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Szász-Baskakov type operators based on q-integers
Journal of Inequalities and Applications volume 2014, Article number: 441 (2014)
Abstract
In the present paper, we introduce the q-analog of the Stancu variant of Szász-Baskakov operators. We establish the moments of the operators by using the q-derivatives and then prove the basic convergence theorem. Next, the Voronovskaja type theorem and some direct results for the above operators are discussed. Also, the rate of convergence and weighted approximation by these operators in terms of modulus of continuity are studied. Then we obtain a point-wise estimate using the Lipschitz type maximal function. Lastly, we study the A-statistical convergence of these operators and also, in order to obtain a better approximation, we study a King type modification of the above operators.
MSC:41A25, 26A15, 40A35.
1 Introduction
In recent years, there has been intensive research on the approximation of functions by positive linear operators introduced by making use of q-calculus. Lupas [1] and Phillips [2] pioneered the study in this direction by introducing q-analogs of the well-known Bernstein polynomials. Derriennic [3] discussed modified Bernstein polynomials with Jacobi weights. Gupta [4] and Gupta and Heping [5] proposed the q-analogs of usual and discretely defined Durrmeyer operators and studied their approximation properties. In [6], Stancu introduced the positive linear operators by modifying the Bernstein polynomials as
where , is the Bernstein basis function and α, β are any two real numbers satisfying . Recently, Buyukyazici [7] introduced the Stancu type generalization of certain q-Baskakov operators and studied some local direct results for these operators. Subsequently, several authors (cf. [8–11]etc.) have considered such a modification for some other sequences of positive linear operators.
To approximate Lebesgue integrable functions defined on , Gupta et al. [12] introduced the following positive linear operators:
where , , , and studied the asymptotic approximation and error estimation in simultaneous approximation. Subsequently, for , Gupta [13] introduced the following operators:
by considering the value of the function at zero explicitly, and studied an estimate of error in terms of the higher order modulus of continuity in simultaneous approximation for a linear combination of the operators (1.2), introduced by May [14]. Later on, Gupta and Noor [15] discussed some direct results in a simultaneous approximation for the operators (1.2).
In [16], Aral introduced Szász-Mirakjan operators based on q-integers and established some direct results and gave two representations of the r th q-derivative of these operators. Aral and Gupta [17] studied the basic convergence theorem for the r th order q-derivatives, a Voronovskaja type theorem, and some more properties of these operators. Agratini and Doǧru [18] constructed Szász-King type operators and investigated the statistical convergence and the rate of local and global convergence for functions with a polynomial growth. Örkcü and Doǧru [19] proposed two different modifications of q-Szász-Mirakjan Kantorovich operators and obtained the rate of convergence in terms of the modulus of continuity. In [20], they introduced Kantorovich type generalization of q-Szász-Mirakjan operators and discussed their A-statistical approximation properties.
Let the space be endowed with the norm then for , , and each positive integer n, we introduce the following Stancu type modification of the operators (1.2) based on q-integers:
where and .
For , we denote by .
Clearly, if and , the operators defined by (1.3) reduce to the operators given by (1.2).
The purpose of the present paper is to study the basic convergence theorem, Voronovskaja type asymptotic formula, local approximation, rate of convergence, weighted approximation, point-wise estimation and A-statistical convergence of the operators (1.3). Further, to obtain a better approximation we also propose a modification of these operators by using a King type approach.
We recall that the q-analog of beta function of second kind [21, 22] is defined by
where , and , .
In particular, for any positive integers n, m, we have
and
2 Moment estimates
Lemma 1 For , , one has
-
(1)
;
-
(2)
, for ;
-
(3)
, for .
Proof We observe that are well defined for the functions 1, t, . Thus, for every , using (1.4) and (1.5), we obtain
Next, for , again applying (1.4) and (1.5), we get
Proceeding similarly, we have
by using . □
Lemma 2 For the operators as defined in (1.3), the following equalities hold:
-
(1)
;
-
(2)
, for ;
-
(3)
, for .
Proof This lemma is an immediate consequence of Lemma 1. Hence the details of its proof are omitted. □
Lemma 3 For (the space of all bounded and uniform continuous functions on endowed with norm ), one has
Proof In view of (1.3) and Lemma 2, the proof of this lemma easily follows. □
Remark 1 For every , we have
and
say.
3 Main results
Theorem 1 Let and . Then for each , the sequence converges to f uniformly on if and only if .
Proof First, we assume that .
We have to show that converges to f uniformly on .
From Lemma 2, we see that
Therefore, the well-known property of the Korovkin theorem implies that converges to f uniformly on provided .
We show the converse part by contradiction. Assume that does not converge to 1. Then it must contain a subsequence such that , as .
Thus, as . Choosing , in , from Lemma 2, we have
which leads us to a contradiction. Hence, . This completes the proof. □
Theorem 2 (Voronovskaja type theorem)
Let and be a sequence such that and as . Suppose that exists at a point , then we have
Proof By Taylor’s formula we have
where is the Peano form of the remainder and .
Applying to the both sides of (3.1), we have
In view of Remark 1, we have
and
Now, we shall show that
when .
By using the Cauchy-Schwarz inequality, we have
We observe that and . Then it follows from Theorem 1 that
in view of the fact that . Now, from (3.4) and (3.5), we get
and from (3.2), (3.3), and (3.6), we get the required result. □
Theorem 3 (Voronovskaja type theorem)
Let and be a sequence such that and as . If exists on , then
holds uniformly on , where .
Proof Let . The remainder part of the proof of this theorem is similar to that of the proof of the previous theorem. So we omit it. □
3.1 Local approximation
For , let us consider the following K-functional:
where and . By [23] there exists an absolute constant such that
where
is the second order modulus of smoothness of f. By
we denote the usual modulus of continuity of .
Theorem 4 Let and . Then, for every and , we have
where C is an absolute constant and
Proof For , we consider the auxiliary operators defined by
From Lemma 2, we observe that the operators are linear and reproduce the linear functions.
Hence
Let . By Taylor’s theorem, we have
Applying to the both sides of the above equation and using (3.11), we have
Thus, by (3.10) we get
On other hand, by (3.10) and Lemma 3, we have
Using (3.12) and (3.13) in (3.10), we obtain
Hence, taking the infimum on the right hand side over all and using (3.8), we get the required result. □
Theorem 5 Let , and be its modulus of continuity on the finite interval , where . Then, for every ,
where is defined in Remark 1.
Proof From [24], for and , we get
Thus, by applying the Cauchy-Schwarz inequality, we have
on choosing . This completes the proof of the theorem. □
3.2 Weighted approximation
Let . Throughout the section, we assume that is a sequence in such that .
Theorem 6 For each , we have
Proof Making use of the Korovkin type theorem on weighted approximation [25], we see that it is sufficient to verify the following three conditions:
Since , the condition in (3.14) holds for .
By Lemma 2, we have for
which implies that the condition in (3.14) holds for .
Similarly, we can write for
which implies that , (3.14) holds for . □
Now, we present a weighted approximation theorem for functions in . Such type of results are proved in [26] for classical Szász operators.
Theorem 7 For each and , we have
Proof Let be arbitrary but fixed. Then
Since , we have .
Let be arbitrary. We can choose to be so large that
In view of Theorem 1, we obtain
Using Theorem 5, we can see that the first term of the inequality (3.15), implies that
Combining (3.16)-(3.18), we get the desired result. □
For , the weighted modulus of continuity is defined as
Lemma 4 [27]
If then
-
(i)
is monotone increasing function of δ,
-
(ii)
,
-
(iii)
for any , .
Theorem 8 If , then for sufficiently large n we have
where , , , , being
respectively, and K is a positive constant independent of f and n.
Proof From the definition of and Lemma 4, we have
where and . Then we obtain
Now, applying the Cauchy-Schwarz inequality to the second term on the right hand side, we get
From Lemma 2,
where is a positive constant.
From (3.20), there exists a positive constant such that , for sufficiently large n.
Proceeding similarly, , for sufficiently large n, where is a positive constant.
So there exists a positive constant , such that , where and n is large enough. Also, we get
Hence, from (3.19), we have
If we take , then we get
Hence, the proof is completed. □
3.3 Point-wise estimates
In this section, we establish some point-wise estimates of the rate of convergence of the operators . First, we give the relationship between the local smoothness of f and local approximation.
We know that a function is in on E, , E be any bounded subset of the interval if it satisfies the condition
where M is a constant depending only on α and f.
Theorem 9 Let , and . Then we have
where M is a constant depending on η and f, and is the distance between x and E defined as
Proof Let be the closure of E in . Then there exists at least one point such that
By our hypothesis and the monotonicity of , we get
Now, applying Hölder’s inequality with and , we obtain
from which the desired result immediate. □
Next, we obtain the local direct estimate of the operators defined in (1.3), using the Lipschitz-type maximal function of order η introduced by Lenze [28] as
Theorem 10 Let and . Then, for all we have
Proof From (3.21), we have
Applying Hölder’s inequality with and , we get
Thus, the proof is completed. □
4 Statistical convergence
Let be a non-negative infinite summability matrix. For a given sequence , the A-transform of x denoted by is defined as
provided the series converges for each n. A is said to be regular if whenever . Then is said to be A-statistically convergent to L i.e. if for every , . If we replace A by then A is a Cesaro matrix of order one and A-statistical convergence is reduced to the statistical convergence. Similarly, if , the identity matrix, then A-statistical convergence is called ordinary convergence. Kolk [29] proved that statistical convergence is better than ordinary convergence. In this direction, significant contributions have been made by (cf. [19, 20, 30–37]etc.).
Let be a sequence such that
Theorem 11 Let be a non-negative regular summability matrix and be a sequence satisfying (4.1). Then, for any compact set and for each function , we have
Proof Let . From Lemma 2, . Again, by Lemma 2, we have
For , let us define the following sets:
which implies that and hence for all , we obtain
Hence, taking the limit as , we have .
Similarly, by using Lemma 2, we have
Now, let us define the following sets:
Then we obtain , which implies that
Thus, as we get . This completes the proof. □
Theorem 12 Let be a non-negative regular summability matrix and be a sequence in satisfying (4.1). Let the operators , , be defined as in (1.3). Then, for each function , we have
where , .
Proof From [[31], p.191, Theorem 3], it is sufficient to prove that , where , .
From Lemma 2, holds.
Again using Lemma 2, we have
For each , let us define the following sets:
which yields in view of (4.2) and therefore for all , we have
Hence, on taking the limit as , . Proceeding similarly,
Now, let us define the following sets:
Then we obtain , which implies that
Hence, taking the limit as we get . This completes the proof of the theorem. □
5 Better estimates
It is well known that the classical Bernstein polynomials preserve constant as well as linear functions. To make the convergence faster, King [38] proposed an approach to modify the Bernstein polynomials, so that the sequence preserves test functions and , where , . As the operator defined in (1.3) reproduces only constant functions, this motivated us to propose the modification of this operator, so that it can preserve constant as well as linear functions.
The modification of the operators given in (1.3) is defined as
where for and .
Lemma 5 For each , by simple computations, we have
-
(1)
;
-
(2)
;
-
(3)
= + + .
Consequently, for each , we have the following equalities:
say.
Theorem 13 Let and . Then there exists a positive constant C such that
where is given by (5.1).
Proof Let , and . Using Taylor’s expansion we have
Applying on both sides and using Lemma 5, we get
Obviously, we have . Therefore
Since , we get
Finally, taking the infimum over all and using (3.7)-(3.9) we obtain
which proves the theorem. □
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Acknowledgements
The authors are extremely grateful to the reviewers for a careful reading of the manuscript and making valuable suggestions leading to a better presentation of the paper. The last author is thankful to the ‘Council of Scientific and Industrial Research’, India, for financial support to carry out the above research work.
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Agrawal, P.N., Karsli, H. & Goyal, M. Szász-Baskakov type operators based on q-integers. J Inequal Appl 2014, 441 (2014). https://doi.org/10.1186/1029-242X-2014-441
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DOI: https://doi.org/10.1186/1029-242X-2014-441