Abstract
The present paper deals mainly with a King type modification of \(( p,\,q) \)-Bernstein operators. By improving the conditions given in Mursaleen et al. (On (p, q)-analogue of Bernstein operators. Appl Math Comput 266:874–882, 2015a), we investigate the Korovkin type approximation of both \(( p,\,q) \)-Bernstein and King type \(( p,\,q) \)-Bernstein operators. We also prove that the error estimation of King type of the operator is better than that of the classical one whenever \(0\le x\le \frac{1}{3}.\)
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In the last two decades quantum calculus has gained a lot of interest in the field of approximation theory. The q-generalizations of several operators have been constructed and studied by many authors. We can refer the book by Kac and Cheung (2002) for the details of q-calculus and the book by Aral et al. (2013) in which one can find the related studies on this area. Recently, a new generalization of q-calculus has been appeared in the approximation theory, namely \(( p,\,q) \) calculus. Before mentioning the studies on this topic, we recall some notations on \((p,\,q)\)-integers. Let \(0<q<p\le 1.\) For each nonnegative integer n, the \((p,\,q)\)-numbers is denoted by \( [ n] _{p,q}\) and given by
For each \(k,\,n\in {\mathbb {N}} ,\,n\ge k\ge 0,\) the \((p,\,q) \)-factorial \([ n] _{p,q}!\) and \((p,\,q)\)-binomial are defined by
The \(( p,\,q) \)-power basis and \(( p,\,q) \)-binomial expansion are defined as
and
The relation between q-analogues and \((p,\,q)\)-analogues can be described as:
Note that as taking \(p=1,\,(p,\,q)\) integers turns out to be q-integers. So q-integers can be regarded as a special case of \((p,\,q)\)-integers.
Lupas (1987) was the first to study q-calculus in approximation theory by defining a q-generalization of Bernstein operators. Later in 1997, Phillips (1996) constructed another modification of Bernstein operators for \(n\in {\mathbb {N}} \) and \(0<q<1\) as
which is known as q-Bernstein operators in the literature.
For the study of \((p,\,q)\)-calculus in approximation theory, pioneer works are due to Mursaleen et al. (2015a, b). They defined the \((p,\,q)\)-Bernstein operators by
for \(n\in {\mathbb {N}} \) and \(0<q<p\le 1.\) Very recently, same authors defined the \(( p,\,q) \)-analogue of Bernstein–Stancu operators in Mursaleen et al. (2015c). Mursaleen et al. (2015d) also studied the approximation properties of \( ( p,\,q) \)-Bernstein–Shurer operators. Since the use of \(( p,\,q) \)-calculus in approximation theory is a very new issue and open for development, lately, the \(( p,\,q) \)-generalizations of several operators are being studied by many authors. We refer to Acar et al. (2016), Cai and Zhou (2016), Cai (2017), Gupta (2016a, b) and Karaisa (2016) for some studies about the \((p,\,q)\)-analogues of some generalizations of Bernstein operators. See also Acar (2016), Gupta (2016c), Mishra and Pandey (2016), Mursaleen et al. (2016) and references therein for some other recent works on the \(( p,\,q) \) generalizations of some other operators.
In the present paper our aim is to construct the King type modification of \( ( p,\,q) \)-Bernstein operators and investigate its approximation properties. Before constructing the mentioned operator, we first give the Korovkin type approximation of the \(( p,\,q) \)-Bernstein operators under the weaker conditions than that of the ones given by Mursaleen et al. (2015a). In Sect. 2, we construct King type \(( p,\,q) \)-Bernstein operator and give some auxiliary lemmas and results. Section 3 is devoted to our main results. We give the uniform convergence of the operators via Korovkin’s theorem and obtain rate of convergence by means of modulus of continuity. We also compare the error estimates of the classical and King type \((p,\,q)\)-Bernstein operators.
2 Construction of the Operators and Auxiliary Lemmas
Before constructing our new operators recall that q-Bernstein operators defined by (1.2) and \(( p,\,q)\)-Bernstein operators defined by (1.3) satisfy the following identities (see Mursaleen et al. 2015b; Phillips 1996 for details):
and
Remark 1
Since \(\dfrac{1}{[ n] _{q/p}}=\dfrac{p^{n-1}}{ [ n] _{p,q}}\) and \(x\in [ 0,\,1] ,\) if we take p and q as sequences \(p_{n}\) and \(q_{n}\) satisfying the conditions
then the \(( p,\,q)\)-Bernstein operator (1.3) converges uniformly to the function f on \([ 0,\,1] .\) Note that with these conditions \([ n] _{q/p}\) tends to infinity as \(n\rightarrow \infty ,\) and hence uniform convergence holds from Korovkin’s theorem.
Example 1
We can always find such sequences: for example, letting \(( q_{n}) = \left( 1-\frac{1}{n}\right) \) and \(( p_{n}) =\left( 1-\frac{1}{n+1}\right) \) we get \(\lim \nolimits _{n\rightarrow \infty }\frac{( q_{n}) }{( p_{n}) }=\lim \nolimits _{n\rightarrow \infty }\frac{\left( 1-\frac{1}{n} \right) }{\left( 1-\frac{1}{n+1}\right) }= 1\) and \(\lim \nolimits _{n\rightarrow \infty }[ n] _{q/p}=\lim \nolimits _{n\rightarrow \infty }\dfrac{[ n] _{p,q}}{p^{n-1}}=\lim \nolimits _{n\rightarrow \infty } \left( 1-\frac{1}{n+1}\right) ^{1-n}\frac{ \left( 1-\frac{1}{n+1}\right) ^{n}-\left( 1-\frac{1}{n}\right) ^{n}}{\left( 1-\frac{1}{n+1}\right) -\left( 1-\frac{1}{n}\right) } =\infty .\) Similarly, if we take \(( q_{n}) =\frac{1}{2}-\frac{1}{2n+0.7}\) and \(( p_{n}) =\frac{1}{2}-\frac{1}{e^{n}}\) we get, \(\lim \nolimits _{n\rightarrow \infty }\frac{( q_{n}) }{( p_{n}) }=\lim \nolimits _{n\rightarrow \infty }\frac{\left( \frac{1}{2}- \frac{1}{2n+0.7}\right) }{\left( \frac{1}{2}-\frac{1}{e^{n}}\right) } =1\) and \(\lim \nolimits _{n\rightarrow \infty }[ n] _{q/p}=\lim \nolimits _{n\rightarrow \infty }\dfrac{[ n] _{p,q}}{p^{n-1}}=\lim \nolimits _{n\rightarrow \infty }\left( \frac{1}{2}-\frac{1}{e^{n}}\right) ^{1-n}\dfrac{\left( \frac{1}{2}-\frac{1}{e^{n}}\right) ^{n}-\left( \frac{1}{2 }-\frac{1}{2n+0.7}\right) ^{n}}{\left( \frac{1}{2}-\frac{1}{e^{n}}\right) -\left( \frac{1}{2}-\frac{1}{2n+0.7}\right) }= \infty .\) Note that in the second example, \(( q_{n}) \) and \(( p_{n}) \) tends to 1/2 separately, as n tends to infinity.
Motivated by the method of King (2003), we now modify \((p,\,q)\)-Bernstein operators in a way that the new operator preserves \(x^{2}.\) For this purpose, for each \(n\in {\mathbb {N}}\) and \(0<q<p\le 1,\) we define the King type \(( p,\,q) \)-Bernstein operators \(B_{n}^{*}:C[ 0,\,1] \rightarrow C[ 0,\,1] \) as,
for \(0\le x\le 1,\) where
Remark 2
\([ n] _{p,q}-p^{n-1}=q[n-1]_{p,q}\ge 0,\, \sqrt{ \frac{[ n] _{p,q}}{( [ n] _{p,q}-p^{n-1}) } x^{2}+\frac{p^{2n-2}}{4( [ n] _{p,q}-p^{n-1}) ^{2}}} \ge \frac{p^{n-1}}{2( [ n] _{p,q}-p^{n-1}) }\) from which we have \(v_{n,p,q}( x) \ge 0.\) On the other hand, since \( x\in [ 0,\,1], \) we get \(\sqrt{\frac{[ n] _{p,q}}{( [ n] _{p,q}-p^{n-1}) }x^{2}+\frac{p^{2n-2}}{4( [ n] _{p,q}-p^{n-1}) ^{2}}}=\frac{\sqrt{4( [ n] _{p,q}-p^{n-1}) [n]x^{2}+p^{2n-2}}}{2( [ n] _{p,q}-p^{n-1}) }\le \frac{2[n] _{p,q}-p^{n-1}}{2( [ n] _{p,q}-p^{n-1}) }.\) Using this inequality in (2.5), we obtain \( v_{n,p,q}( x) \le 1.\) Hence, \(0\le v_{n,p,q}( x) \le 1\) and \(B_{n}^{*}\) is a linear and positive operator.
Lemma 1
For the operators\(B_{n}^{*}\)defined by (2.4), the following identities hold:
Proof
With the help of the identities in (2.2) and the test functions of q-Bernstein operators given by (2.1), we can write
and
Using the identity \(q[n-1]_{p,q}=[n]_{p,q}-p^{n-1}\) and arranging the terms one can easily obtain that
\(\square \)
3 Main Results
3.1 Korovkin Type Approximation
Before giving the Korovkin type approximation theorem for the operator (2.4), let us remind the following remark:
Remark 3
For fixed \(p,\,q\) with \(0<q<p\le 1,\) one can see that \(\lim \nolimits _{n\rightarrow \infty }\,[ n] _{p,q}=\lim \nolimits _{n\rightarrow \infty }\frac{p^{n}-q^{n}}{p-q}=0\) and \(\lim \nolimits _{n\rightarrow \infty }\,p^{1-n}[ n] _{p,q}=\lim \nolimits _{n\rightarrow \infty }\frac{p}{p-q}\frac{p^{n}-q^{n}}{p^{n}}=\lim \nolimits _{n\rightarrow \infty }\frac{p}{p-q}\left( 1-\left( \frac{q}{p} \right) ^{n}\right) =\frac{p}{p-q}\) which is different from infinity. So, \( v_{n,p,q}( x) \) cannot converge to x as \(n\rightarrow \infty \) and this means that we are not able to investigate approximation properties of the operators \(B_{n}^{*}( f,\,p,\,q;\,\cdot ) \) given by (2.4) via Korovkin’s theorem. To obtain convergency results, we assume p and q as sequences \(p_{n}\) and \(q_{n}\) satisfying the conditions given in (2.3).
Theorem 1
Let\(p_{n}\)and\(q_{n}\)be sequences satisfying the conditions given in (2.3). Then for every\(f\in C[ 0,\,1] ,\, B_{n}^{*}( f,\,p_{n},\,q_{n};\,\cdot ) \)converges uniformly to f.
Proof
We know that \(B_{n}^{*}\) is a linear and positive operator. Let us examine the second moment of the operator.
Using the identity in (1.1), we can rewrite the above equality as
Hence, from the conditions given in (2.3), we get
Finally the result follows from Korovkin’s theorem. \(\square \)
3.2 Rate of Convergence
Let \(f\in C[0,\,1].\) The modulus of continuity of f, denoted by \(\omega _{f}( \delta ),\) is defined as
Note that for \(f\in C[0,\,1],\,\lim \nolimits _{\delta \rightarrow 0 }\omega _{f}( \delta ) =0\) and for each \(t,\,x\in [0,\,1]\)
We now give the rate of convergence of the operators \(B_{n}^{*}( f,\,p,\,q,\,\cdot ) \) by means of modulus of continuity.
Theorem 2
Let f be a function in\(C[0,\,1]\)and\(x\in [0,\,1].\)The operators\( B_{n}^{*},\,n\in {\mathbb {N}} ,\)defined by (2.4) satisfy
where\(\delta _{n,p,q,x}=\sqrt{2x( x-v_{n,p,q}( x) ) }\)and\(v_{n,p,q}\)is given by (2.5).
Proof
Note that based on Cauchy’s inequality we can write \(( B_{n}^{*}( e_{1},\,p,\,q;\,x) ) ^{2}\le B_{n}^{*}( e_{0},\,p,\,q;\,x) B_{n}^{*}( e_{2},\,p,\,q;\,x).\) So, by Lemma 1, we have \(B_{n}^{*}( e_{1},\,p,\,q;\,x) \le x\) for each \(x\in [0,\,1],\) from which we get \(2x( x-v_{n,p,q}(x)) \ge 0.\) From (3.1) and (3.2) we have
Cauchy’s inequality and the identities given by (2.6) imply
Choosing \(\delta =\delta _{n,p,q,x}=\sqrt{2x( x-v_{n,p,q}) },\) the proof is completed. \(\square \)
Theorem 3
Let f be a function in\(C^{1}[0,\,1]\)and\(x\in [0,\,1].\)The operators\(B_{n}^{*},\,n\in {\mathbb {N}} ,\)defined by (2.4) verify for\(\delta >0\)
where\(v_{n,p,q}\)is given in (2.5).
Proof
For every \(f\in C^{1}[0,\,1]\) and \(x,\,t\in [0,\,1],\) we have
On the other hand, we can write
Applying \(B_{n}^{*}\) to both the sides of (3.3) and using the above inequality we get,
Using Cauchy’s inequality, we have
So, from the second central moment of the operator \(B_{n}^{*},\) we obtain
which is the desired result. \(\square \)
Note that for the \((p,\,q)\)-Bernstein operators, for \(f\in C[0,\,1],\, x\in [0,\,1],\) one has
We claim that the error estimation in Theorem 2 is better than that of (3.4) provided that \(0\le x\le 1/3.\) Indeed, if we can show that the inequality
is satisfied for all \(0\le x\le 1/3,\) then we are done. So we are dealing with finding \(x\ge 0\) such that
is satisfied for all \(n\in {\mathbb {N}}.\) We may rewrite the above inequality in the form
After some calculations one can see that, for each \(n\ge 2,\) the equality holds for the values \(x_{n}=\frac{[ n] _{q/p}+1}{3[ n] _{q/p}+1}\) or \(x=1.\) One can also see that
as \(n\rightarrow \infty .\) The inequality given by (3.5) is satisfied by those x values in the interval \(\left[ 0,\,\frac{[ n ] _{q/p}+1}{3[ n] _{q/p}+1}\right] .\) Since (3.6) is satisfied, (3.5) holds for all \(0\le x\le \frac{1}{3}\) (see Mahmudov 2009, for q-analogue). So, the error estimation for the King type \( (p,\,q)\)-Bernstein operators is better than that of the classical \((p,\,q)\)-Bernstein operators whenever \(x\in \left[ 0,\,\frac{1}{3}\right] .\) The converse inequality in (3.5) is satisfied for those x values in the interval \(A_{n}:=\left[ \frac{[ n] _{q/p}+1}{3[ n] _{q/p}+1},\,1\right] \) for all \(n\ge 2.\) We claim that the error estimation for the classical \((p,\,q)\)-Bernstein operators is better than that of the King type \((p,\,q)\)-Bernstein operators in the interval \(\left( \frac{1}{3},\,1 \right] .\) Note that \(A_{1}\subset A_{2}\subset A_{3}\cdots \) holds. Let \(x\in \bigcup _{n\ge 2}A_{n}.\) Since (3.6) is satisfied, \([ x_{n},\,1 ] \subset \left( \frac{1}{3},\,1\right] \) for all \(n\ge 2.\) Hence, the union of these intervals also satisfy
Now let \(x\in \left( \frac{1}{3},\,1\right] .\) Since (3.6) holds, there exists \(N_{0}\) such that for all \(n\ge N_{0}\, \left| x_{n}-\frac{1}{3}\right| <\delta =\left| x-\frac{1}{3}\right| . \) This implies \(x\in [x_{n},\,1]\) and eventually \(x\in \bigcup _{n\ge 2}A_{n}.\) So we obtain
From (3.7) and (3.8) we get \(\left( \frac{1}{3},\,1\right] =\bigcup _{n\ge 2}A_{n}\) and our claim is true.
References
Acar T (2016) (p, q)-Generalization of Szász–Mirakyan operators. Math Methods Appl Sci 39(10):2685–2695
Acar T, Aral A, Mohiuddine SA (2016) Approximation by bivariate (p, q)-Bernstein–Kantorovich operators. Iran J Sci Technol Trans A. https://doi.org/10.1007/s40995-016-0045-4
Aral A, Gupta V, Agarwal RP (2013) Applications of q-calculus in operator theory. Springer, New York
Cai QB (2017) On (p, q)-analogue of modified Bernstein–Schurer operators for functions of one and two variables. J Appl Math Comput 54(1–2):1–21
Cai QB, Zhou G (2016) On (p, q)-analogue of Kantorovich type Bernstein–Stancu–Schurer operators. Appl Math Comput 276:12–20
Gupta V (2016) (p, q)-Szász–Mirakyan–Baskakov operators. Complex Anal Oper Theory. https://doi.org/10.1007/s11785-015-0521-4
Gupta V (2016) Bernstein Durrmeyer operators based on two parameters. Facta Univ Ser Math Inform 31(1):79–95
Gupta V (2016) (p, q)-Genuine Bernstein Durrmeyer operators. Boll Unione Mat Ital 9(3):399–409
Kac VG, Cheung P (2002) Quantum calculus. Universitext. Springer, New York
Karaisa A (2016) On the approximation properties of bivariate (p, q)-Bernstein operators. arXiv:1601.05250
King JP (2003) Positive linear operators which preserve \( x^{2}\). Acta Math Hung 99(3):203–208
Lupas A (1987) A \(q\)-analogue of the Bernstein operator. Univ Cluj-Napoca Semin Numer Stat Calc Prepr 9:85–92
Mahmudov N (2009) Korovkin-type theorems and applications. Open Math 7(2):348–356
Mishra VN, Pandey S (2016) Chlodowsky variant of (p, q) Kantorovich–Stancu–Schurer operators. Int J Anal Appl 11(1):28–39
Mursaleen M, Ansari KJ, Khan A (2015) On (p, q)-analogue of Bernstein operators. Appl Math Comput 266:874–882
Mursaleen M, Nasiruzzaman M, Khan A, Ansari KJ (2016) Some approximation results on Bleimann–Butzer–Hahn operators defined by (p, q)-integers. Filomat 30(3):639–648
Mursaleen M, Ansari KJ, Khan A (2015) On (p, q)-analogue of Bernstein operators (revised). arXiv preprint arXiv:1503.07404
Mursaleen M, Ansari KJ, Khan A (2015) Some approximation results by (p, q)-analogue of Bernstein–Stancu operators. Appl Math Comput 264(2015):392–402 [Corrigendum Appl Math Comput 269:744–746]
Mursaleen M, Nasiruzzaman M, Ashirbayev N (2015) Some approximation results on Bernstein–Schurer operators defined by (p, q)-integers. J Inequal Appl. https://doi.org/10.1186/s13660-015-0767-4
Phillips GM (1996) Bernstein polynomials based on the q-integers. Ann Numer Math 4:511–518
Acknowledgements
We would like to thank the referees for their suggestions that improved the presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dalmanoğlu, Ö., Örkcü, M. Approximation Properties of King Type \((p,\,q)\)-Bernstein Operators. Iran J Sci Technol Trans Sci 43, 249–254 (2019). https://doi.org/10.1007/s40995-017-0434-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40995-017-0434-3