Abstract
In the present article, we consider the q-analogue of generalized Bernstein–Kantorovich operators. For the proposed operators, we studied some convergence properties by using first- and second-order modulus of continuity.
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2010 Mathematics Subject Classification
1 Introduction
In the year 1912, Bernstein [5] introduced the Bernstein operators and provided the constructive proof of Weierstrass theorem. Later, several researchers have generalized Bernstein operators using different parameters and studied various convergence properties. For more (see [6, 7, 16]).
Recently, Chen et al. [7] defined a family of Bernstein operators, for the functions \(f \in \left[ {0,1} \right] \), \(\alpha \) is fixed and \(n \in \mathbb {N}\) are as follows:
where \({f_k} = f\left( {\frac{k}{n}} \right) \). For \(n>2\) the \(\alpha \)-Bernstein polynomial \({p_{n,k}^{(\alpha )}(x)}\) of degree n is defined by
and
For the first time in 1987, Bernstein operators based on q-integers were introduced by Lupas [12] and they are rational functions. Again in 1997, Phillips [14] introduced the q-Bernstein polynomials known as Phillips q-Bernstein operators. In past decade, linear positive operators based on q-integers is an active area of research. For more (see [4, 8, 11]).
Chai et al. [8] have considered the q-analouge of (1.1) is as follows:
where
\(q \in (0,1]\) and \({\mathrm{{f}}_k} = f\left( {\frac{{{{[k]}_q}}}{{{{[n]}_q}}}} \right) \). For detailed explanation (see [3]).
Dhamija et al. [10] proposed the Kantorovich form of modified Szász–Mirakyan operators. Several researchers have also studied Kantorovich form of different linear positive operators and established local and global approximation results. More details (see [1, 2, 13, 15]).
Mohiuddine et al. [13] proposed the Kantorovich form of the operators (1.1), which is given as
where \(p_{n,k}^{\left( \alpha \right) }(x)\) is defined in (1.1).
For \(\alpha =1\) and \(q=1\) the operators (1.4) reduces to Bernstein–Kantorovich operators.
Motivated from the above stated work, we consider the q-analogue of the operators (1.3) as follows:
and \(p_{n,q,k}^{\left( \alpha \right) }(x)\) is given in (1.2).
In this paper, we estimated the moments of the proposed operators and discuss the rate of convergence using modulus of continuity.
2 Basic Results
In this section, we prove some auxiliary result to prove our main results.
Lemma 2.1
From [8], we have \(B_{n,q}^{(\alpha )}(1;x) = 1\), \(B_{n,q}^{(\alpha )}(t;x) = x\) and
Lemma 2.2
-
(i)
\(K_{n,q}^{(\alpha )}(1;x) = 1\);
-
(ii)
\(K_{n,q}^{(\alpha )}(t;x) = \frac{{2q{{[n]}_q}}}{{{{\left[ 2 \right] }_q}{{[n + 1]}_q}}}x +\frac{1}{{{{\left[ 2 \right] }_q}{{\left[ {n + 1} \right] }_q}}}\);
-
(iii)
\(K_{n,q}^{(\alpha )}({t^2};x) = \frac{{3{q^2}[n]_q^2}}{{{{\left[ 3 \right] }_q}[n + 1]_q^2}}{x^2} + \frac{{3{q^2}}}{{{{\left[ 3 \right] }_q}[n + 1]_q^2}}\left( {{{[n]}_q} + (1 - \alpha ){q^{n - 1}}{[2]}_q} \right) x(1 - x)\) \(+\,\frac{{3q{{[n]}_q}x}}{{{{\left[ 3 \right] }_q}\left[ {n + 1} \right] _q^2}} + \frac{1}{{{{\left[ 3 \right] }_q}\left[ {n + 1} \right] _q^2}}.\)
Proof
From [15], \(\int \limits _{\frac{{q{{\left[ k \right] }_q}}}{{{{\left[ {n + 1} \right] }_q}}}}^{\frac{{{{\left[ {k + 1} \right] }_q}}}{{{{\left[ {n + 1} \right] }_q}}}} {1{d_q}t = \frac{1}{{{{\left[ {n + 1} \right] }_q}}}}\), \(\int \limits _{\frac{{q{{\left[ k \right] }_q}}}{{{{\left[ {n + 1} \right] }_q}}}}^{\frac{{{{\left[ {k + 1} \right] }_q}}}{{{{\left[ {n + 1} \right] }_q}}}} {t{d_q}t = \frac{{2q{{\left[ k \right] }_q}}}{{{{\left[ 2 \right] }_q}\left[ {n + 1} \right] _q^2}}} + \frac{1}{{{{\left[ 2 \right] }_q}\left[ {n + 1} \right] _q^2}}\) and
It is easy to say that \(K_{n,q}^{(\alpha )}(1;x) = 1\).
For \(f(t)=t\) and using Lemma 2.1, we have
Similarly, for \(f(t)=t^2\), we can estimate. So here we skip. \(\square \)
Lemma 2.3
The central moments for the operators (1.4) are as follows:
-
(i)
\(K_{n,q}^{(\alpha )}(t - x;x) = \frac{{2q{{[n]}_q}}}{{{{[2]}_q}{{[n + 1]}_q}}}x + \frac{1}{{{{[2]}_q}{{[n + 1]}_q}}}\);
-
(ii)
\(K_{n,q}^{(\alpha )}({(t - x)^2};x) = \left( {\frac{{3{q^2}{{[n]}_q^2}}}{{{{[3]}_q}[n + 1]_q^2}} - \frac{{4q{{[n]}_q}}}{{{{[2]}_q}{{[n + 1]}_q}}} + 1} \right) {x^2}\) \(+\,\frac{{3{q^2}}}{{{{[3]}_q}{{[n + 1]}_q}}}\left( {{{[n]}_q} + {{[2]}_q}(1 - \alpha ){q^{n - 1}}} \right) x(1 - x) + \left( {\frac{{3q{{[n]}_q}}}{{{{[3]}_q}[n + 1]_q^2}} - \frac{2}{{{{[3]}_q}{{[n + 1]}_q}}}} \right) x\) \( +\, \frac{1}{{{{[3]}_q}[n + 1]_q^2}}\).
Proof
Using linearity property of the operators (1.4) and Lemma 2.2, we get the required results. \(\square \)
Lemma 2.4
Let \(0<q<1\) and \(c \in [0,q d]\), \(d>0\). Then the inequality
Proof
For the proof of the Lemma (see [15]). \(\square \)
3 Main Results
Let C[0, 1] be the space of all continuous functions on [0, 1] with sup-norm \(\left\| f \right\| : = {\sup _{x \in [0,1]}}\left| {f(x)} \right| \). Let \(f \in C\left[ {0,1} \right] \) and \(\delta > 0\). Then the modulus of continuity \(\omega \left( {f,\delta } \right) \) is given as:
It is well-known \(\mathop {\lim }\limits _{\delta \rightarrow 0} \omega (f;\delta ) = 0\). For \(f \in C[0,1]\) and \(x,t \in [0,1]\), we have
For \(f \in C[0,1]\) the Peetre K-functional is given by
where \(\delta >0\) and \({W^2} = \left\{ {g \in C [0,1]:{g^{\prime }},{g^{\prime \prime }} \in C [0,1]} \right\} \). In [9], there exists an absolute constant \(\lambda >0\), such that
and the second-order modulus of continuity \({\omega _2}(.;\delta )\) for \(f\in C [0,1]\) as follows:
Theorem 3.1
For \(0 <q\le 1\), \(q=\left\{ {{q_n}} \right\} \) be a sequence converging to 1 as \(n\rightarrow \infty \). Then, for all \(f \in C [0,1]\) and \(\alpha \in [0,1]\), it implies \(K_{n,q}^{(\alpha )}(f;x)\) converges to f(x) uniformly on [0, 1] for sufficiently large n.
Proof
From Lemma 2.2, \(\mathop {\lim }\nolimits _{n \rightarrow \infty } {q_n} = 1\), we have \(\mathop {\lim }\nolimits _{n\rightarrow \infty } K_{n,q}^{(\alpha )}(1;x) = 1\), \(\mathop {\lim }\nolimits _{n\rightarrow \infty } K_{n,q}^{(\alpha )}(t;x) = x\) and \(\mathop {\lim }\nolimits _{n\rightarrow \infty } K_{n,q}^{(\alpha )}(t^2;x) = x^2\). Then by Bohaman–Korovkin theorem \(\mathop {\lim }\nolimits _{n\rightarrow \infty } K_{n,q}^{(\alpha )}(f(t);x) = f(x)\) converges uniformly on [0, 1]. \(\square \)
Theorem 3.2
For \(f \in C [0,1]\), \(q \in (0,1)\) and \(\alpha \in [0,1]\), we have
where \(\mu _{n,2}^q(x)\) and \(\mu _{n,1}^q(x)\) are second- and first-central moments of the operators (1.4).
Proof
We define an auxiliary operators
For the operators \(\hat{K}_{n,q}^{(\alpha )}(.;x)\), we get
Suppose, \(g \in W^2\), \(x,t \in [0,1]\). Then by Tylor’s expansion, we have
Applying \(\hat{K}_{n,q}^{(\alpha )}(.;x)\) in above equation, we have
Therefore,
From (3.3), we have
From (3.3), (3.5) and (3.6), we have
Now taking infimum on the right-hand side of the above inequality over \(g \in W^2\), we get
From (3.2), we get
Hence, this is our required result. \(\square \)
Theorem 3.3
Let \(q_{n} \in (0,1)\) be a sequence converging to 1 and \(\alpha \) is fixed. Then for \(f \in C [0,1]\), we have
where \({\delta _n}(x) = {\left( {K_{n,q}^{(\alpha )}({{(t - x)}^2};x)} \right) ^{\frac{1}{2}}}\).
Proof
For nondecreasing function \(f \in C [0,1]\). Using linearity and monotonicity of \(K_{n,q}^{(\alpha )}\), we have
Applying Lemma 2.4 with \(c = \frac{{q{{[k]}_q}}}{{{{[n + 1]}_q}}}\) and \(d = \frac{{{{[k+1]}_q}}}{{{{[n + 1]}_q}}}\), we get
Using H\(\ddot{o}\)lder’s inequality for sums, we have
By choosing \(\delta = \delta _{n}(x)\), we get the required result. \(\square \)
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Pratap, R., Deo, N. (2020). Q-Analogue of Generalized Bernstein–Kantorovich Operators. In: Deo, N., Gupta, V., Acu, A., Agrawal, P. (eds) Mathematical Analysis I: Approximation Theory . ICRAPAM 2018. Springer Proceedings in Mathematics & Statistics, vol 306. Springer, Singapore. https://doi.org/10.1007/978-981-15-1153-0_6
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