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Some approximation results for generalized Kantorovich-type operators
Journal of Inequalities and Applications volume 2013, Article number: 585 (2013)
In this paper, we construct a new family of operators, prove some approximation results in A-statistical sense and establish some direct theorems for Kantorovich-type integral operators.
MSC: Primary 41A10, 41A25, 41A36.
1 Introduction and preliminaries
Statistical convergence  and its variants, extensions and generalizations have been proved to be an active area of recent research in summability theory, e.g., lacunary statistical convergence , λ-statistical convergence , A-statistical convergence , statistical A-summability , statistical summability , statistical summability , statistical summability  and statistical σ-summability etc. Following the work of Gadjiv and Orhan , these statistical summability methods have been used in establishing many approximation theorems (e.g., [5, 11–20] and ). Recently, the statistical approximation properties have also been investigated for several operators. For instance, in  Butzer and Hahn operators; in  and q-analogue of Stancu-Beta operators; in  Bleimann, Butzer and Hahn operators; in  Baskakov-Kantorovich operators; in  Szász-Mirakjan operators; in  analogues of Bernstein-Kantorovich operators; and in q-Lagrange polynomials were defined and their statistical approximation properties were investigated. Most recently, the statistical summability of Walsh-Fourier series has been discussed in . In this paper, we construct a new family of operators with the help of Erkuş-Srivastava polynomials, establish some A-statistical approximation properties and direct theorems.
Let us recall the following definitions.
Let ℕ denote the set of all natural numbers. Let and . Then the natural density of K is defined by if the limit exists, where denotes the cardinality of the set . A sequence of real numbers is said to be statistically convergent to L (cf. Fast ) provided that for every the set has natural density zero, i.e., for each ,
In this case, we write . Note that every convergent sequence is statistically convergent but not conversely.
Let , , be an infinite matrix. For a given sequence , the A-transform of x is defined by , where , provided the series converges for each n. We say that A is regular if . Let A be a regular matrix.
We say that a sequence is A-statistically convergent to a number L (cf. Kolk ) if for every ,
In this case, we denote this limit by .
Note that for , the Cesàro matrix of order 1, A-statistical convergence reduces to the statistical convergence.
2 Construction of a new operator and its properties
The well-known (two-variable) polynomials , which are generated by
are the Lagrange polynomials which occur in certain problems in statistics . Recently, Chan  introduced and systematically investigated the multivariable extension of the classical Lagrange polynomials . These multivariable Lagrange polynomials, which are popularly known in the literature as the Chan-Chyan-Srivastava polynomials, are generated by (see  and )
Clearly, the defined generating function (2.2) yields the explicit representation given by [, p.140, Eq. (6)]
or, equivalently, by [, p.522, Eq. (17)]
On the other hand, Altin and Erkuş  presented a multivariable extension of the so-called Lagrange-Hermite polynomials generated by
The case of the polynomials given by (2.5) corresponds to the familiar (two-variable) Lagrange-Hermite polynomials considered by Dattoli et al. .
The multivariable polynomials
which are defined by the following generating function [, p.268, Eq. (3)]:
are a unification (and generalization) of several known families of multivariable polynomials including (for example) Chan-Chyan-Srivastava polynomials
defined by (2.2) (see  for details). Obviously, the Chan-Chyan-Srivastava polynomials
follow as a special case of the polynomials due to Erkuş and Srivastava 
where (as well as in what follows)
Moreover, the Lagrange-Hermite polynomials
follow as a special case of the polynomials 
The generating function (2.6) yields the following explicit representation ([, p.268, Eq. (4)]):
which, in the special case when
corresponds to (2.3).
The following relationship is established between the polynomials due to Erkuş and Srivastava  and the Chan-Chyan-Srivastava polynomials by applying the generating functions (2.2) and (2.6) in .
where it is tacitly assumed that the following set:
which depends upon the distinct values of the factor occurring in the expression
exists such that
Thus, by assertion (2.8), we obtain the desired relationship as follows:
Now by using the Erkuş-Srivastava multivariable polynomials given by (2.2), we introduce the following family of positive linear operators on :
Throughout this paper, we assume that
are sequences of real numbers such that
For convenience, taking , , in (2.9), we have
Lemma 2.1 For each and ,
Lemma 2.2 For each and ,
Proof Let each be fixed. Then from (2.10) we get
Lemma 2.3 For each and ,
Proof Let each be fixed. Then from (2.10) we get
On the other hand, since
it follows from Lemma 2.1 and Lemma 2.2 that
Combining (2.11) and (2.12), we have
Then, taking supremum over , we have
Remark 2.2 Let , since is linear, we get
3 A-statistical approximation
Let be a linear space of all real-valued continuous functions f on , and let T be a linear operator which maps into itself. We say that T is positive if for every non-negative , we have for all . We know that is a Banach space with the norm
For typographical convenience, we will write in place of if no confusion arises.
Theorem 3.1 Let be a non-negative regular summability matrix. Then
if and only if for all ,
Proof Suppose that (3.2) holds for all . Then we have
since . By Lemma 2.2, we have
By (3.3) and (3.4), we immediately get
Conversely, suppose that (3.1) holds. Then from Lemma 2.1 we have . Hence
Also from Lemma 2.2 it follows that
Therefore, by using (3.1), we get
Now we claim that
By Lemma 2.3, we have
Now, for a given , we define the following sets:
From (3.8), it is easy to see that . Then, for each , we get
Using (3.3), we get
Now, using the above facts and taking the limit as in (3.9), we conclude that
which gives (3.7). Now, combining (3.5)-(3.7), and using the statistical version of the Korovkin approximation theorem (see Gadjiv and Orhan , Theorem 1), we get the desired result.
This completes the proof of the theorem. □
In a similar manner, we can extend Theorem 3.1 to the -dimensional case for the operators given by (2.9) as follows.
Theorem 3.2 Let be a non-negative regular summability matrix. Then
if and only if for all ,
Remark 3.1 If in Theorem 3.2 we replace by the identity matrix, we immediately get the following theorem which is a classical case of Theorem 3.2.
Theorem 3.3 if and only if for all , the sequence
is uniformly convergent to f on .
Finally, we display an example which satisfies all the hypotheses of Theorem 3.2, but not of Theorem 3.3. Therefore, this indicates that our A-statistical approximation in Theorem 3.2 is stronger than its classical case.
Take , the Cesàro matrix of order 1 and
are sequences of real numbers defined by
We then observe that
and also that
Therefore, by Theorem 3.2, we have that for all ,
However, since the sequence defined by (3.10) is non-convergent, Theorem 3.3 does not hold in this case.
4 Direct theorems
By , we denote the space of all real-valued continuous bounded functions f on the interval , the norm on the space is given by
Peetre’s K-functional is defined by
By  there exists a positive constant s.t.
where the second-order modulus of smoothness is
Also, for , the usual modulus of continuity is given by
Theorem 4.1 Let and . Then, for all and , there exists an absolute constant s.t.
Proof Let . From Taylor’s expansion
and from Lemmas (2.1), (2.2) and (2.3), we get
Using Remark 2.2, we obtain
On the other hand, by the definition of , we have
Hence, taking infimum on the right-hand side over all , we get
In view of the property of K-functional, for every , we get
This completes the proof of the theorem. □
Theorem 4.2 Let be such that and , , such that as . Then the following equality holds:
uniformly on .
Proof By the Taylor’s formula, we may write
where is the remaining term and . Applying to (4.1), we obtain
By the Cauchy-Schwartz inequality, we have
Observe that and . Then it follows from Theorem 4.1 that
uniformly with respect to .
Now, from (4.2), (4.3) and Remark 2.2, we get
Finally, using Remark 2.1, we get the following:
Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241–244.
Fridy JA, Orhan C: Lacunary statistical convergence. Pac. J. Math. 1993, 160: 43–51. 10.2140/pjm.1993.160.43
Mursaleen M, Alotaibi A, Mohiuddine SA: Statistical convergence through de la Vallée-Poussin mean in locally solid Riesz spaces. Adv. Differ. Equ. 2013., 2013: Article ID 66 10.1186/1687-1847-2013-66
Kolk E: Matrix summability of statistically convergent sequences. Analysis 1993, 13: 77–83.
Edely OHH, Mursaleen M: On statistical A -summability. Math. Comput. Model. 2009, 49: 672–680. 10.1016/j.mcm.2008.05.053
Moricz F:Tauberian conditions under which statistical convergence follows from statistical summability . J. Math. Anal. Appl. 2002, 275: 277–287. 10.1016/S0022-247X(02)00338-4
Moricz F: Theorems relating to statistical harmonic summability and ordinary convergence of slowly decreasing or oscillating sequences. Analysis 2004, 24: 127–145.
Moricz F, Orhan C: Tauberian conditions under which statistical convergence follows from statistical summability by weighted means. Studia Sci. Math. Hung. 2004, 41(4):391–403. 10.1556/SScMath.41.2004.4.3
Mursaleen M, Edely OHH: On the invariant mean and statistical convergence. Appl. Math. Lett. 2009, 22: 1700–1704. 10.1016/j.aml.2009.06.005
Gadjiv AD, Orhan C: Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 2002, 32: 129–138. 10.1216/rmjm/1030539612
Demirci K, Dirik F: Approximation for periodic functions via statistical σ -convergence. Math. Commun. 2011, 16: 77–84.
Edely OHH, Mohiuddine SA, Noman AK: Korovkin type approximation theorems obtained through generalized statistical convergence. Appl. Math. Lett. 2010, 23: 1382–1387. 10.1016/j.aml.2010.07.004
Mohiuddine SA, Alotaibi A, Mursaleen M:Statistical summability and a Korovkin type approximation theorem. J. Inequal. Appl. 2012., 2012: Article ID 172 10.1186/1029-242X-2012-172
Devore RA, Lorentz GG: Constructive Approximation. Springer, Berlin; 1993.
Mursaleen M, Alotaibi A: Statistical summability and approximation by de la Vallée-Poussin mean. Appl. Math. Lett. 2011, 24: 320–324. Erratum: Appl. Math. Lett. 25, 665 (2012) 10.1016/j.aml.2010.10.014
Mursaleen M, Karakaya V, Ertürk M, Gürsoy F: Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl. Math. Comput. 2012, 218: 9132–9137. 10.1016/j.amc.2012.02.068
Srivastava HM, Mursaleen M, Khan A: Generalized equi-statistical convergence of positive linear operators and associated approximation theorems. Math. Comput. Model. 2012, 55: 2040–2051. 10.1016/j.mcm.2011.12.011
Belen C, Mursaleen M, Yildirim M: Statistical A -summability of double sequences and a Korovkin type approximation theorem. Bull. Korean Math. Soc. 2012, 49(4):851–861. 10.4134/BKMS.2012.49.4.851
Demirci K, Dirik F: Four-dimensional matrix transformation and rate of A -statistical convergence of periodic functions. Math. Comput. Model. 2010, 52: 1858–1866. 10.1016/j.mcm.2010.07.015
Mursaleen M, Alotaibi A: Korovkin type approximation theorem for functions of two variables through statistical A -summability. Adv. Differ. Equ. 2012., 2012: Article ID 65 10.1186/1687-1847-2012-65
Mursaleen M, Kiliçman A: Korovkin second theorem via B -statistical A -summability. Abstr. Appl. Anal. 2013., 2013: Article ID 598963 10.1155/2013/598963
Aral A, Doğru O: Bleimann, Butzer and Hahn operators based on q -integers. J. Inequal. Appl. 2007., 2007: Article ID 79410 10.1155/2007/79410
Aral A, Gupta V: On q -analogue of Stancu-Beta operators. Appl. Math. Lett. 2012, 25: 67–71. 10.1016/j.aml.2011.07.009
Mursaleen M, Khan A: Statistical approximation properties of modified q -Stancu-Beta operators. Bull. Malays. Math. Soc. 2013, 36(3):683–690.
Ersan S, Doğru O: Statistical approximation properties of q -Bleimann, Butzer and Hahn operators. Math. Comput. Model. 2009, 49: 1595–1606. 10.1016/j.mcm.2008.10.008
Gupta V, Radu C: Statistical approximation properties of q -Baskokov-Kantorovich operators. Cent. Eur. J. Math. 2009, 7(4):809–818. 10.2478/s11533-009-0055-y
Örkcü M, Doğru O: Weighted statistical approximation by Kantorovich type q -Szász-Mirakjan operators. Appl. Math. Comput. 2011, 217: 7913–7919. 10.1016/j.amc.2011.03.009
Radu C: Statistical approximation properties of Kantorovich operators based on q -integers. Creative Math. Inform. 2008, 17(2):75–84.
Mursaleen M, Khan A, Srivastava HM, Nisar KS: Operators constructed by means of q -Lagrange polynomials and A -statistical approximation. Appl. Math. Comput. 2013, 219: 6911–6918. 10.1016/j.amc.2013.01.028
Alotaibi A, Mursaleen M: A -Statistical summability of Fourier series and Walsh-Fourier series. Appl. Math. Inform. Sci. 2012, 6(3):535–538.
Erdélyi A, Magnus W, Berhettinger F, Tricomi FG III. In Higher Transcendental Functions. McGraw-Hill, New York; 1953.
Chan WC-C, Chyan C-J, Srivastava HM: The Lagrange polynomials of several variables. Integral Transforms Spec. Funct. 2001, 12: 139–148. 10.1080/10652460108819340
Chen KY, Liu SJ, Srivastava HM: Some new results for the Lagrange polynomials in several variables. ANZIAM J. 2007, 49: 243–258. 10.1017/S1446181100012815
Altin A, Erkuş E: On a multivariable extension of the Lagrange-Hermite polynomials. Integral Transforms Spec. Funct. 2006, 17: 239–244. 10.1080/10652460500432006
Erkuş E, Srivastava HM: A unified presentation of some families of multivariable polynomials. Integral Transforms Spec. Funct. 2006, 17: 267–273. 10.1080/10652460500444928
Liu S-J, Lin S-D, Srivastava HM, Wong M-M: Bilateral generating functions for the Erkus-Srivastava polynomials and the generalized lauricella functions. Appl. Math. Comput. 2012, 218: 7685–7693. 10.1016/j.amc.2012.01.005
The authors declare that they have no competing interests.
The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
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Mursaleen, M., Khan, F., Khan, A. et al. Some approximation results for generalized Kantorovich-type operators. J Inequal Appl 2013, 585 (2013). https://doi.org/10.1186/1029-242X-2013-585
- Lagrange polynomial
- Korovkin approximation theorems
- A-statistical convergence
- Kantorovich-type operators
- positive linear operators
- modulus of continuity
- Peetre’s K-functional
- Lipschitz class