 minus 119909)2

+ (120592 minus 119910)2

)

le 120596 (119891 120575)(radic(119906 minus 119909)

2

+ (120592 minus 119910)2

120575+ 1)

(34)

Then we have the following result

Theorem 5 Let (119879119895119896) be a double sequence of positive linear

operators acting from 119862lowast(R2) into 119862lowast(R2) Let (120572119895119896) and (120573

119895119896)

be two positive non-increasing sequences Suppose that

(i) 119879119895119896(1198910) minus 1198910119862lowast(R2)

= 1199041199052(119862 1 1)-119900(120572

119898119899)

(ii) 120596(119891 120582119895119896) = 119904119905

2(119862 1 1) minus 119900(120573

119895119896) where 120582

119898119895119896=

radic 119879119895119896(120593)119862lowast(R2) and

120593 (119906 120592) = sin2 (119906 minus 1199092

) + sin2 (120592 minus 119910

2)

119891119900119903 119890119886119888ℎ (119906 120592) (119909 119910) isin R2

(35)

Then for all 119891 isin 119862lowast(R2)

10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)= 1199041199052(119862 1 1) -119900 (120574

119895119896) (36)

where 120574119895119896= max120572

119895119896 120573119895119896

Proof Let 119891 isin 119862lowast(R2) and (119909 119910) isin minus[120587 120587]times [minus120587 120587] Let 120575 gt

0 we have the following cases

Case I If 120575 lt |119906 minus 119909| le 120587 120575 lt |120592 minus 119910| le 120587 then |119906 minus 119909| le

120587| sin((119906 minus 119909)2)| and |120592 minus 119910| le 120587| sin((120592 minus 119910)2)| Thereforeby (34) we have1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)

1003816100381610038161003816

le 120596 (119891 120575) (1205872sin2 ((119906 minus 119909) 2) + sin2 ((120592 minus 119910) 2)

1205752+ 1)

(37)

Case II If |119906 minus 119909| gt 120587 |120592 minus 119910| le 120587 Let 119896 be an integer suchthat |119906 + 2119896120587 minus 119909| le 120587 then

1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816

=1003816100381610038161003816119891 (119906 + 2119896120587 120592) minus 119891 (119909 119910)

1003816100381610038161003816

le 120596 (119891 120575)

times(1205872sin2 ((119906 + 2119896120587 minus 119909) 2) + sin2 ((120592 minus 119910) 2)

1205752+ 1)

= 120596 (119891 120575) (1205872sin2 ((119906 minus 119909) 2) + sin2 ((120592 minus 119910) 2)

1205752+ 1)

(38)

Similarly in the other two cases when |119906 minus119909| le 120587 |120592 minus119910| gt 120587

and |119906 minus 119909| gt 120587 |120592 minus 119910| gt 120587 we obtain (37)Now using the definition of modulus of continuity and

the linearity and the positivity of the operators 119879119895119896we get

10038161003816100381610038161003816119879119895119896(119891 119909 119910) minus 119891 (119909 119910)

10038161003816100381610038161003816

le 119879119895119896(1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)

1003816100381610038161003816 119909 119910)

+1003816100381610038161003816119891 (119909 119910)

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

le 120596 (119891 120575) 119879119895119896(1198910 119909 119910) + 120587

2120596 (119891 120575)

1205752119879119895119896(120593 119909 119910)

+1003816100381610038161003816119891 (119909 119910)

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

(39)

Taking supremum over (119909 119910) on both sides of the aboveinequality and let

120575 = 120575119895119896= radic

10038171003817100381710038171003817119879119895119896(120593)

10038171003817100381710038171003817119862lowast(R2) (40)

We obtain10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)

le 120596 (119891 120575119895119896)10038171003817100381710038171003817119879119895119896(1198910) minus 1198910

10038171003817100381710038171003817119862lowast(R2)+ (1 + 120587

2

)

times 120596 (119891 120575119895119896) +119872

10038171003817100381710038171003817119879119895119896(1198910) minus 1198910

10038171003817100381710038171003817119862lowast(R2)

(41)

Journal of Function Spaces and Applications 5

where 119872 = 119891119862lowast(R2) Let 119871

119898119899= (1(119898 + 1)(119899 +

1))sum119898

119895=0sum119899

119896=0119879119895119896 Now for a given 120576 gt 0 define the following

sets

119863 = (119898 119899) 1003817100381710038171003817119871119898119899 (119891) minus 119891

1003817100381710038171003817119862lowast(R2) ge 120576

1198631= (119898 119899)

1003817100381710038171003817119871119898119899 (1198910) minus 11989101003817100381710038171003817119862lowast(R2) ge

120576

3

1198632= (119898 119899) 120596 (119891 120575

119898119899) ge

120576

3 (1 + 1205872)

1198633= (119898 119899)

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) ge

120576

3119872

(42)

Then 119863 sub 1198631cup 1198632cup 1198633 Further define

1198634= (119898 119899) 120596 (119891 120575

119898119899) ge radic

120576

3

1198635= (119898 119899)

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) ge radic

120576

3

(43)

We see that 1198631sub 1198634cup 1198635 Therefore 119863 sub ⋃

5

119894=2119863119894 There-

fore since 120574119898119899

= max120572119898119899 120573119898119899 we conclude that for every

(119895 119896) isin N times N

1205752(119863) le

5

sum119894=2

1205752(119863119894) (44)

Using conditions (i) and (ii) we get 119871119898119899(119891) minus 119891

119862lowast(R2) =

1199041199052(119862 1 1)-119900(120574

119898119899)

Acknowledgment

The authors gratefully acknowledge the financial supportfrom King Abdulaziz University Jeddah Saudi Arabia

References

[1] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951

[2] H Steinhaus ldquoSur la convergence ordinaire et la convergenceasymptotiquerdquo Colloquium Mathematicum vol 2 pp 34ndash731951

[3] F Moricz ldquoStatistical convergence of multiple sequencesrdquoArchiv der Mathematik vol 81 no 1 pp 82ndash89 2003

[4] M Mursaleen and O H H Edely ldquoStatistical convergence ofdouble sequencesrdquo Journal of Mathematical Analysis and Appli-cations vol 288 no 1 pp 223ndash231 2003

[5] F Moricz ldquoTauberian theorems for double sequences that arestatistically summable (119862 1 1)rdquo Journal of Mathematical Ana-lysis and Applications vol 286 no 1 pp 340ndash350 2003

[6] F Altomare ldquoKorovkin-type theorems and approximation bypositive linear operatorsrdquo Surveys in ApproximationTheory vol5 pp 92ndash164 2010

[7] P P Korovkin Linear Operators and Approximation TheoryHindustan New Delhi India 1960

[8] O H H Edely S A Mohiuddine and A K Noman ldquoKorovkintype approximation theorems obtained through generalizedstatistical convergencerdquoAppliedMathematics Letters vol 23 no11 pp 1382ndash1387 2010

[9] R F Patterson andE Savas ldquoKorovkin andWeierstrass approxi-mation via lacunary statistical sequencesrdquo Journal ofMathemat-ics and Statistics vol 1 no 2 pp 165ndash167 2005

[10] M Mursaleen and A Alotaibi ldquoStatistical summability andapproximation by de la Vallee-Poussin meanrdquo Applied Mathe-matics Letters vol 24 no 3 pp 320ndash324 2011 ErratumAppliedMathematics Letters vol 25 p 665 2012

[11] ADGadjiev andCOrhan ldquoSome approximation theorems viastatistical convergencerdquoThe Rocky Mountain Journal of Mathe-matics vol 32 no 1 pp 129ndash138 2002

[12] C Belen M Mursaleen and M Yildirim ldquoStatistical 119860-sum-mability of double sequences and a Korovkin type approxima-tion theoremrdquo Bulletin of the Korean Mathematical Society vol49 no 4 pp 851ndash861 2012

[13] K Demirci and S Karakus ldquoKorovkin-type approximationtheorem for double sequences of positive linear operators viastatistical 119860-summabilityrdquo Results in Mathematics vol 63 no1-2 pp 1ndash13 2013

[14] F Dirik and K Demirci ldquoKorovkin type approximation theo-rem for functions of two variables in statistical senserdquo TurkishJournal of Mathematics vol 34 no 1 pp 73ndash83 2010

[15] K Demirci and F Dirik ldquoFour-dimensional matrix transforma-tion and rate of119860-statistical convergence of periodic functionsrdquoMathematical and Computer Modelling vol 52 no 9-10 pp1858ndash1866 2010

[16] S A Mohiuddine and A Alotaibi ldquoStatistical convergence andapproximation theorems for functions of two variablesrdquo Journalof Computational Analysis and Applications vol 15 no 2 pp218ndash223 2013

[17] G A AnastassiouMMursaleen and S AMohiuddine ldquoSomeapproximation theorems for functions of two variables throughalmost convergence of double sequencesrdquo Journal of Compu-tational Analysis and Applications vol 13 no 1 pp 37ndash46 2011

[18] S A Mohiuddine ldquoAn application of almost convergence inapproximation theoremsrdquo Applied Mathematics Letters vol 24no 11 pp 1856ndash1860 2011

[19] S A Mohiuddine A Alotaibi and M Mursaleen ldquoStatisticalsummability (119862 1) and a Korovkin type approximation theo-remrdquo Journal of Inequalities and Applications vol 2012 article172 2012

[20] M Mursaleen and A Alotaibi ldquoKorovkin type approximationtheorem for functions of two variables through statistical 119860-summabilityrdquoAdvances inDifference Equations vol 2012 article65 2012

[21] M Mursaleen and A Alotaibi ldquoKorovkin type approximationtheorem for statistical 119860-summability of double sequencesrdquoJournal of Computational Analysis and Applications vol 15 no6 pp 1036ndash1045 2013

[22] C Belen and S A Mohiuddine ldquoGeneralized weighted statis-tical convergence and applicationrdquo Applied Mathematics andComputation vol 219 no 18 pp 9821ndash9826 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 3

where 120595(119906 120592) = sin2((119906 minus 119909)2) + sin2((120592 minus 119910)2) Sincethe function 119891 isin 119862lowast(R2) is 2120587-periodic the inequality (19)holds for (119906 120592) isin R2 Then we obtain

10038161003816100381610038161003816119879119895119896(119891 119909 119910) minus 119891 (119909 119910)

10038161003816100381610038161003816

le 119879119895119896(1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)

1003816100381610038161003816 119909 119910)

+1003816100381610038161003816119891 (119909 119910)

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816119879119895119896(120576 +

2119872119891

sin2 (1205752)120595 (119906 120592) 119909 119910)

100381610038161003816100381610038161003816100381610038161003816

+ 119872119891

10038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

le 120576 + ((120576 +119872119891)10038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816)

+119872119891

sin2 (1205752)

times 210038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

+ |sin119909| 10038161003816100381610038161003816119879119895119896 (1198911 119909 119910) minus 1198911 (119909 119910)10038161003816100381610038161003816

+1003816100381610038161003816sin119910

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198912 119909 119910) minus 119891

3(119909 119910)

10038161003816100381610038161003816

+ |cos119909| 10038161003816100381610038161003816119879119895119896 (1198913 119909 119910) minus 1198913 (119909 119910)10038161003816100381610038161003816

+1003816100381610038161003816cos119910

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198914 119909 119910) minus 119891

4(119909 119910)

10038161003816100381610038161003816

lt 120576 + 119870

4

sum119894=0

10038161003816100381610038161003816119879119895119896(119891119894 119909 119910) minus 119891

119894(119909 119910)

10038161003816100381610038161003816

(20)

where 119870 = 120576 + 119872119891+ (2119872

119891sin2(1205752)) Now taking

sup(119909119910)isin119868times119869

we get

10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)lt 120576 + 119870

4

sum119894=0

10038171003817100381710038171003817119879119895119896(119891119894) minus 119891119894

10038171003817100381710038171003817119862lowast(R2) (21)

Now for a given 119903 gt 0 choose 1205761015840 gt 0 such that 1205761015840 lt 119903 Definethe following sets

119863 = (119898 119899) 1003817100381710038171003817119871119898119899 (119891) minus 119891

1003817100381710038171003817119862lowast(R2) ge 119903

119863119894= (119898 119899)

1003817100381710038171003817119871119898119899 (119891119894) minus 1198911198941003817100381710038171003817119862lowast(R2) ge

119903 minus 1205761015840

5119870

(119894 = 0 1 2 3 4)

(22)

where 119871119898119899

= (1(119898 + 1)(119899 + 1))sum119898

119895=0sum119899

119896=0119879119895119896 Then by (21)

119863 sube

4

⋃119894=0

119863119894 (23)

and so

120575(2)

119860(119863) le

4

sum119894=0

120575(2)

119860(119863119894) (24)

Now using (16) we get

1199041199052(119862 1 1) - lim

119895119896rarrinfin

10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)= 0 (25)

Example 4 Now we present an example of double sequencesof positive linear operators showing thatCorollary 2 does notwork but our approximation theoremworksWe consider thedouble sequence of Fejer operators on 119862

lowast(R2)

120590119898119899

(119891 119909 119910) =1

(119899120587)sdot

1

(119899120587)

times int120587

minus120587

int120587

minus120587

119891 (119906 120592) 119865119898(119906) 119865119899(120592) 119889119906 119889120592

(26)

where

119865119898(119906) =

sin2 (119898 (119906 minus 119909) 2)

sin2 ((119906 minus 119909) 2)

1

120587int120587

minus120587

119865119898(119906) 119889119906 = 1 (27)

Observe that

120590119898119899

(1198910 119909 119910) = 119891

0(119909 119910)

120590119898119899

(1198911 119909 119910) =

119898 minus 1

1198981198911(119909 119910)

120590119898119899

(1198912 119909 119910) =

119899 minus 1

1198991198912(119909 119910)

120590119898119899

(1198913 119909 119910) =

119898 minus 1

1198981198913(119909 119910)

120590119898119899

(1198914 119909 119910) =

119899 minus 1

1198991198914(119909 119910)

(28)

Define a double sequence 120572 = (120572119898119899) by 120572

119898119899= (minus1)

119898+119899119898 119899 isin N

We observe that 120572 = (120572119898119899) is neither 119875-convergent nor

statistically convergent but

1199041199052(119862 1 1) -lim120572 = 0 (29)

Let us define the operators 119871119898119899

119862lowast(R2) rarr 119862lowast(R2) by

119871119898119899

(119891 119909 119910) = (1 + 120572119898119899) 120590119898119899

(119891 119909 119910) (30)

Then observe that the double sequence of positive linearoperators (119871

119898119899) defined by (30) satisfies all hypotheses of

Theorem 3 Hence by (28) we have for all 119891 isin 119862lowast(R2)

1199041199052(119862 1 1) - lim

119898119899rarrinfin

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) = 0 (31)

Since (120572119898119899) is neither 119875-convergent nor statistically conver-

gent the sequence (119871119898119899) given by (30) is also neither 119875-

convergent nor statistically convergent to the function 119891 isin

119862lowast(R2) So we conclude that Corollary 2 and Theorem 1do not work for the operators (119871

119898119899) given by (30) while

Theorem 3 still works Hence we conclude that 1199041199052(119862 1 1)-

version is stronger than that of 119875-version as well as statisticalversion

4 Journal of Function Spaces and Applications

3 Rate of Statistical Summability (1198621 1)

Let (120573119898119899) be a positive nonincreasing double sequence We

say that a double sequence 119909 = (119909119898119899) is statistically sum-

mable (119862 1 1) to the number 119871 with the rate 119900(120573119898119899) if for

every 120576 gt 0

119875- lim119898119899rarrinfin

1

120573119898119899

10038161003816100381610038161003816119895 le 119898 119896 le 119899

10038161003816100381610038161003816120590119895119896minus ℓ

10038161003816100381610038161003816ge 120598

10038161003816100381610038161003816= 0 (32)

In this case wewrite 119909119898119899minus119871 = 119904119905

2(119862 1 1)-119900(120573

119898119899) as 119898 119899 rarr

infinNow we recall the notion of modulus of continuity The

modulus of continuity of 119891 isin 119862lowast(R2) denoted by 120596(119891 120575) for120575 gt 0 is defined by

120596 (119891 120575) = sup 1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816 (119906 120592) (119909 119910) isin R

2

radic(119906 minus 119909)2

+ (120592 minus 119910)2

le 120575

(33)

It is well known that

1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816

le 120596 (119891radic(119906 minus 119909)2

+ (120592 minus 119910)2

)

le 120596 (119891 120575)(radic(119906 minus 119909)

2

+ (120592 minus 119910)2

120575+ 1)

(34)

Then we have the following result

Theorem 5 Let (119879119895119896) be a double sequence of positive linear

operators acting from 119862lowast(R2) into 119862lowast(R2) Let (120572119895119896) and (120573

119895119896)

be two positive non-increasing sequences Suppose that

(i) 119879119895119896(1198910) minus 1198910119862lowast(R2)

= 1199041199052(119862 1 1)-119900(120572

119898119899)

(ii) 120596(119891 120582119895119896) = 119904119905

2(119862 1 1) minus 119900(120573

119895119896) where 120582

119898119895119896=

radic 119879119895119896(120593)119862lowast(R2) and

120593 (119906 120592) = sin2 (119906 minus 1199092

) + sin2 (120592 minus 119910

2)

119891119900119903 119890119886119888ℎ (119906 120592) (119909 119910) isin R2

(35)

Then for all 119891 isin 119862lowast(R2)

10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)= 1199041199052(119862 1 1) -119900 (120574

119895119896) (36)

where 120574119895119896= max120572

119895119896 120573119895119896

Proof Let 119891 isin 119862lowast(R2) and (119909 119910) isin minus[120587 120587]times [minus120587 120587] Let 120575 gt

0 we have the following cases

Case I If 120575 lt |119906 minus 119909| le 120587 120575 lt |120592 minus 119910| le 120587 then |119906 minus 119909| le

120587| sin((119906 minus 119909)2)| and |120592 minus 119910| le 120587| sin((120592 minus 119910)2)| Thereforeby (34) we have1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)

1003816100381610038161003816

le 120596 (119891 120575) (1205872sin2 ((119906 minus 119909) 2) + sin2 ((120592 minus 119910) 2)

1205752+ 1)

(37)

Case II If |119906 minus 119909| gt 120587 |120592 minus 119910| le 120587 Let 119896 be an integer suchthat |119906 + 2119896120587 minus 119909| le 120587 then

1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816

=1003816100381610038161003816119891 (119906 + 2119896120587 120592) minus 119891 (119909 119910)

1003816100381610038161003816

le 120596 (119891 120575)

times(1205872sin2 ((119906 + 2119896120587 minus 119909) 2) + sin2 ((120592 minus 119910) 2)

1205752+ 1)

= 120596 (119891 120575) (1205872sin2 ((119906 minus 119909) 2) + sin2 ((120592 minus 119910) 2)

1205752+ 1)

(38)

Similarly in the other two cases when |119906 minus119909| le 120587 |120592 minus119910| gt 120587

and |119906 minus 119909| gt 120587 |120592 minus 119910| gt 120587 we obtain (37)Now using the definition of modulus of continuity and

the linearity and the positivity of the operators 119879119895119896we get

10038161003816100381610038161003816119879119895119896(119891 119909 119910) minus 119891 (119909 119910)

10038161003816100381610038161003816

le 119879119895119896(1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)

1003816100381610038161003816 119909 119910)

+1003816100381610038161003816119891 (119909 119910)

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

le 120596 (119891 120575) 119879119895119896(1198910 119909 119910) + 120587

2120596 (119891 120575)

1205752119879119895119896(120593 119909 119910)

+1003816100381610038161003816119891 (119909 119910)

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

(39)

Taking supremum over (119909 119910) on both sides of the aboveinequality and let

120575 = 120575119895119896= radic

10038171003817100381710038171003817119879119895119896(120593)

10038171003817100381710038171003817119862lowast(R2) (40)

We obtain10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)

le 120596 (119891 120575119895119896)10038171003817100381710038171003817119879119895119896(1198910) minus 1198910

10038171003817100381710038171003817119862lowast(R2)+ (1 + 120587

2

)

times 120596 (119891 120575119895119896) +119872

10038171003817100381710038171003817119879119895119896(1198910) minus 1198910

10038171003817100381710038171003817119862lowast(R2)

(41)

Journal of Function Spaces and Applications 5

where 119872 = 119891119862lowast(R2) Let 119871

119898119899= (1(119898 + 1)(119899 +

1))sum119898

119895=0sum119899

119896=0119879119895119896 Now for a given 120576 gt 0 define the following

sets

119863 = (119898 119899) 1003817100381710038171003817119871119898119899 (119891) minus 119891

1003817100381710038171003817119862lowast(R2) ge 120576

1198631= (119898 119899)

1003817100381710038171003817119871119898119899 (1198910) minus 11989101003817100381710038171003817119862lowast(R2) ge

120576

3

1198632= (119898 119899) 120596 (119891 120575

119898119899) ge

120576

3 (1 + 1205872)

1198633= (119898 119899)

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) ge

120576

3119872

(42)

Then 119863 sub 1198631cup 1198632cup 1198633 Further define

1198634= (119898 119899) 120596 (119891 120575

119898119899) ge radic

120576

3

1198635= (119898 119899)

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) ge radic

120576

3

(43)

We see that 1198631sub 1198634cup 1198635 Therefore 119863 sub ⋃

5

119894=2119863119894 There-

fore since 120574119898119899

= max120572119898119899 120573119898119899 we conclude that for every

(119895 119896) isin N times N

1205752(119863) le

5

sum119894=2

1205752(119863119894) (44)

Using conditions (i) and (ii) we get 119871119898119899(119891) minus 119891

119862lowast(R2) =

1199041199052(119862 1 1)-119900(120574

119898119899)

Acknowledgment

The authors gratefully acknowledge the financial supportfrom King Abdulaziz University Jeddah Saudi Arabia

References

[1] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951

[2] H Steinhaus ldquoSur la convergence ordinaire et la convergenceasymptotiquerdquo Colloquium Mathematicum vol 2 pp 34ndash731951

[3] F Moricz ldquoStatistical convergence of multiple sequencesrdquoArchiv der Mathematik vol 81 no 1 pp 82ndash89 2003

[4] M Mursaleen and O H H Edely ldquoStatistical convergence ofdouble sequencesrdquo Journal of Mathematical Analysis and Appli-cations vol 288 no 1 pp 223ndash231 2003

[5] F Moricz ldquoTauberian theorems for double sequences that arestatistically summable (119862 1 1)rdquo Journal of Mathematical Ana-lysis and Applications vol 286 no 1 pp 340ndash350 2003

[6] F Altomare ldquoKorovkin-type theorems and approximation bypositive linear operatorsrdquo Surveys in ApproximationTheory vol5 pp 92ndash164 2010

[7] P P Korovkin Linear Operators and Approximation TheoryHindustan New Delhi India 1960

[8] O H H Edely S A Mohiuddine and A K Noman ldquoKorovkintype approximation theorems obtained through generalizedstatistical convergencerdquoAppliedMathematics Letters vol 23 no11 pp 1382ndash1387 2010

[9] R F Patterson andE Savas ldquoKorovkin andWeierstrass approxi-mation via lacunary statistical sequencesrdquo Journal ofMathemat-ics and Statistics vol 1 no 2 pp 165ndash167 2005

[10] M Mursaleen and A Alotaibi ldquoStatistical summability andapproximation by de la Vallee-Poussin meanrdquo Applied Mathe-matics Letters vol 24 no 3 pp 320ndash324 2011 ErratumAppliedMathematics Letters vol 25 p 665 2012

[11] ADGadjiev andCOrhan ldquoSome approximation theorems viastatistical convergencerdquoThe Rocky Mountain Journal of Mathe-matics vol 32 no 1 pp 129ndash138 2002

[12] C Belen M Mursaleen and M Yildirim ldquoStatistical 119860-sum-mability of double sequences and a Korovkin type approxima-tion theoremrdquo Bulletin of the Korean Mathematical Society vol49 no 4 pp 851ndash861 2012

[13] K Demirci and S Karakus ldquoKorovkin-type approximationtheorem for double sequences of positive linear operators viastatistical 119860-summabilityrdquo Results in Mathematics vol 63 no1-2 pp 1ndash13 2013

[14] F Dirik and K Demirci ldquoKorovkin type approximation theo-rem for functions of two variables in statistical senserdquo TurkishJournal of Mathematics vol 34 no 1 pp 73ndash83 2010

[15] K Demirci and F Dirik ldquoFour-dimensional matrix transforma-tion and rate of119860-statistical convergence of periodic functionsrdquoMathematical and Computer Modelling vol 52 no 9-10 pp1858ndash1866 2010

[16] S A Mohiuddine and A Alotaibi ldquoStatistical convergence andapproximation theorems for functions of two variablesrdquo Journalof Computational Analysis and Applications vol 15 no 2 pp218ndash223 2013

[17] G A AnastassiouMMursaleen and S AMohiuddine ldquoSomeapproximation theorems for functions of two variables throughalmost convergence of double sequencesrdquo Journal of Compu-tational Analysis and Applications vol 13 no 1 pp 37ndash46 2011

[18] S A Mohiuddine ldquoAn application of almost convergence inapproximation theoremsrdquo Applied Mathematics Letters vol 24no 11 pp 1856ndash1860 2011

[19] S A Mohiuddine A Alotaibi and M Mursaleen ldquoStatisticalsummability (119862 1) and a Korovkin type approximation theo-remrdquo Journal of Inequalities and Applications vol 2012 article172 2012

[20] M Mursaleen and A Alotaibi ldquoKorovkin type approximationtheorem for functions of two variables through statistical 119860-summabilityrdquoAdvances inDifference Equations vol 2012 article65 2012

[21] M Mursaleen and A Alotaibi ldquoKorovkin type approximationtheorem for statistical 119860-summability of double sequencesrdquoJournal of Computational Analysis and Applications vol 15 no6 pp 1036ndash1045 2013

[22] C Belen and S A Mohiuddine ldquoGeneralized weighted statis-tical convergence and applicationrdquo Applied Mathematics andComputation vol 219 no 18 pp 9821ndash9826 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces and Applications

3 Rate of Statistical Summability (1198621 1)

Let (120573119898119899) be a positive nonincreasing double sequence We

say that a double sequence 119909 = (119909119898119899) is statistically sum-

mable (119862 1 1) to the number 119871 with the rate 119900(120573119898119899) if for

every 120576 gt 0

119875- lim119898119899rarrinfin

1

120573119898119899

10038161003816100381610038161003816119895 le 119898 119896 le 119899

10038161003816100381610038161003816120590119895119896minus ℓ

10038161003816100381610038161003816ge 120598

10038161003816100381610038161003816= 0 (32)

In this case wewrite 119909119898119899minus119871 = 119904119905

2(119862 1 1)-119900(120573

119898119899) as 119898 119899 rarr

infinNow we recall the notion of modulus of continuity The

modulus of continuity of 119891 isin 119862lowast(R2) denoted by 120596(119891 120575) for120575 gt 0 is defined by

120596 (119891 120575) = sup 1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816 (119906 120592) (119909 119910) isin R

2

radic(119906 minus 119909)2

+ (120592 minus 119910)2

le 120575

(33)

It is well known that

1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816

le 120596 (119891radic(119906 minus 119909)2

+ (120592 minus 119910)2

)

le 120596 (119891 120575)(radic(119906 minus 119909)

2

+ (120592 minus 119910)2

120575+ 1)

(34)

Then we have the following result

Theorem 5 Let (119879119895119896) be a double sequence of positive linear

operators acting from 119862lowast(R2) into 119862lowast(R2) Let (120572119895119896) and (120573

119895119896)

be two positive non-increasing sequences Suppose that

(i) 119879119895119896(1198910) minus 1198910119862lowast(R2)

= 1199041199052(119862 1 1)-119900(120572

119898119899)

(ii) 120596(119891 120582119895119896) = 119904119905

2(119862 1 1) minus 119900(120573

119895119896) where 120582

119898119895119896=

radic 119879119895119896(120593)119862lowast(R2) and

120593 (119906 120592) = sin2 (119906 minus 1199092

) + sin2 (120592 minus 119910

2)

119891119900119903 119890119886119888ℎ (119906 120592) (119909 119910) isin R2

(35)

Then for all 119891 isin 119862lowast(R2)

10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)= 1199041199052(119862 1 1) -119900 (120574

119895119896) (36)

where 120574119895119896= max120572

119895119896 120573119895119896

Proof Let 119891 isin 119862lowast(R2) and (119909 119910) isin minus[120587 120587]times [minus120587 120587] Let 120575 gt

0 we have the following cases

Case I If 120575 lt |119906 minus 119909| le 120587 120575 lt |120592 minus 119910| le 120587 then |119906 minus 119909| le

120587| sin((119906 minus 119909)2)| and |120592 minus 119910| le 120587| sin((120592 minus 119910)2)| Thereforeby (34) we have1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)

1003816100381610038161003816

le 120596 (119891 120575) (1205872sin2 ((119906 minus 119909) 2) + sin2 ((120592 minus 119910) 2)

1205752+ 1)

(37)

Case II If |119906 minus 119909| gt 120587 |120592 minus 119910| le 120587 Let 119896 be an integer suchthat |119906 + 2119896120587 minus 119909| le 120587 then

1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)1003816100381610038161003816

=1003816100381610038161003816119891 (119906 + 2119896120587 120592) minus 119891 (119909 119910)

1003816100381610038161003816

le 120596 (119891 120575)

times(1205872sin2 ((119906 + 2119896120587 minus 119909) 2) + sin2 ((120592 minus 119910) 2)

1205752+ 1)

= 120596 (119891 120575) (1205872sin2 ((119906 minus 119909) 2) + sin2 ((120592 minus 119910) 2)

1205752+ 1)

(38)

Similarly in the other two cases when |119906 minus119909| le 120587 |120592 minus119910| gt 120587

and |119906 minus 119909| gt 120587 |120592 minus 119910| gt 120587 we obtain (37)Now using the definition of modulus of continuity and

the linearity and the positivity of the operators 119879119895119896we get

10038161003816100381610038161003816119879119895119896(119891 119909 119910) minus 119891 (119909 119910)

10038161003816100381610038161003816

le 119879119895119896(1003816100381610038161003816119891 (119906 120592) minus 119891 (119909 119910)

1003816100381610038161003816 119909 119910)

+1003816100381610038161003816119891 (119909 119910)

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

le 120596 (119891 120575) 119879119895119896(1198910 119909 119910) + 120587

2120596 (119891 120575)

1205752119879119895119896(120593 119909 119910)

+1003816100381610038161003816119891 (119909 119910)

100381610038161003816100381610038161003816100381610038161003816119879119895119896(1198910 119909 119910) minus 119891

0(119909 119910)

10038161003816100381610038161003816

(39)

Taking supremum over (119909 119910) on both sides of the aboveinequality and let

120575 = 120575119895119896= radic

10038171003817100381710038171003817119879119895119896(120593)

10038171003817100381710038171003817119862lowast(R2) (40)

We obtain10038171003817100381710038171003817119879119895119896(119891) minus 119891

10038171003817100381710038171003817119862lowast(R2)

le 120596 (119891 120575119895119896)10038171003817100381710038171003817119879119895119896(1198910) minus 1198910

10038171003817100381710038171003817119862lowast(R2)+ (1 + 120587

2

)

times 120596 (119891 120575119895119896) +119872

10038171003817100381710038171003817119879119895119896(1198910) minus 1198910

10038171003817100381710038171003817119862lowast(R2)

(41)

Journal of Function Spaces and Applications 5

where 119872 = 119891119862lowast(R2) Let 119871

119898119899= (1(119898 + 1)(119899 +

1))sum119898

119895=0sum119899

119896=0119879119895119896 Now for a given 120576 gt 0 define the following

sets

119863 = (119898 119899) 1003817100381710038171003817119871119898119899 (119891) minus 119891

1003817100381710038171003817119862lowast(R2) ge 120576

1198631= (119898 119899)

1003817100381710038171003817119871119898119899 (1198910) minus 11989101003817100381710038171003817119862lowast(R2) ge

120576

3

1198632= (119898 119899) 120596 (119891 120575

119898119899) ge

120576

3 (1 + 1205872)

1198633= (119898 119899)

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) ge

120576

3119872

(42)

Then 119863 sub 1198631cup 1198632cup 1198633 Further define

1198634= (119898 119899) 120596 (119891 120575

119898119899) ge radic

120576

3

1198635= (119898 119899)

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) ge radic

120576

3

(43)

We see that 1198631sub 1198634cup 1198635 Therefore 119863 sub ⋃

5

119894=2119863119894 There-

fore since 120574119898119899

= max120572119898119899 120573119898119899 we conclude that for every

(119895 119896) isin N times N

1205752(119863) le

5

sum119894=2

1205752(119863119894) (44)

Using conditions (i) and (ii) we get 119871119898119899(119891) minus 119891

119862lowast(R2) =

1199041199052(119862 1 1)-119900(120574

119898119899)

Acknowledgment

The authors gratefully acknowledge the financial supportfrom King Abdulaziz University Jeddah Saudi Arabia

References

[1] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951

[2] H Steinhaus ldquoSur la convergence ordinaire et la convergenceasymptotiquerdquo Colloquium Mathematicum vol 2 pp 34ndash731951

[3] F Moricz ldquoStatistical convergence of multiple sequencesrdquoArchiv der Mathematik vol 81 no 1 pp 82ndash89 2003

[4] M Mursaleen and O H H Edely ldquoStatistical convergence ofdouble sequencesrdquo Journal of Mathematical Analysis and Appli-cations vol 288 no 1 pp 223ndash231 2003

[5] F Moricz ldquoTauberian theorems for double sequences that arestatistically summable (119862 1 1)rdquo Journal of Mathematical Ana-lysis and Applications vol 286 no 1 pp 340ndash350 2003

[6] F Altomare ldquoKorovkin-type theorems and approximation bypositive linear operatorsrdquo Surveys in ApproximationTheory vol5 pp 92ndash164 2010

[7] P P Korovkin Linear Operators and Approximation TheoryHindustan New Delhi India 1960

[8] O H H Edely S A Mohiuddine and A K Noman ldquoKorovkintype approximation theorems obtained through generalizedstatistical convergencerdquoAppliedMathematics Letters vol 23 no11 pp 1382ndash1387 2010

[9] R F Patterson andE Savas ldquoKorovkin andWeierstrass approxi-mation via lacunary statistical sequencesrdquo Journal ofMathemat-ics and Statistics vol 1 no 2 pp 165ndash167 2005

[10] M Mursaleen and A Alotaibi ldquoStatistical summability andapproximation by de la Vallee-Poussin meanrdquo Applied Mathe-matics Letters vol 24 no 3 pp 320ndash324 2011 ErratumAppliedMathematics Letters vol 25 p 665 2012

[11] ADGadjiev andCOrhan ldquoSome approximation theorems viastatistical convergencerdquoThe Rocky Mountain Journal of Mathe-matics vol 32 no 1 pp 129ndash138 2002

[12] C Belen M Mursaleen and M Yildirim ldquoStatistical 119860-sum-mability of double sequences and a Korovkin type approxima-tion theoremrdquo Bulletin of the Korean Mathematical Society vol49 no 4 pp 851ndash861 2012

[13] K Demirci and S Karakus ldquoKorovkin-type approximationtheorem for double sequences of positive linear operators viastatistical 119860-summabilityrdquo Results in Mathematics vol 63 no1-2 pp 1ndash13 2013

[14] F Dirik and K Demirci ldquoKorovkin type approximation theo-rem for functions of two variables in statistical senserdquo TurkishJournal of Mathematics vol 34 no 1 pp 73ndash83 2010

[15] K Demirci and F Dirik ldquoFour-dimensional matrix transforma-tion and rate of119860-statistical convergence of periodic functionsrdquoMathematical and Computer Modelling vol 52 no 9-10 pp1858ndash1866 2010

[16] S A Mohiuddine and A Alotaibi ldquoStatistical convergence andapproximation theorems for functions of two variablesrdquo Journalof Computational Analysis and Applications vol 15 no 2 pp218ndash223 2013

[17] G A AnastassiouMMursaleen and S AMohiuddine ldquoSomeapproximation theorems for functions of two variables throughalmost convergence of double sequencesrdquo Journal of Compu-tational Analysis and Applications vol 13 no 1 pp 37ndash46 2011

[18] S A Mohiuddine ldquoAn application of almost convergence inapproximation theoremsrdquo Applied Mathematics Letters vol 24no 11 pp 1856ndash1860 2011

[19] S A Mohiuddine A Alotaibi and M Mursaleen ldquoStatisticalsummability (119862 1) and a Korovkin type approximation theo-remrdquo Journal of Inequalities and Applications vol 2012 article172 2012

[20] M Mursaleen and A Alotaibi ldquoKorovkin type approximationtheorem for functions of two variables through statistical 119860-summabilityrdquoAdvances inDifference Equations vol 2012 article65 2012

[21] M Mursaleen and A Alotaibi ldquoKorovkin type approximationtheorem for statistical 119860-summability of double sequencesrdquoJournal of Computational Analysis and Applications vol 15 no6 pp 1036ndash1045 2013

[22] C Belen and S A Mohiuddine ldquoGeneralized weighted statis-tical convergence and applicationrdquo Applied Mathematics andComputation vol 219 no 18 pp 9821ndash9826 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 5

where 119872 = 119891119862lowast(R2) Let 119871

119898119899= (1(119898 + 1)(119899 +

1))sum119898

119895=0sum119899

119896=0119879119895119896 Now for a given 120576 gt 0 define the following

sets

119863 = (119898 119899) 1003817100381710038171003817119871119898119899 (119891) minus 119891

1003817100381710038171003817119862lowast(R2) ge 120576

1198631= (119898 119899)

1003817100381710038171003817119871119898119899 (1198910) minus 11989101003817100381710038171003817119862lowast(R2) ge

120576

3

1198632= (119898 119899) 120596 (119891 120575

119898119899) ge

120576

3 (1 + 1205872)

1198633= (119898 119899)

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) ge

120576

3119872

(42)

Then 119863 sub 1198631cup 1198632cup 1198633 Further define

1198634= (119898 119899) 120596 (119891 120575

119898119899) ge radic

120576

3

1198635= (119898 119899)

1003817100381710038171003817119871119898119899 (119891) minus 1198911003817100381710038171003817119862lowast(R2) ge radic

120576

3

(43)

We see that 1198631sub 1198634cup 1198635 Therefore 119863 sub ⋃

5

119894=2119863119894 There-

fore since 120574119898119899

= max120572119898119899 120573119898119899 we conclude that for every

(119895 119896) isin N times N

1205752(119863) le

5

sum119894=2

1205752(119863119894) (44)

Using conditions (i) and (ii) we get 119871119898119899(119891) minus 119891

119862lowast(R2) =

1199041199052(119862 1 1)-119900(120574

119898119899)

Acknowledgment

The authors gratefully acknowledge the financial supportfrom King Abdulaziz University Jeddah Saudi Arabia

References

[1] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951

[2] H Steinhaus ldquoSur la convergence ordinaire et la convergenceasymptotiquerdquo Colloquium Mathematicum vol 2 pp 34ndash731951

[3] F Moricz ldquoStatistical convergence of multiple sequencesrdquoArchiv der Mathematik vol 81 no 1 pp 82ndash89 2003

[4] M Mursaleen and O H H Edely ldquoStatistical convergence ofdouble sequencesrdquo Journal of Mathematical Analysis and Appli-cations vol 288 no 1 pp 223ndash231 2003

[5] F Moricz ldquoTauberian theorems for double sequences that arestatistically summable (119862 1 1)rdquo Journal of Mathematical Ana-lysis and Applications vol 286 no 1 pp 340ndash350 2003

[6] F Altomare ldquoKorovkin-type theorems and approximation bypositive linear operatorsrdquo Surveys in ApproximationTheory vol5 pp 92ndash164 2010

[7] P P Korovkin Linear Operators and Approximation TheoryHindustan New Delhi India 1960

[8] O H H Edely S A Mohiuddine and A K Noman ldquoKorovkintype approximation theorems obtained through generalizedstatistical convergencerdquoAppliedMathematics Letters vol 23 no11 pp 1382ndash1387 2010

[9] R F Patterson andE Savas ldquoKorovkin andWeierstrass approxi-mation via lacunary statistical sequencesrdquo Journal ofMathemat-ics and Statistics vol 1 no 2 pp 165ndash167 2005

[10] M Mursaleen and A Alotaibi ldquoStatistical summability andapproximation by de la Vallee-Poussin meanrdquo Applied Mathe-matics Letters vol 24 no 3 pp 320ndash324 2011 ErratumAppliedMathematics Letters vol 25 p 665 2012

[11] ADGadjiev andCOrhan ldquoSome approximation theorems viastatistical convergencerdquoThe Rocky Mountain Journal of Mathe-matics vol 32 no 1 pp 129ndash138 2002

[12] C Belen M Mursaleen and M Yildirim ldquoStatistical 119860-sum-mability of double sequences and a Korovkin type approxima-tion theoremrdquo Bulletin of the Korean Mathematical Society vol49 no 4 pp 851ndash861 2012

[13] K Demirci and S Karakus ldquoKorovkin-type approximationtheorem for double sequences of positive linear operators viastatistical 119860-summabilityrdquo Results in Mathematics vol 63 no1-2 pp 1ndash13 2013

[14] F Dirik and K Demirci ldquoKorovkin type approximation theo-rem for functions of two variables in statistical senserdquo TurkishJournal of Mathematics vol 34 no 1 pp 73ndash83 2010

[15] K Demirci and F Dirik ldquoFour-dimensional matrix transforma-tion and rate of119860-statistical convergence of periodic functionsrdquoMathematical and Computer Modelling vol 52 no 9-10 pp1858ndash1866 2010

[16] S A Mohiuddine and A Alotaibi ldquoStatistical convergence andapproximation theorems for functions of two variablesrdquo Journalof Computational Analysis and Applications vol 15 no 2 pp218ndash223 2013

[17] G A AnastassiouMMursaleen and S AMohiuddine ldquoSomeapproximation theorems for functions of two variables throughalmost convergence of double sequencesrdquo Journal of Compu-tational Analysis and Applications vol 13 no 1 pp 37ndash46 2011

[18] S A Mohiuddine ldquoAn application of almost convergence inapproximation theoremsrdquo Applied Mathematics Letters vol 24no 11 pp 1856ndash1860 2011

[19] S A Mohiuddine A Alotaibi and M Mursaleen ldquoStatisticalsummability (119862 1) and a Korovkin type approximation theo-remrdquo Journal of Inequalities and Applications vol 2012 article172 2012

[20] M Mursaleen and A Alotaibi ldquoKorovkin type approximationtheorem for functions of two variables through statistical 119860-summabilityrdquoAdvances inDifference Equations vol 2012 article65 2012

[21] M Mursaleen and A Alotaibi ldquoKorovkin type approximationtheorem for statistical 119860-summability of double sequencesrdquoJournal of Computational Analysis and Applications vol 15 no6 pp 1036ndash1045 2013

[22] C Belen and S A Mohiuddine ldquoGeneralized weighted statis-tical convergence and applicationrdquo Applied Mathematics andComputation vol 219 no 18 pp 9821ndash9826 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of