Approximation for periodic functions via weighted statistical convergence
Applied Mathematics and Computation 219 (2013) 8231–8236
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Approximation for periodic functions via weighted statistical convergence Osama H.H. Edely a, M. Mursaleen b,⇑, Asif Khan b a b
Department of Mathematics and Computer, Tafila Technical University, Jordan Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
a r t i c l e
i n f o
Keywords: Density Statistical convergence Weighted statistical convergence Positive linear operator Korovkin type approximation theorem
a b s t r a c t Korovkin type approximation theorems are useful tools to check whether a given sequence ðLn ÞnP1 of positive linear operators on C½0; 1 of all continuous functions on the real interval ½0; 1 is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1; x and x2 in the space C½0; 1 as well as for the functions 1, cos and sin in the space of all continuous 2p-periodic functions on the real line. In this paper, we use the notion of weighted statistical convergence to prove the Korovkin approximation theorem for the functions 1, cos and sin in the space of all continuous 2p-periodic functions on the real line and show that our result is stronger. We also study the rate of weighted statistical convergence. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction and preliminaries Let N be the set of all natural numbers and K # N and K n ¼ fk 6 n : k 2 K g. Then the natural density of K is defined by dðKÞ ¼ limn n1 jK n j if the limit exists, where the vertical bars indicate the number of elements in the enclosed set. The sequence x ¼ ðxk Þ is said to be statistically convergent to L if for every e > 0, the set K e :¼ fk 2 N : jxk Lj P eg has natural density zero (cf. Fast [6]), i.e. for each e > 0,
1 lim jfj 6 n : jxj Lj P egj ¼ 0: n n In this case, we write L ¼ st-lim x. Note that every convergent sequence is statistically convergent but not conversely. Recently, Karakaya and Chishti [7] has defined the concept of weighted statistial convergence which was further modified/corrected in [15]. Pn Let p ¼ ðpk Þ1 k¼0 be a sequence of nonnegative numbers such that p0 > 0 and P n ¼ k¼0 pk ! 1 as n ! 1. Set Pn 1 t n ¼ Pn k¼0 pk xk ; n ¼ 0; 1; 2; . . .. We define the weighted density of K by dN ðKÞ ¼ limn P1n jK n j if the limit exists. We say that the sequence x ¼ ðxk Þ is weighted statistically convergent (or SN –convergent) to L if for every e > 0, the set fk 2 N : pk jxk Lj P g has weighted density zero, i.e.
lim n
1 jfk 6 Pn : pk jxk Lj P gj ¼ 0: Pn
⇑ Corresponding author. E-mail addresses: [email protected] (O.H.H. Edely), [email protected], [email protected] (M. Mursaleen), [email protected] (A. Khan). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.02.024
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O.H.H. Edely et al. / Applied Mathematics and Computation 219 (2013) 8231–8236
In this case, we write L ¼ SN -lim x. Let x ¼ ðxk Þ be a sequence defined by
xk ¼
( pffiffiffi k if k ¼ n2 ; n 2 N; if k – n2 :
0
ð1:1:1Þ
That is ðxk Þ ¼ ð1; 0; 0; 2; 0; 0; 0; 0; 3; 0; . . . ; 0; 4; 0; 0; . . .Þ. Let pk ¼ k. Then pk xk ¼ ð1; 0; 0; 8; 0; 0; 0; 0; 27; 0; . . . ; 0; 64; 0; 0; . . .Þ. Since
1 1 pffiffiffiffiffi lim fk 6 P n : pk jxk 0j P eg6 lim Pn ! 0; n Pn Pn
ðxk Þ is weighted statistical convergent to 0 but not convergent. Let FðRÞ denote the linear space of all real-valued functions defined on R. Let CðRÞ be the space of all functions f continuous on R. We know that CðRÞ is a Banach space with norm
kf k1 :¼ sup jf ðxÞj;
f 2 CðRÞ:
x2R
We denote by C 2p ðRÞ the space of all 2p-periodic functions f 2 CðRÞ which is a Banach spaces with
kf k2p ¼ sup jf ðtÞj: t2R
The classical Korovkin first and second theorems state as follows [9,10]: Theorem I. Let ðT n Þ be a sequence of positive linear operators from C½0; 1 into F½0; 1. Then limn kT n ðf ; xÞ f ðxÞk1 ¼ 0, for all f 2 C½0; 1 if and only if limn kT n ðfi ; xÞ fi ðxÞk1 ¼ 0, for i ¼ 0; 1; 2, where f0 ðxÞ ¼ 1; f 1 ðxÞ ¼ x and f2 ðxÞ ¼ x2 . Theorem II. Let ðT n Þ be a sequence of positive linear operators from C 2p ðRÞ into FðRÞ. Then limn kT n ðf ; xÞ f ðxÞk1 ¼ 0, for all f 2 C 2p ðRÞ if and only if limn kT n ðfi ; xÞ fi ðxÞk1 ¼ 0, for i ¼ 0; 1; 2, where f0 ðxÞ ¼ 1; f 1 ðxÞ ¼ cos x and f2 ðxÞ ¼ sin x. Several mathematicians have worked on extending or generalizing the Korovkin’s theorems in many ways and to several settings, including function spaces, abstract Banach lattices, Banach algebras, Banach spaces and so on. This theory is very useful in real analysis, functional analysis, harmonic analysis, measure theory, probability theory, summability theory and partial differential equations. But the foremost applications are concerned with constructive approximation theory which uses it as a valuable tool. Even today, the development of Korovkin-type approximation theory is far from complete. Note that the first and the second theorems of Korovkin are actually equivalent to the algebraic and the trigonometric version, respectively, of the classical Weierstrass approximation theorem [1]. Recently, the Korovkin second theorem is proved in [4] by using the concept of statistical convergence. Quite recently, Korovkin second theorem is proved by Demirci and Dirik [3] for statistical r-convergence [12]. For some recent work on this topic, we refer to [2,5,8,11,13–17]. In this paper, we prove Korovkin second theorem by applying the notion of weighted statistical convergence. We also give an example to justify that our result is stronger than Theorem II. 2. Main result We write Ln ðf ; xÞ for Ln ðf ðsÞ; xÞ; and we say that L is a positive operator if Lðf ; xÞ P 0 for all f ðxÞ P 0. Theorem 2.1. Let ðT k Þ be a sequence of positive linear operators from C 2p ðRÞ into C 2p ðRÞ. Then for all f 2 C 2p ðRÞ
SN lim kT k ðf ; xÞ f ðxÞk2p ¼ 0; k!1
ð2:1:0Þ
if and only if
SN lim kT k ð1; xÞ 1k2p ¼ 0;
ð2:1:1Þ
SN lim kT k ðcos t; xÞ cos xk2p ¼ 0;
ð2:1:2Þ
SN lim kT k ðsin t; xÞ sin xk2p ¼ 0:
ð2:1:3Þ
k!1
k!1
k!1
Proof. Since each of f0 ; f 1 ; f 2 belongs to C 2p ðRÞ, conditions (2.1.1)–(2.1.3) follow immediately from (2.1.0). Let the conditions (2.1.1)–(2.1.3) hold and f 2 C 2p ðRÞ. Let I be a closed subinterval of length 2p of R. Fix x 2 I. By the continuity of f at x, it follows that for given e > 0 there is a number d > 0 such that for all t
jf ðtÞ f ðxÞj < e;
ð2:1:4Þ
O.H.H. Edely et al. / Applied Mathematics and Computation 219 (2013) 8231–8236
8233
whenever jt xj < d. Since f is bounded, it follows that
jf ðtÞ f ðxÞj 6 2kf k2p ;
ð2:1:5Þ
for all t 2 R. For all t 2 ðx d; 2p þ x d, it is well-known that
jf ðtÞ f ðxÞj < e þ
2kf k2p 2 d 2
sin
ð2:1:6Þ
wðtÞ;
2
where wðtÞ ¼ sin ðtx Þ: Since the function f 2 C 2p ðRÞ is 2p-periodic, the inequality (2.1.6) holds for t 2 R. 2 Now, operating T k ð1; xÞ to this inequality, we obtain
jT k ðf ; xÞ f ðxÞj 6 ðe þ jf ðxÞjÞjT k ð1; xÞ 1j þ e þ
kf k2p 2 d 2
sin
fjT k ð1; xÞ 1j þ j cos xjjT k ðcos t; xÞ cos xj
þ j sin xjjT k ðsin t; xÞ sin xjg 6 e þ ðe þ jf ðxÞj þ
kf k2p 2 d 2
sin
ÞfjT k ð1; xÞ 1j þ jT k ðcos t; xÞ cos xj þ jT k ðsin t; xÞ sin xjg:
ð2:1:7Þ
Now, taking supx2I , we get
kT k ðf ; xÞ f ðxÞk2p 6 e þ K kT k ð1; xÞ 1k2p þ kT k ðcos t; xÞ cos xk2p þ kT k ðsin t; xÞ sin xk2p ; where K :¼
e þ kf k2p þ kfsink22pd . Hence
2 kT k ðf ; xÞpk f ðxÞk2p 6 e þ K kT k ð1; xÞpk 1k2p þ kT k ðcos t; xÞpk cos xk2p þ kT k ðsin t; xÞpk sin xk2p :
For a given r > 0 choose
ð2:1:8Þ
e > 0 such that e < r . Define the following sets 0
0
D ¼ fk 6 n : jjT k ðf ; xÞpk f ðxÞjj2p P rg; D1 ¼
r e0 ; k 6 n : jjT k ð1; xÞpk 1jj2p P 4K
D2 ¼
r e0 k 6 n : jjT k ðcos t; xÞpk cos xjj2p P ; 4K
D3 ¼
r e0 k 6 n : jjT k ðsin t; xÞpk sin xjj2p P : 4K
Then
D D1 [ D2 [ D3 and so
dN ðDÞ 6 dN ðD1 Þ þ dN ðD2 Þ þ dN ðD3 Þ: Therefore, using conditions (2.1.1)–(2.1.3), we get
SN —limjjT n ðf ; xÞ f ðxÞjj2p ¼ 0: n
This completes the proof of the theorem.
3. Rate of weighted statistical convergence In this section, we study the rate of weighted statistical convergence of a sequence of positive linear operators defined from C 2p ðRÞ into C 2p ðRÞ. Definition 3.1. Let ðan Þ be a positive non-increasing sequence. We say that the sequence x ¼ ðxk Þ is weighted statistically convergent to the number L with the rate oðan Þ if for every e > 0,
lim n
1 jfk 6 Pn : pk jxk Lj P egj ¼ 0: an Pn
In this case, we write xk L ¼ SN –oðan Þ. As usual we have the following auxiliary result whose proof is standard.
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Lemma 3.1. Let ðan Þ and ðbn Þ be two positive non-increasing sequences. Let x ¼ ðxk Þ and y ¼ ðyk Þ be two sequences such that xk L1 ¼ SN –oðan Þ and yk L2 ¼ SN –oðbn Þ. Then (i) aðxk L1 Þ ¼ SN –oðan Þ, for any scalar a, (ii) ðxk L1 Þ ðyk L2 Þ ¼ SN –oðcn Þ, (iii) ðxk L1 Þðyk L2 Þ ¼ SN –oðan bn Þ, where cn ¼ maxfan ; bn g. Now, we recall the notion of modulus of continuity. The modulus of continuity of f 2 C 2p ðRÞ, denoted by xðf ; dÞ is defined by
xðf ; dÞ ¼ sup jf ðxÞ f ðyÞj: jxyj
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Approximation for periodic functions via weighted statistical convergence Osama H.H. Edely a, M. Mursaleen b,⇑, Asif Khan b a b
Department of Mathematics and Computer, Tafila Technical University, Jordan Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
a r t i c l e
i n f o
Keywords: Density Statistical convergence Weighted statistical convergence Positive linear operator Korovkin type approximation theorem
a b s t r a c t Korovkin type approximation theorems are useful tools to check whether a given sequence ðLn ÞnP1 of positive linear operators on C½0; 1 of all continuous functions on the real interval ½0; 1 is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1; x and x2 in the space C½0; 1 as well as for the functions 1, cos and sin in the space of all continuous 2p-periodic functions on the real line. In this paper, we use the notion of weighted statistical convergence to prove the Korovkin approximation theorem for the functions 1, cos and sin in the space of all continuous 2p-periodic functions on the real line and show that our result is stronger. We also study the rate of weighted statistical convergence. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction and preliminaries Let N be the set of all natural numbers and K # N and K n ¼ fk 6 n : k 2 K g. Then the natural density of K is defined by dðKÞ ¼ limn n1 jK n j if the limit exists, where the vertical bars indicate the number of elements in the enclosed set. The sequence x ¼ ðxk Þ is said to be statistically convergent to L if for every e > 0, the set K e :¼ fk 2 N : jxk Lj P eg has natural density zero (cf. Fast [6]), i.e. for each e > 0,
1 lim jfj 6 n : jxj Lj P egj ¼ 0: n n In this case, we write L ¼ st-lim x. Note that every convergent sequence is statistically convergent but not conversely. Recently, Karakaya and Chishti [7] has defined the concept of weighted statistial convergence which was further modified/corrected in [15]. Pn Let p ¼ ðpk Þ1 k¼0 be a sequence of nonnegative numbers such that p0 > 0 and P n ¼ k¼0 pk ! 1 as n ! 1. Set Pn 1 t n ¼ Pn k¼0 pk xk ; n ¼ 0; 1; 2; . . .. We define the weighted density of K by dN ðKÞ ¼ limn P1n jK n j if the limit exists. We say that the sequence x ¼ ðxk Þ is weighted statistically convergent (or SN –convergent) to L if for every e > 0, the set fk 2 N : pk jxk Lj P g has weighted density zero, i.e.
lim n
1 jfk 6 Pn : pk jxk Lj P gj ¼ 0: Pn
⇑ Corresponding author. E-mail addresses: [email protected] (O.H.H. Edely), [email protected], [email protected] (M. Mursaleen), [email protected] (A. Khan). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.02.024
8232
O.H.H. Edely et al. / Applied Mathematics and Computation 219 (2013) 8231–8236
In this case, we write L ¼ SN -lim x. Let x ¼ ðxk Þ be a sequence defined by
xk ¼
( pffiffiffi k if k ¼ n2 ; n 2 N; if k – n2 :
0
ð1:1:1Þ
That is ðxk Þ ¼ ð1; 0; 0; 2; 0; 0; 0; 0; 3; 0; . . . ; 0; 4; 0; 0; . . .Þ. Let pk ¼ k. Then pk xk ¼ ð1; 0; 0; 8; 0; 0; 0; 0; 27; 0; . . . ; 0; 64; 0; 0; . . .Þ. Since
1 1 pffiffiffiffiffi lim fk 6 P n : pk jxk 0j P eg6 lim Pn ! 0; n Pn Pn
ðxk Þ is weighted statistical convergent to 0 but not convergent. Let FðRÞ denote the linear space of all real-valued functions defined on R. Let CðRÞ be the space of all functions f continuous on R. We know that CðRÞ is a Banach space with norm
kf k1 :¼ sup jf ðxÞj;
f 2 CðRÞ:
x2R
We denote by C 2p ðRÞ the space of all 2p-periodic functions f 2 CðRÞ which is a Banach spaces with
kf k2p ¼ sup jf ðtÞj: t2R
The classical Korovkin first and second theorems state as follows [9,10]: Theorem I. Let ðT n Þ be a sequence of positive linear operators from C½0; 1 into F½0; 1. Then limn kT n ðf ; xÞ f ðxÞk1 ¼ 0, for all f 2 C½0; 1 if and only if limn kT n ðfi ; xÞ fi ðxÞk1 ¼ 0, for i ¼ 0; 1; 2, where f0 ðxÞ ¼ 1; f 1 ðxÞ ¼ x and f2 ðxÞ ¼ x2 . Theorem II. Let ðT n Þ be a sequence of positive linear operators from C 2p ðRÞ into FðRÞ. Then limn kT n ðf ; xÞ f ðxÞk1 ¼ 0, for all f 2 C 2p ðRÞ if and only if limn kT n ðfi ; xÞ fi ðxÞk1 ¼ 0, for i ¼ 0; 1; 2, where f0 ðxÞ ¼ 1; f 1 ðxÞ ¼ cos x and f2 ðxÞ ¼ sin x. Several mathematicians have worked on extending or generalizing the Korovkin’s theorems in many ways and to several settings, including function spaces, abstract Banach lattices, Banach algebras, Banach spaces and so on. This theory is very useful in real analysis, functional analysis, harmonic analysis, measure theory, probability theory, summability theory and partial differential equations. But the foremost applications are concerned with constructive approximation theory which uses it as a valuable tool. Even today, the development of Korovkin-type approximation theory is far from complete. Note that the first and the second theorems of Korovkin are actually equivalent to the algebraic and the trigonometric version, respectively, of the classical Weierstrass approximation theorem [1]. Recently, the Korovkin second theorem is proved in [4] by using the concept of statistical convergence. Quite recently, Korovkin second theorem is proved by Demirci and Dirik [3] for statistical r-convergence [12]. For some recent work on this topic, we refer to [2,5,8,11,13–17]. In this paper, we prove Korovkin second theorem by applying the notion of weighted statistical convergence. We also give an example to justify that our result is stronger than Theorem II. 2. Main result We write Ln ðf ; xÞ for Ln ðf ðsÞ; xÞ; and we say that L is a positive operator if Lðf ; xÞ P 0 for all f ðxÞ P 0. Theorem 2.1. Let ðT k Þ be a sequence of positive linear operators from C 2p ðRÞ into C 2p ðRÞ. Then for all f 2 C 2p ðRÞ
SN lim kT k ðf ; xÞ f ðxÞk2p ¼ 0; k!1
ð2:1:0Þ
if and only if
SN lim kT k ð1; xÞ 1k2p ¼ 0;
ð2:1:1Þ
SN lim kT k ðcos t; xÞ cos xk2p ¼ 0;
ð2:1:2Þ
SN lim kT k ðsin t; xÞ sin xk2p ¼ 0:
ð2:1:3Þ
k!1
k!1
k!1
Proof. Since each of f0 ; f 1 ; f 2 belongs to C 2p ðRÞ, conditions (2.1.1)–(2.1.3) follow immediately from (2.1.0). Let the conditions (2.1.1)–(2.1.3) hold and f 2 C 2p ðRÞ. Let I be a closed subinterval of length 2p of R. Fix x 2 I. By the continuity of f at x, it follows that for given e > 0 there is a number d > 0 such that for all t
jf ðtÞ f ðxÞj < e;
ð2:1:4Þ
O.H.H. Edely et al. / Applied Mathematics and Computation 219 (2013) 8231–8236
8233
whenever jt xj < d. Since f is bounded, it follows that
jf ðtÞ f ðxÞj 6 2kf k2p ;
ð2:1:5Þ
for all t 2 R. For all t 2 ðx d; 2p þ x d, it is well-known that
jf ðtÞ f ðxÞj < e þ
2kf k2p 2 d 2
sin
ð2:1:6Þ
wðtÞ;
2
where wðtÞ ¼ sin ðtx Þ: Since the function f 2 C 2p ðRÞ is 2p-periodic, the inequality (2.1.6) holds for t 2 R. 2 Now, operating T k ð1; xÞ to this inequality, we obtain
jT k ðf ; xÞ f ðxÞj 6 ðe þ jf ðxÞjÞjT k ð1; xÞ 1j þ e þ
kf k2p 2 d 2
sin
fjT k ð1; xÞ 1j þ j cos xjjT k ðcos t; xÞ cos xj
þ j sin xjjT k ðsin t; xÞ sin xjg 6 e þ ðe þ jf ðxÞj þ
kf k2p 2 d 2
sin
ÞfjT k ð1; xÞ 1j þ jT k ðcos t; xÞ cos xj þ jT k ðsin t; xÞ sin xjg:
ð2:1:7Þ
Now, taking supx2I , we get
kT k ðf ; xÞ f ðxÞk2p 6 e þ K kT k ð1; xÞ 1k2p þ kT k ðcos t; xÞ cos xk2p þ kT k ðsin t; xÞ sin xk2p ; where K :¼
e þ kf k2p þ kfsink22pd . Hence
2 kT k ðf ; xÞpk f ðxÞk2p 6 e þ K kT k ð1; xÞpk 1k2p þ kT k ðcos t; xÞpk cos xk2p þ kT k ðsin t; xÞpk sin xk2p :
For a given r > 0 choose
ð2:1:8Þ
e > 0 such that e < r . Define the following sets 0
0
D ¼ fk 6 n : jjT k ðf ; xÞpk f ðxÞjj2p P rg; D1 ¼
r e0 ; k 6 n : jjT k ð1; xÞpk 1jj2p P 4K
D2 ¼
r e0 k 6 n : jjT k ðcos t; xÞpk cos xjj2p P ; 4K
D3 ¼
r e0 k 6 n : jjT k ðsin t; xÞpk sin xjj2p P : 4K
Then
D D1 [ D2 [ D3 and so
dN ðDÞ 6 dN ðD1 Þ þ dN ðD2 Þ þ dN ðD3 Þ: Therefore, using conditions (2.1.1)–(2.1.3), we get
SN —limjjT n ðf ; xÞ f ðxÞjj2p ¼ 0: n
This completes the proof of the theorem.
3. Rate of weighted statistical convergence In this section, we study the rate of weighted statistical convergence of a sequence of positive linear operators defined from C 2p ðRÞ into C 2p ðRÞ. Definition 3.1. Let ðan Þ be a positive non-increasing sequence. We say that the sequence x ¼ ðxk Þ is weighted statistically convergent to the number L with the rate oðan Þ if for every e > 0,
lim n
1 jfk 6 Pn : pk jxk Lj P egj ¼ 0: an Pn
In this case, we write xk L ¼ SN –oðan Þ. As usual we have the following auxiliary result whose proof is standard.
8234
O.H.H. Edely et al. / Applied Mathematics and Computation 219 (2013) 8231–8236
Lemma 3.1. Let ðan Þ and ðbn Þ be two positive non-increasing sequences. Let x ¼ ðxk Þ and y ¼ ðyk Þ be two sequences such that xk L1 ¼ SN –oðan Þ and yk L2 ¼ SN –oðbn Þ. Then (i) aðxk L1 Þ ¼ SN –oðan Þ, for any scalar a, (ii) ðxk L1 Þ ðyk L2 Þ ¼ SN –oðcn Þ, (iii) ðxk L1 Þðyk L2 Þ ¼ SN –oðan bn Þ, where cn ¼ maxfan ; bn g. Now, we recall the notion of modulus of continuity. The modulus of continuity of f 2 C 2p ðRÞ, denoted by xðf ; dÞ is defined by
xðf ; dÞ ¼ sup jf ðxÞ f ðyÞj: jxyj
