1 Introduction
In the field of mathematical analysis, Karl Weierstrass established an elegant theorem, the first Weierstrass approximation theorem, in 1885. This theorem has specially a big role in polynomial interpolation corresponding to every continuous function \(f(x)\) on interval \([a,b]\). The proof given by Weierstrass was rigorous and difficult to understand. In 1912, Bernstein [1] gave a simple proof of this theorem by introducing the Bernstein polynomials with the aid of the binomial distribution, hence for \(f\in C[0,1]\), we have
where \(\mathcal{S}_{n,k}(x)=\binom{n}{k}x^{k}(1-x)^{n-k}\). Many mathematicians researched in this direction and studied various modifications in several functional spaces using different error optimization techniques, i.e., Acar et al. [2‐7], Acu et al. [8, 9], Barbosu [10], Agrawal et al. [11], Aral [12], Mursaleen et al. [13‐17], Srivastava et al. [18‐20]; for more details see also the references therein and [21‐30].
$$ B_{n}(f;x)=\sum_{k=0}^{n} \mathcal{S}_{n,k}(x)f \biggl(\frac{k}{n} \biggr),\quad n\in \mathbb{N}, 0\leq x\leq 1, $$
(1.1)
2 Construction of the α-Baskakov–Durrmeyer operators and estimation of their moments
Recently, Cai, Lian and Zhou [31] presented a new sequence of α-Bernstein operators with \(\alpha \in [-1,1]\). Later, Ali Aral et al. [32] gave a sequence of α-Bernstein operators as follows:
where \(f\in C_{B}[0,\infty )\) which denotes the set of all continuous and bounded functions and
with
The operators defined by (2.1) are restricted for continuous functions only. To approximate the functions in Lebesgue measurable space, we design a new sequence of operators:
where \(\mathcal{Q}_{n,k}(t)=\frac{1}{B(k+1,n)} \frac{t^{k}}{(1+t)^{(n+k+1)}}\). Note that, simply in the case of \(\alpha =1\), the operators reduced to Baskakov–Durrmeyer type operators; for details see [33].
$$ L_{n,\alpha }(f;x)=\sum_{k=0}^{\infty }f \biggl(\frac{k}{n} \biggr) \mathcal{S}_{n,k}^{(\alpha )}(x), \quad n\in \mathbb{N}, x\in [0,\infty ), $$
(2.1)
$$\begin{aligned} \mathcal{S}_{n,k}^{(\alpha )}(x) =&\frac{x^{k-1}}{(1+x)^{n+k-1}} \biggl\{ \frac{\alpha x}{1+x}\binom{n+k-1}{k}-(1-\alpha ) (1+x) \binom{n+k-3}{k-2} \\ &{}+(1-\alpha )y\binom{n+k-1}{k} \biggr\} \end{aligned}$$
$$ \binom{n-3}{-2}=\binom{n-2}{-1}=0. $$
$$ L_{n,\alpha }^{*}(f;x)=\sum _{k=0}^{\infty }\mathcal{S}_{n,k}^{(\alpha )}(x) \int _{0}^{\infty }\mathcal{Q}_{n,k}(t)f(t)\,dt, $$
(2.2)
Anzeige
For \(r\in \{0,1,2,3,4\}\), we consider the test functions and central moments,
$$ e_{r}=t^{r} \quad \mbox{and} \quad \psi _{y}^{r}(t;x)=(t-x)^{r}. $$
(2.3)
Lemma 2.1
([31])
We have
$$\begin{aligned}& L_{n,\alpha }(e_{0};x) = 1, \\& L_{n,\alpha }(e_{1};x) = x+\frac{2}{n}(\alpha -1), \\& L_{n,\alpha }(e_{2};x) = x^{2}+\frac{4\alpha -3}{n}x+ \frac{1}{n^{2}}(n+4 \alpha -4). \end{aligned}$$
Lemma 2.2
Let the test functions
\(e_{r}\)
defined by (2.3), then, for all
\(L_{n,\alpha }^{*}\), we have
$$\begin{aligned}& L_{n,\alpha }^{*}(e_{0};x) = 1, \\& L_{n,\alpha }^{*}(e_{1};x) = \biggl(\frac{n}{n-1}+ \frac{2(\alpha -1)}{n-1} \biggr)x+\frac{1}{n-1}, \\& L_{n,\alpha }^{*}(e_{2};x) = \biggl(\frac{n^{2}}{(n-2)(n-1)}+ \frac{n(4 \alpha -3)}{(n-2)(n-1)} \biggr)x^{2}+ \frac{(4n+10\alpha -10)}{(n-2)(n-1)}x+ \frac{2}{(n-2)(n-1)}. \end{aligned}$$
Anzeige
Proof
Take \(f=e_{0}\), then from Lemma 2.1, we have
For \(r=1\)
For \(r=2\)
□
$$\begin{aligned} L_{n,\alpha }^{*}(e_{0};x) =&\sum _{k=0}^{\infty }\mathcal{S}_{n,k} ^{(\alpha )}(x) \int _{0}^{\infty }\mathcal{Q}_{n,k}(t)\,dt \\ =&\sum_{k=0}^{\infty }\mathcal{S}_{n,k}^{(\alpha )}(x) \frac{B(k+1,n)}{B(k+1,n)} \\ =&\sum_{k=0}^{\infty }\mathcal{S}_{n,k}^{(\alpha )}(x) \\ =&1. \end{aligned}$$
$$\begin{aligned} L_{n,\alpha }^{*}(e_{1};x) =&\sum _{k=0}^{\infty }\mathcal{S}_{n,k} ^{(\alpha )}(x) \int _{0}^{\infty }t \mathcal{Q}_{n,k}(t)\,dt \\ =&\sum_{k=0}^{\infty }\mathcal{S}_{n,k}^{(\alpha )}(x) \frac{B(k+2,n-1)}{B(k+1,n)} \\ =&\sum_{k=0}^{\infty }\mathcal{S}_{n,k}^{(\alpha )}(x) \frac{(k+1)B(k+1,n)}{(n-1)B(k+1,n)} \\ =&\sum_{k=0}^{\infty }\mathcal{S}_{n,k}^{(\alpha )}(x) \frac{(k+1)}{(n-1)} \\ =& \biggl(\frac{n}{n-1}+\frac{2(\alpha -1)}{n-1} \biggr)x+ \frac{1}{n-1}. \end{aligned}$$
$$\begin{aligned} L_{n,\alpha }^{*}(e_{2};x) =&\sum _{k=0}^{\infty }\mathcal{S}_{n,k} ^{(\alpha )}(x) \int _{0}^{\infty }t^{2} \mathcal{Q}_{n,k}(t) \,dt \\ =&\sum_{k=0}^{\infty }\mathcal{S}_{n,k}^{(\alpha )}(x) \frac{B(k+3,n-2)}{B(k+1,n)} \\ =&\sum_{k=0}^{\infty }\mathcal{S}_{n,k}^{(\alpha )}(x) \frac{(k+2)(k+1)B(k+1,n)}{(n-2)(n-1)B(k+1,n)} \\ =&\sum_{k=0}^{\infty }\mathcal{S}_{n,k}^{(\alpha )}(x) \frac{(k+2)(k+1)}{(n-2)(n-1)} \\ =&\frac{n^{2}+n(4\alpha -3)}{(n-2)(n-1)}x^{2}+ \frac{(4n+10\alpha -10)}{(n-2)(n-1)}x+\frac{2}{(n-2)(n-1)}. \end{aligned}$$
Lemma 2.3
Let the operators given by (2.2). Then we have
$$\begin{aligned}& L_{n,\alpha }^{*}\bigl(\psi _{x}^{0};x\bigr) = 1, \\& L_{n,\alpha }^{*}\bigl(\psi _{x}^{1};x\bigr) = \frac{2\alpha -1}{n-1}x+ \frac{1}{n-1}, \\& L_{n,\alpha }^{*}\bigl(\psi _{x}^{2};x\bigr) = \frac{2n+2(4\alpha -3)}{(n-2)(n-1)}x^{2}+ \frac{2n+2(5\alpha -3)}{(n-2)(n-1)}x+\frac{2}{(n-2)(n-1)}. \end{aligned}$$
3 Approximation in Korovkin and weighted Korovkin spaces
Take \(C_{B}(\mathbb{R^{+}})\) be the space of all bounded and continuous functions defined on the set \(\mathbb{R^{+}}\), where \(\mathbb{R^{+}}=[0, \infty )\) and a normed defined on \(C_{B}\) as
Let
$$ \Vert f \Vert _{C_{B}}=\sup _{x\geq 0} \bigl\vert f(x) \bigr\vert . $$
$$ E:=\biggl\{ f:x\in \mathbb{R^{+}} \text{ and } \lim _{x\rightarrow \infty } \biggl(\frac{f(x)}{1+x^{2}} \biggr)< \infty \biggr\} . $$
Lemma 3.1
For every
\(f\in C[0,\infty )\cap E\)
the operators
\(L_{n,\alpha } ^{*}\)
given in (2.2) are uniformly convergent to
f
on each compact subset of
\([0,A]\), whenever
\(A\in (0,\infty )\).
Proof
In the view of Korovkin-type property, it is enough to show that
From Lemma 2.2, obviously \(L_{n,\alpha }^{*}(e_{0};y)\rightarrow e_{0}(x)\) as \(n\rightarrow \infty \) and for \(s=1\)
Similarly, we can prove for \(s=2\) that \(L_{n,\alpha }^{*}(e_{2};x) \rightarrow e_{2}\), which proves Proposition 3.1. □
$$ L_{n,\alpha }^{*}(e_{s};x)\rightarrow e_{s}(x), \quad \text{for } s=0,1,2. $$
$$ \lim_{n\rightarrow \infty } L_{n,\alpha }^{*}(e_{1};x)= \lim_{n\rightarrow \infty } \biggl(\frac{n+2(\alpha -1)}{n-1}x+ \frac{1}{n-1} \biggr)=e_{1}(x). $$
Suppose \(C[0,\infty )\) is the set of all continuous functions and \(f\in C[0,\infty )\) with the weight function \(\sigma (x)=1+x^{2}\),
where the norm defined on weight function σ such as \(\Vert f \Vert _{\sigma }=\sup_{x\in [0,\infty )}\frac{ \vert f(x) \vert }{\sigma (x)}\) and the constant \(\mathcal{M}_{f}\) depends only on f.
$$\begin{aligned}& \mathfrak{P}_{\sigma }(x) = \bigl\{ f: \bigl\vert f(x) \bigr\vert \leq \mathcal{M}_{f}\sigma (x), x\in [0,\infty ) \bigr\} , \\& \mathfrak{Q}_{\sigma }(x) = \bigl\{ f:f\in C[0,\infty )\cap \mathfrak{P}_{\sigma }(x) , x\in [0,\infty ) \bigr\} , \\& \mathfrak{Q}_{\sigma }^{m}(x) = \biggl\{ f:f\in \mathfrak{Q}_{\sigma }(x), \lim_{x\rightarrow \infty }\frac{f(x)}{\sigma (x)}=m, x\in [0,\infty ) \biggr\} , \end{aligned}$$
Theorem 3.2
For all
\(f\in \mathfrak{Q}_{\sigma }^{m}(x)\)
the operators
\(L_{n,\alpha }^{*}( \cdot\, ; \cdot )\)
defined by (2.2) satisfy
$$ \lim_{n\to \infty } \bigl\Vert L_{n,\alpha }^{*}(f;x)-f \bigr\Vert _{ \sigma }=0. $$
Proof
Take \(f(t) \in \mathfrak{Q}_{\sigma }^{m}(x)\) with \(x\in [0,\infty )\) and \(f(t)=e_{\nu }\) for \(\nu =0,1,2\). Then from the well-known Korovkin theorem \(L_{n,\alpha }^{*}(e_{\nu };x)\rightarrow x^{\nu }\), satisfying the properties of uniformly behaving as \(n \to \infty \). Since for \(\nu =0\), from Lemma 2.2
\(L_{n,\alpha }^{*}(e_{0};x)=1\), thus we have
$$ \bigl\Vert L_{n,\alpha }^{*} (e_{0};x ) -1 \bigr\Vert _{\sigma } =0. $$
(3.1)
For \(\nu =1\), we have
$$\begin{aligned} \bigl\Vert L_{n,\alpha }^{*} (e_{1};x ) -x \bigr\Vert _{\sigma } &= \sup_{x \in [0,\infty )}\frac{ \vert L _{n,\alpha }^{*}(e_{1};x)-x \vert }{1+x^{2}} \\ &= \biggl(\frac{n+2(\alpha -1)}{n-1}-1 \biggr)\sup_{x \in [0,\infty )} \frac{x}{1+x ^{2}}+\frac{1}{(n-1)}\sup_{x \in [0,\infty )} \frac{1}{1+x^{2}}. \end{aligned}$$
As \(n \to \infty \),
In a similar way for \(\nu =2\),
This completes the proof. □
$$ \bigl\Vert L_{n,\alpha }^{*} (e_{1};x ) -x \bigr\Vert _{\sigma } =0. $$
(3.2)
$$\begin{aligned} & \bigl\Vert L_{n,\alpha }^{*} (e_{2};x ) -x^{2} \bigr\Vert _{\sigma } \\ &\quad = \sup_{y \in [0,\infty )}\frac{ \vert L_{n,\alpha }^{*}(e_{2};x)-x ^{2} \vert }{1+x^{2}} \\ &\quad = \biggl(\frac{n^{2}+n(4\alpha -3)}{(n-2)(n-1)}-1 \biggr) \sup_{x \in [0,\infty )} \frac{x^{2}}{1+x^{2}} \\ &\qquad {} + \biggl(\frac{4n+10\alpha -10}{(n-2)(n-1)} \biggr)\sup_{x \in [0, \infty )} \frac{x}{1+x^{2}}+ \frac{2}{(n-2)(n-1)}\sup_{x \in [0, \infty )} \frac{1}{1+x^{2}}, \\ & \bigl\Vert L_{n,\alpha }^{*} (e_{2};x ) -x^{2} \bigr\Vert _{\sigma } =0 \quad \text{when } n \to \infty. \end{aligned}$$
(3.3)
4 Pointwise approximation properties by \(L_{n,\alpha }^{*}\)
Here, we study the order of approximation of a function f with the aid of positive linear operators \(L_{n,\alpha }^{*}(f;x)\) defined by (2.2) in terms of the classical modulus of continuity, the second-order modulus of continuity, Peetres K-functional and the Lipschitz class. A well-known property is the modulus of continuity of order one and of order two defined as follows. For \(\delta >0\) and \(f\in C[a,b]\) the classical modulus of continuity of order one is given by
and of order two it is given by
$$ \omega (f;\delta )=\sup_{x_{1},x_{2}\in [a,b], |x_{1}-x_{2}|\leq \delta } \bigl\vert f(x_{1})-f(x _{2}) \bigr\vert , $$
$$ \omega _{2}\bigl(f;\delta ^{\frac{1}{2}}\bigr)=\sup _{0< h< \delta ^{\frac{1}{2}}} \sup_{x\in \mathbb{R}^{+}} \bigl\vert f(x)-2f(x+h)+f(x+2h) \bigr\vert . $$
(4.1)
Let \(C_{B}[0,\infty )\) denote the space of all bounded and continuous functions on \([0,\infty )\) and
with the norm
also
$$ C_{B}^{2}[0,\infty )=\bigl\{ \psi \in C_{B}[0, \infty ):\psi ^{\prime }, \psi ^{\prime \prime }\in C_{B}[0,\infty ) \bigr\} , $$
(4.2)
$$ \Vert \psi \Vert _{C_{B}^{2}[0,\infty )}= \Vert \psi \Vert _{C_{B}[0,\infty )}+ \bigl\Vert \psi ^{\prime } \bigr\Vert _{C_{B}[0,\infty )}+ \bigl\Vert \psi ^{\prime \prime } \bigr\Vert _{C_{B}[0,\infty )}, $$
(4.3)
$$ \Vert \psi \Vert _{C_{B}[0,\infty )}=\sup_{x\in [0,\infty )} \bigl\vert \psi (x) \bigr\vert . $$
(4.4)
Lemma 4.1
([31])
Let
\(\{P_{n}\}_{n\geq 1}\)
be the sequence for the positive integer
n
with
\(P_{n}(1;x)=1\). Then for every
\(\psi \in C_{B}^{2}[0,\infty )\)
$$ \bigl\vert P_{n}(\psi ;x)-\psi (x) \bigr\vert \leq \bigl\Vert g' \bigr\Vert \sqrt{P_{n}\bigl((s-x)^{2};x \bigr)}+ \frac{1}{2} \bigl\Vert \psi '' \bigr\Vert P_{n}\bigl((s-x)^{2};x\bigr). $$
Lemma 4.2
([31])
For all
\(f\in C[a,b]\)
and
\(h\in (0,\frac{b-a}{2} ) \), we have the following inequalities:
where
\(f_{h}\)
denotes the second-order Steklov function.
$$\begin{aligned} (\mathrm{i})&\quad \Vert f_{h}-f \Vert \leq \frac{3}{4} \omega _{2}(f,h), \\ (\mathrm{ii})&\quad \bigl\Vert f_{h}'' \bigr\Vert \leq \frac{3}{2h^{2}}\omega _{2}(f,h), \end{aligned}$$
Theorem 4.3
For all
\(f\in C_{B}[0,\infty )\)
and
\(x\in [0,a]\), \(a>0\)
we have
where
\(\varTheta _{n}(x)=L_{n,\alpha }^{*}(\psi _{x}^{2};x)\)
and
\(L_{n,\alpha }^{*}(\psi _{x}^{2};x)\)
is defined by Lemma
2.3.
$$ \bigl\vert L_{n,\alpha }^{*}(f;x)-f(x) \bigr\vert \leq 2\omega \bigl(f; \sqrt{\varTheta _{n}(x)} \bigr), $$
Proof
In view of the classical modulus of continuity, we have
In the light of the Cauchy–Schwartz inequality, we get
Choosing \(\delta = (\varTheta _{n}(x) )^{\frac{1}{2}}=\sqrt{ L_{n,\alpha }^{*}(\psi _{x}^{2};x)}\), we arrive at the desired result. □
$$\begin{aligned} \bigl\vert L_{n,\alpha }^{*}(f;x)-f(x) \bigr\vert \leq & \sum _{k=0}^{\infty }\mathcal{S} _{n,k}^{(\alpha )}(x) \int _{0}^{\infty }\mathcal{Q}_{n,k}(t) \bigl\vert f(t)-f(x) \bigr\vert \,dt \\ \leq & \Biggl\{ 1+\frac{1}{\delta }\sum_{k=0}^{\infty } \mathcal{S}_{n,k} ^{(\alpha )}(x) \int _{0}^{\infty }\mathcal{Q}_{n,k}(t) \vert t-x \vert \,dt \Biggr\} \omega (f;\delta ). \end{aligned}$$
$$\begin{aligned} \bigl\vert L_{n,\alpha }^{*}(f;x)-f(x) \bigr\vert \leq & \Biggl\{ 1+\frac{1}{\delta } \Biggl(\sum_{k=0}^{\infty } \mathcal{S}_{n,k}^{(\alpha )}(x) \int _{0}^{\infty } \mathcal{Q}_{n,k}(t) (t-x)^{2}\,dt \Biggr)^{\frac{1}{2}} \Biggr\} \omega (f;\delta ) \\ =& \biggl\{ 1+\frac{1}{\delta } \sqrt{L_{n,\alpha }^{*} \bigl(\psi _{x}^{2};x\bigr)} \biggr\} \omega (f;\delta ). \end{aligned}$$
Theorem 4.4
For every
\(f\in C[0,a]\), \(a>0\)
the operators
\(L_{n,\alpha }^{*}( \cdot\, ; \cdot )\)
defined by (2.2) satisfy
where
\(\delta = (\varTheta _{n}(x) )^{\frac{1}{2}}\)
is defined by Theorem
4.3
and
\(\omega _{2}(f;\delta )\)
is by (4.1) equipped with the norm
\(\Vert f \Vert =\max_{x\in [a,b]}|f(x)|\).
$$ \bigl\vert L_{n,\alpha }^{*}(f;x)-f(x) \bigr\vert \leq \frac{2}{a} \Vert f \Vert \delta ^{2}+ \frac{3}{4} \bigl(a+2+h^{2}\bigr)\omega _{2}(f;\delta ), $$
Proof
Consider \(f_{h}\) is the Steklov function define in Lemma 4.2. Using Lemma 2.2, we obtain
In view of the fact that \(f_{h}\in C^{2}[0,a]\) and using Lemma 4.1, we obtain
From the Landau inequality and Lemma 4.2, we have
On choosing \(\delta = (\varTheta _{n}(x) )^{\frac{1}{4}}\), one has
Combining (4.6), (4.5) and Lemma 4.2, we obtain the required result. □
$$\begin{aligned} \bigl\vert L_{n,\alpha }^{*}(f;x)-f(x) \bigr\vert \leq & \bigl\vert L_{n,\alpha }^{*}(f-f_{h};x) \bigr\vert + \bigl\vert f _{h}-f(x) \bigr\vert + \bigl\vert L_{n,\alpha }^{*}(f_{h};x)-f_{h}(x) \bigr\vert \\ \leq & 2 \Vert f_{h}-f \Vert + \bigl\vert L_{n,\alpha }^{*}(f_{h};x)-f_{h}(x) \bigr\vert . \end{aligned}$$
$$ \bigl\vert L_{n,\alpha }^{*}(f;x)-f(x) \bigr\vert \leq \bigl\Vert f_{h}' \bigr\Vert \sqrt {L_{n,\alpha }^{*}\bigl((e _{1}-x)^{2};x \bigr)}+\frac{1}{2} \bigl\Vert f''_{h} \bigr\Vert L_{n,\alpha }^{*}\bigl((e_{1}-x)^{2};x \bigr). $$
(4.5)
$$\begin{aligned} \Vert f_{h} \Vert \leq &\frac{2}{a} \Vert f_{h} \Vert +\frac{a}{2} \bigl\Vert f_{h}'' \bigr\Vert \\ \leq &\frac{2}{a} \Vert f_{h} \Vert +\frac{3a}{4} \frac{1}{h^{2}}\omega _{2}(f;h). \end{aligned}$$
$$ \bigl\vert L_{n,\alpha }^{*}(f_{h};x)-f_{h}(x) \bigr\vert \leq \frac{2}{a} \Vert f \Vert h^{2}+ \frac{3a}{4}\omega _{2}(f;h)+\frac{3}{4}h^{2} \omega _{2}(f;h). $$
(4.6)
Theorem 4.5
Let
\(L_{n,\alpha }^{*}( \cdot\, ; \cdot )\)
be the operators defined by (2.2). Then, for every
\(f\in C_{B}^{2}[0,\infty )\),
uniformly for
\(0\leq x\leq a\), \(a>0\).
$$ \lim_{n\rightarrow \infty }(n-1) \bigl(L_{n,\alpha }^{*}(f;x)-f(x) \bigr)=\bigl(1+2 \alpha x-x^{2}\bigr)f'(x)+2 \bigl(x+x^{2}\bigr)f''(x), $$
Proof
Let \(x_{0}\in [0,\infty )\) be a fixed number; all \(x\in [0,\infty )\). Then using Taylor’s series, we have
where \(\varphi (x,x_{0})\in C_{B}[0,\infty )\) and \(\lim_{x\rightarrow x_{0}}\varphi (x,x_{0})=0\).
$$ f(x)-f(x_{0})=(x-x_{0})f'(x_{0})+ \frac{1}{2}(x-x_{0})^{2}f''(x_{0})+ \varphi (x,x_{0}) (x-x_{0})^{2}, $$
(4.7)
By applying the operators \(L_{n,\alpha }^{*}\) on (4.7), we deduce
In view of the Cauchy–Schwartz inequality for the last term of Eq. (4.8), we get
We have
This completes the proof. □
$$\begin{aligned} L_{n,\alpha }^{*}(f;x_{0})-f(x_{0}) =&f'(x_{0})L_{n,\alpha }^{*}(e _{1}-x_{0};x_{0})+\frac{1}{2}L_{n,\alpha }^{*} \bigl((x-x_{0})^{2};x_{0}\bigr)f''(x _{0}) \\ &{}+L_{n,\alpha }^{*}\bigl(\varphi (x,x_{0}) (x-x_{0})^{2}\bigr). \end{aligned}$$
(4.8)
$$ (n-1)L_{n,\alpha }^{*}\bigl(\varphi (x,x_{0}) (t-x_{0})^{2}\bigr)\leq (n-1)^{2}\sqrt{L _{n,\alpha }^{*} \bigl((e_{1}-x_{0})^{2}\bigr)L_{n,\alpha }^{*} \bigl(\varphi ^{2}(x,x _{0})\bigr)}. $$
(4.9)
$$\begin{aligned}& \lim_{n\rightarrow \infty }(n-1) \bigl(L_{n,\alpha }^{*}(e_{0}-x _{0};x) \bigr) = \bigl(1+2\alpha x-x^{2} \bigr)f'(x), \\& \lim_{n\rightarrow \infty }(n-1) \bigl(L_{n,\alpha }^{*}\bigl((e _{0}-x_{0})^{2};x\bigr) \bigr) = 2 \bigl(x+x^{2}\bigr)f''(x), \\& \lim_{n\rightarrow \infty } \bigl(L_{n,\alpha }^{*} \bigl((e_{0}-x _{0})^{4};x\bigr) \bigr) = 0. \end{aligned}$$
Now here we estimate the rate of convergence in terms of the usual Lipschitz class \(\operatorname{Lip}_{M}(\nu )\). Let \(f\in C[0,a )\), \(a>0\) and M be a positive constant, and, for any \(\nu \in (0,1]\), the Lipschitz class \(\operatorname{Lip}_{M}(\nu )\) is as follows:
$$ \operatorname{Lip}_{M}(\nu )= \bigl\{ f: \bigl\vert f(\varsigma _{1})-f(\varsigma _{2}) \bigr\vert \leq M \vert \varsigma _{1}-\varsigma _{2} \vert ^{\nu }\ \bigl( \varsigma _{1}, \varsigma _{2}\in [ 0,\infty)\bigr) \bigr\} . $$
(4.10)
Theorem 4.6
Let
\(f\in \operatorname{Lip}_{M}(\nu )\)
with
\(M>0\)
and
\(0<\nu \leq 1\). Then the operators
\(L_{n,\alpha }^{*}( \cdot\, ; \cdot )\)
satisfy
where
\(n>2\)
and
\(\varTheta _{n}(x)\)
defined by Theorem
4.3.
$$ \bigl\vert L_{n,\alpha }^{*}(f;x)-f(x) \bigr\vert \leq M \bigl( \varTheta _{n}(x) \bigr) ^{\frac{\nu }{2}}, $$
Proof
From the Hölder inequality and (4.10), we conclude
Hence
This completes the proof. □
$$\begin{aligned} \bigl\vert L_{n,\alpha }^{*}(f;x)-f(x) \bigr\vert \leq & \bigl\vert L_{n,\alpha }^{*}\bigl(f(t)-f(x);x\bigr) \bigr\vert \\ \leq &L_{n,\alpha }^{*} \bigl( \bigl\vert f(t)-f(x) \bigr\vert ;x \bigr) \\ \leq & ML_{n,\alpha }^{*} \bigl( \vert t-x \vert ^{\nu };x \bigr) . \end{aligned}$$
$$\begin{aligned}& \bigl\vert L_{n,\alpha }^{*}(f;x)-f(x) \bigr\vert \\& \quad \leq M \sum_{k=0}^{\infty } \mathcal{S}_{n,k}^{(\alpha )}(x) \int _{0}^{\infty }\mathcal{Q}_{n,k}(t) \vert t-x \vert ^{\nu }\,dt \\& \quad \leq M \sum_{k=0}^{\infty } \bigl( \mathcal{S}_{n,k}^{( \alpha )}(x) \bigr)^{\frac{2-\nu }{2}} \\& \qquad {} \times \bigl(\mathcal{S}_{n,k}^{(\alpha )}(x) \bigr)^{\frac{ \nu }{2}} \int _{0}^{\infty }\mathcal{Q}_{n,k}(t) \vert t-x \vert ^{\nu }\,dt \\& \quad \leq M \Biggl(\sum_{k=0}^{\infty } \mathcal{S}_{n,k}^{(\alpha )}(x) \int _{0}^{\infty }\mathcal{Q}_{n,k}(t)\,dt \Biggr)^{\frac{2-\nu }{2}} \\& \qquad {} \times \Biggl(\sum_{k=0}^{\infty } \mathcal{S}_{n,k}^{(\alpha )}(x) \int _{0}^{\infty }\mathcal{Q}_{n,k}(t) \vert t-x \vert ^{2} \,dt \Biggr) ^{\frac{\nu }{2}} \\& \quad = M \bigl(L_{n,\alpha }^{*}\bigl(\psi _{x}^{2};x \bigr) \bigr)^{\frac{ \nu }{2}}. \end{aligned}$$
Theorem 4.7
For all
\(\psi \in C_{B}^{2}{}[ 0,\infty )\)
and
\(n>2\),
where
\(\Delta _{n}(x)= (\frac{2\alpha -1}{n-1}x+\frac{1}{n-1} )\)
and
\(\varTheta _{n}(x)\)
is defined by Theorem
4.3.
$$ \bigl\vert L_{n,\alpha }^{*}(\psi ;x)-\psi (x) \bigr\vert \leq \biggl(\Delta _{n}(x)+\frac{ \varTheta _{n}(x)}{2} \biggr) \Vert \psi \Vert _{C_{B}^{2}{}[ 0,\infty )}, $$
Proof
Let \(\psi \in C_{B}^{2}(\mathbb{R}^{+})\); for all \(\varphi \in (x,t)\) a Taylor series expansion is
On applying \(L_{n,\alpha }^{*}\), using linearity,
which implies that
From (4.3) we have \(\Vert \psi ^{\prime } \Vert _{C_{B}{}[ 0,\infty )}\leq \Vert \psi \Vert _{C_{B}^{2}{}[ 0,\infty )}\), \(\Vert \psi ^{\prime \prime } \Vert _{C_{B}{}[ 0,\infty )}\leq \Vert \psi \Vert _{C_{B}^{2}{}[ 0,\infty )}\).
This completes the proof. □
$$ \psi (t)=\frac{(t-x)^{2}}{2}\psi ^{\prime \prime }(\varphi )+(t-x) \psi ^{\prime }(x)+\psi (x). $$
$$ L_{n,\alpha }^{*}(\psi ;x)-\psi (x)=\psi ^{\prime }(x)L_{n,\alpha } ^{*} \bigl( (t-x);x \bigr) + \frac{\psi ^{\prime \prime }(\varphi )}{2}L_{n,\alpha }^{*} \bigl( (t-x)^{2};x \bigr) , $$
$$\begin{aligned}& \bigl\vert L_{n,\alpha }^{*}(\psi ;x)-\psi (x) \bigr\vert \\& \quad \leq \biggl(\frac{2\alpha -1}{n-1}x+\frac{1}{n-1} \biggr) \bigl\Vert \psi ^{\prime } \bigr\Vert _{C_{B}{}[ 0,\infty )} \\& \qquad {}+ \biggl\{ \frac{2n+2(4\alpha -3)}{(n-2)(n-1)}x^{2}+\frac{2n+2(5 \alpha -3)}{(n-2)(n-1)}x+ \frac{2}{(n-2)(n-1)} \biggr\} \frac{ \Vert \psi ^{\prime \prime } \Vert _{C_{B}{}[ 0,\infty )}}{2}. \end{aligned}$$
$$\begin{aligned}& \bigl\vert L_{n,\alpha }^{*}(\psi ;x)-\psi (x) \bigr\vert \\& \quad \leq \biggl(\frac{2\alpha -1}{n-1}x+\frac{1}{n-1} \biggr) \Vert \psi \Vert _{C_{B}^{2}{}[ 0,\infty )} \\& \qquad {}+ {\biggl\{ } \frac{2n+2(4\alpha -3)}{(n-2)(n-1)}x^{2}+\frac{2n+2(5 \alpha -3)}{(n-2)(n-1)}x+ \frac{2}{(n-2)(n-1)} {\biggr\} } \frac{ \Vert \psi \Vert _{{}[ 0,\infty )}}{2}. \end{aligned}$$
In 1968 [34] for investigating the interpolation between two Banach spaces Peetre introduced the K-functional by
and a positive constant \(\mathfrak{D}\) exists such that \(K_{2}(f; \delta )\leq \mathfrak{D} \omega _{2}(f;\delta ^{\frac{1}{2}})\) with \(\delta >0\) and \(\omega _{2}(f;\delta )\) is the second-order modulus of continuity.
$$ K_{2}(f;\delta )=\inf_{C_{B}^{2}{}[ 0,\infty )} \bigl\{ \bigl( \Vert f-\psi \Vert _{C_{B}{}[ 0,\infty )}+\delta \Vert \psi \Vert _{C_{B}^{2}{}[ 0,\infty )} \bigr) :\psi \in C_{B} ^{2}{}[ 0,\infty ) \bigr\} $$
(4.11)
Theorem 4.8
Suppose
\(C_{B}{}[ 0,\infty )\)
is the set of all bounded and continuous functions on
\({}[ 0,\infty )\). Then for every
\(f\in C_{B}{}[ 0,\infty )\)
where
\(\mathfrak{K}_{n}(x)=\frac{2\Delta _{n}(x)+\varTheta _{n}(x)}{4}\)
is defined by Theorem
4.7.
$$ \bigl\vert L_{n,\alpha }^{*}(f;x)-f(x) \bigr\vert \leq 2 \mathfrak{D} \bigl\{ \omega _{2} \bigl( f;\sqrt{\mathfrak{K}_{n}(x)} \bigr) + \min \bigl( 1,\mathfrak{K}_{n}(x) \bigr) \Vert f \Vert _{C_{B}{}[ 0,\infty )}\bigr\} , $$
Proof
In the light of results obtained by Theorem 4.7, we prove the desired theorem; hence
$$\begin{aligned} \bigl\vert L_{n,\alpha }^{*}(f;x)-f(x) \bigr\vert \leq& \bigl\vert L_{n,\alpha }^{*}(f- \psi ;x) \bigr\vert + \bigl\vert f(x)-\psi (x) \bigr\vert + \bigl\vert L_{n,\alpha }^{*}(\psi ;x)- \psi (x) \bigr\vert \\ \leq& 2 \Vert f-\psi \Vert _{C_{B}{}[ 0,\infty )}+ \biggl(\frac{ \varTheta _{n}(x)}{2}+ \Delta _{n}(x) \biggr) \Vert \psi \Vert _{C_{B}^{2}{}[ 0,\infty )} \\ =& 2 \biggl( \Vert f-\psi \Vert _{C_{B}{}[ 0,\infty )}+ \biggl(\frac{\varTheta _{n}(x)}{4}+ \frac{\Delta _{n}(x)}{2} \biggr) \Vert \psi \Vert _{C_{B}^{2}{}[ 0,\infty )} \biggr). \end{aligned}$$
If we take the infimum over all \(\psi \in C_{B}^{2}{}[ 0,\infty )\) and we use (4.11), we get
Now from [35] we use the relation for an absolute constant \(\mathfrak{D}>0\)
This completes the proof. □
$$ \bigl\vert L_{n,\alpha }^{*}(f;x)-f(x) \bigr\vert \leq 2K_{2} \biggl( f; \biggl(\frac{ \varTheta _{n}(x)}{4}+\frac{\Delta _{n}(x)}{2} \biggr) \biggr). $$
$$ K_{2}(f;\delta )\leq \mathfrak{D}\bigl\{ \omega _{2}(f; \sqrt{\delta })+ \min (1,\delta ) \Vert f \Vert \bigr\} . $$
5 Conclusion and observations
The manuscript parametric variant of Baskakov–Durrmeyer operators is a new extension of Baskakov Durrmeyer type operators. In the present investigation in our manuscript in order to get uniform convergence for the operators of the α-type extended version we study the order of approximation, the rate of convergence, the Korovkin-type, the weighted Korovkin-type approximation theorems, Peetres K-functional, Lipschitz functions and a set of direct theorems. It must be noted that we have more modeling flexibility when adding the parameter α to the Baskakov–Durrmeyer operators.
Competing interests
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