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Approximation by Jakimovski–Leviatan-beta operators in weighted space

Abstract

The main purpose of this article is to introduce a more generalized version of Jakimovski–Leviatan-beta operators through the Appell polynomials. We present some uniform convergence results of these operators via Korovkin’s theorem and obtain the rate of convergence by using the modulus of continuity and Lipschitz class. Moreover, we obtain the approximation with the help of Peetre’s K-functional and give some direct theorems.

Introduction

Approximation theory basically deals with problems to find approximation of functions by simpler functions like polynomial. Bernstein [8] was first to construct a sequence of positive linear operators to provide a constructive proof of the well-known Weierstrass approximation theorem. Since then several operators have been defined to study approximation properties in different spaces.

In [13], Jakimovski and Leviatan introduced the following operators and obtained some of their approximation properties:

$$ R_{m}(h;y)=\frac{e^{-my}}{P(1)}\sum _{k=0}^{\infty}Q_{k}(my){h} \biggl(\frac{k}{m} \biggr). $$
(1.1)

For all \(h\in E[0,\infty)\), the set of functions of exponential type on \([0,\infty)\) with \(|{h}(x)|\leq\beta e^{\alpha x}\), \(\alpha,\beta>0\), where \(Q_{k}(y)=\sum_{j=0}^{k}b_{{j}}\frac{y^{k-j}}{(k-j)!}\) (\(k\in\mathbb{N}\)), are Appell polynomials [5] defined by the identity

$$ P(w)e^{wy}=\sum_{m=0}^{\infty}Q_{m}(y)w^{m}, $$
(1.2)

such that \(P(w)=\sum_{m=0}^{\infty}b_{m}w^{m}\), \(P(1)\neq0\) is an analytic function in the disk \(|w|< r\) (\(r>1\)). Note that, for \(P(1)=1\), \(Q_{{m}}(y)=\frac{y}{{m!}}\) and operators (1.1) are reduced to Favard–Szász operators:

$$ S_{m}(h;y)=e^{-my}\sum_{k=0}^{\infty} \frac{(my)^{k}}{k!}{h} \biggl(\frac{k}{m} \biggr). $$
(1.3)

Recently, Büyükyazıcı et al. [9] studied the following operators:

$$ S_{m}^{\ast}(h;y)=\frac{e^{-\frac{m}{b_{m}}y}}{P(1)}\sum _{k=0}^{\infty}Q_{k} \biggl( \frac{m}{b_{m}}y \biggr) {h} \biggl(\frac{k}{m}b_{m} \biggr). $$
(1.4)

In this paper, we generalize the above operators and study their several approximation properties. We investigate a Korovkin-type theorem and obtain the order of convergence by using the modulus of continuity. Furthermore, we obtain the approximation with the help of Lipschitz continuous functions and give some direct theorems.

For more details on the related work, we refer to [14, 6, 7, 11, 14, 15, 1719, 21, 2326].

We define an integral type modification of Jakimovski–Leviatan operators by introducing the sequences of unbounded and increasing functions \(\{u_{m}\}\), \(\{v_{m}\}\) such that

$$ \lim_{m\rightarrow\infty}\frac{u_{m}}{v_{m}}=1+O \biggl( \frac{1}{v_{m}} \biggr)\quad \mbox{and}\quad \lim _{m\rightarrow\infty}\frac{1}{v_{m}}\rightarrow0. $$
(1.5)

Let \(m\in\mathbb{N}\), \(\phi>m\) and

$$ h\in C_{\phi}[0,\infty)=\bigl\{ h\in C[0,\infty):h(t)=O \bigl(t^{\phi}\bigr)\bigr\} . $$
(1.6)

For \(Q_{r}(y)\geq0\) and \(P(1)\neq0\), we define

$$ L_{m}^{u_{m},v_{m}}(h;y)=\frac{e^{-u_{m}y}}{P(1)}\sum _{r=0}^{\infty }Q_{r}(u_{m}y) \frac{1}{B(r+1,m)} \int_{0}^{\infty}\frac {t^{r}}{(1+t)^{r+m+1}}h \biggl( \frac{mt}{v_{m}} \biggr) \,\mathrm{d}t. $$
(1.7)

Moments

Lemma 2.1

Suppose\(e_{i}=t^{i-1}\) (\(i=1,2,3\)). Then the following hold true for operators (1.7):

  1. (1)

    \(L_{m}^{u_{m},v_{m} }(e_{1};y) =1\);

  2. (2)

    \(L_{m}^{u_{m},v_{m} }(e_{2};y)= \frac{u_{m}}{v_{m}}\frac{m}{(m-1)} y+\frac{1}{v_{m}} \frac{m}{(m-1)} \bigl(\frac{P^{\prime}(1)}{P(1)} +1\bigr)\);

  3. (3)

    \(L_{m}^{u_{m},v_{m} }(e_{3};y)= (\frac{u_{m}}{v_{m}} )^{2}\frac{m^{2}}{(m-2)(m-1)} y^{2}+2\frac{u_{m}}{v_{m}^{2}}\frac{m^{2}}{(m-2)(m-1)} \bigl( \frac {P^{\prime }(1)}{P(1)}+2 \bigr)y+ \frac{1}{v_{m}^{2}}\frac{m^{2}}{(m-2)(m-1)} \bigl( \frac{P^{\prime \prime }(1)}{P(1)}+4\frac{P^{\prime}(1)}{P(1)}+2 \bigr)\).

Proof

We can easily see that

$$\begin{aligned}& \sum_{r=0}^{\infty}Q_{r}(u_{m}y)=P(1)e^{u_{m}y}, \end{aligned}$$
(2.1)
$$\begin{aligned}& \sum_{r=0}^{\infty}rQ_{r}(u_{m}y)= \bigl[ P^{\prime }(1)+(u_{m}y )P(1)\bigr] e^{u_{m}y}, \end{aligned}$$
(2.2)
$$\begin{aligned}& \sum_{r=0}^{\infty}r^{2}Q_{r}(u_{m}y)= \bigl[P^{\prime\prime }(1)+2(u_{m}y)P^{\prime }(1)+P^{\prime}(1)+(u_{m}y)P(1)+{(u_{m}y)}^{2}P(1) \bigr]e^{u_{m}y}. \end{aligned}$$
(2.3)

(1) Take \(h=e_{1}\)

$$\begin{aligned} L_{m}^{u_{m},v_{m}}(e_{1};x) =& \frac{e^{-u_{m}y}}{P(1)}\sum_{r=0}^{\infty }Q_{r}(u_{m}y) \frac{1}{B(r+1,m)} \int_{0}^{\infty}\frac {t^{r}}{(1+t)^{r+m+1}}\,\mathrm{d}t \\ =&\frac{e^{-u_{m}y}}{P(1)}\sum_{r=0}^{\infty}Q_{r}(u_{m}y) \frac {B(r+1,m)}{B(r+1,m)} \\ =&1. \end{aligned}$$

(2) Take \(h=e_{2}\)

$$\begin{aligned} L_{m}^{u_{m},v_{m}}(e_{2};y) =& \frac{me^{-u_{m}y}}{v_{m}P(1)}\sum_{r=0}^{\infty}Q_{r}(u_{m}y) \frac{1}{B(r+1,m)} \int_{0}^{\infty }\frac{t^{r+1}}{(1+t)^{r+m+1}}\,\mathrm{d}t \\ =&\frac{me^{-u_{m}y}}{v_{m}P(1)}\sum_{r=0}^{\infty}Q_{r}(u_{m}y) \frac{B(r+2,m-1)}{B(r+1,m)} \\ =&\frac{m}{v_{m}(m-1)}\frac{e^{-u_{m}y}}{P(1)}\sum_{r=0}^{\infty }(r+1)Q_{r}(u_{m}y) \frac{B(r+1,m)}{B(r+1,m)} \\ =&\frac{m}{v_{m}(m-1)}+\frac{m}{v_{m}(m-1)}\frac{e^{-u_{m}y}}{P(1)}\sum_{r=0}^{\infty}rQ_{r}(u_{m}y) \\ =&\frac{m}{v_{m}(m-1)}+\frac{mu_{m}}{v_{m}(m-1)} \biggl( y+\frac{1}{u_{m}} \frac{P^{\prime}(1)}{P(1)} \biggr) . \end{aligned}$$

(3) Take \(h=e_{3}\)

$$\begin{aligned} L_{m}^{u_{m},v_{m} }(e_{3};y) =& \frac{m^{2}e^{-u_{m}y}}{v_{m}^{2}P(1)}\sum_{r=0}^{\infty }Q_{r}(u_{m}y) \frac{1}{B(r+1,m)} \int_{0}^{\infty}\frac {t^{r+2}}{(1+t)^{r+m+1}}\, \mathrm{d}t \\ =&\frac{m^{2}}{v_{m}^{2}(m-2)(m-1)}\frac{e^{-u_{m}y}}{P(1)}\sum_{r=0}^{\infty }Q_{r}(u_{m}y) \bigl(r^{2}+3r+2\bigr) \\ =&\frac{2m^{2}}{v_{m}^{2}(m-2)(m-1)}+\frac{3m^{2}}{v_{m}^{2}(m-2)(m-1)} \biggl(\frac{P^{\prime}(1)}{P(1)} +u_{m}y \biggr) \\ +&\frac{m^{2}}{v_{m}^{2}(m-2)(m-1)} \biggl(\frac{P^{\prime\prime}(1)}{P(1)} +2u_{m}y\frac{P^{\prime}(1)}{P(1)}+\frac{P^{\prime}(1)}{P(1)}+u_{m}y+u_{m}^{2}y^{2} \biggr). \end{aligned}$$

 □

Lemma 2.2

If\(\varUpsilon_{j}=(e_{2}-y)^{j}\)for\(j=1,2\), then we have

1:

\(L_{m}^{u_{m},v_{m} }(\varUpsilon_{1};y)= \bigl( \frac{u_{m}}{v_{m}}\frac {m}{(m-1)}-1 \bigr) y+\frac{1}{v_{m}}\frac{m}{(m-1)} \bigl(\frac{P^{\prime}(1)}{P(1)}+ 1\bigr) \);

2:

\(L_{m}^{u_{m},v_{m} }(\varUpsilon_{2};y) = \bigl[ (\frac {u_{m}}{v_{m}} )^{2}\frac{m^{2}}{(m-2)(m-1)}-2\frac{u_{m}}{v_{m}}\frac{m}{(m-1)}+1 \bigr] y^{2} + \bigl[ 2\frac{u_{m}}{v_{m}^{2}}\frac{m^{2}}{(m-2)(m-1)} \bigl( \frac{P^{\prime }(1)}{P(1)}+2 \bigr)-2\frac{u_{m}}{v_{m}}\frac{m}{(m-1)} \bigl( \frac{P^{\prime}(1)}{P(1)}+1 \bigr) \bigr]y+ \frac{1}{v_{m}^{2}}\frac{m^{2}}{(m-2)(m-1)} \bigl( \frac{P^{\prime\prime }(1)}{P(1)}+\frac{P^{\prime}(1)}{P(1)}+2\bigr )\).

Approximation in a weighted space

In this section we present approximation results in the space \(C_{B}[0,\infty)\) of all bounded and continuous functions on \([0,\infty)\), which is a linear normed space with the norm

$$ \Vert h \Vert _{C_{B}[0,\infty)}=\sup_{y\geq0} \bigl\vert h(y) \bigr\vert . $$

For our convenience, we rewrite

$$ C_{{\gamma} }[0,\infty)= \bigl\{ h\in C[0,\infty ) : \bigl\vert h(t) \bigr\vert \leq K(1+t)^{{\gamma}} \mbox{ for some $K>0$} \bigr\} . $$

Let

$$ E= \biggl\{ h\in C[0,\infty):\lim_{y\rightarrow\infty} \frac {h(y)}{1+y^{2}}\text{ exists} \biggr\} . $$

Theorem 3.1

Let\(h\in C_{{\gamma} }[0,\infty)\cap E\)and\({\gamma} \geq2\). Then

$$ \lim_{m\rightarrow\infty}L_{m}^{u_{m},v_{m}}(h;y)=h(y)\quad\textit{uniformly.} $$

Proof

In the lighting of Korovkin’s theorem and Lemma 2.1, it is obvious that

$$ \lim_{m\rightarrow\infty}L_{m}^{u_{m},v_{m}}(e_{j};y)=y^{j-1},\quad j=1,2,3. $$

 □

Following Gadžiev [12], we recall the weighted spaces for which the analogues of Korovkin’s theorem hold [20]. Let ϕ be a continuous and strictly increasing function and \(\varrho(y)=1+\phi ^{2}(y)\), \(\lim_{y\rightarrow\infty}\varrho(y)=\infty\). Moreover,

$$ B_{\varrho}[0,\infty)= \bigl\{ h: \bigl\vert h(y) \bigr\vert \leq K_{h}\varrho(y) \bigr\} \quad \text{and}\quad C_{\varrho}[0, \infty)=B_{\varrho}[0,\infty)\cap C[0,\infty) $$

with \(\Vert h\Vert_{\varrho}=\sup_{y\geq0}\frac{|h(y)|}{\varrho(y)}\), where the constant \(K_{h}\) depends only on h. The sequence of positive linear operators \(\{L_{m}^{u_{m},v_{m}}\}_{m\geq1}\) maps \(C_{\varrho }[0,\infty)\) into \(B_{\varrho}[0,\infty)\) if and only if

$$ \bigl\vert L_{m}^{u_{m},v_{m}}(\varrho;y) \bigr\vert \leq K \varrho(y), $$

where \(y\in{}[0,\infty)\) and \(K>0\) is a constant. Finally, let \(C_{\varrho}^{0}[0,\infty)\subset C_{\varrho}[0,\infty)\) satisfying

$$ \lim_{y\rightarrow\infty}\frac{h(y)}{\varrho(y)}=K_{h}. $$

Theorem 3.2

(cf. [16])

Let the sequence of positive linear operators\(K_{m}\), acting from\(C_{\varrho}[0,\infty) \)to\(B_{\varrho}[0,\infty)\), satisfy the conditions

$$ \lim_{m\rightarrow\infty} \bigl\Vert K_{m} \bigl(t^{k};y\bigr)-y^{k} \bigr\Vert _{\varrho}=0 \quad(k=0,1,2). $$

Then, for each\(h\in C_{\varrho}^{0}[0,\infty)\),

$$ \lim_{m\rightarrow\infty} \Vert K_{m}h-h \Vert _{\varrho}=0. $$

Theorem 3.3

For every\(h\in C_{\varrho}^{0}[0,\infty)\), operators\(L_{m}^{u_{m},v_{m}}\)satisfy

$$ \lim_{m\rightarrow\infty} \bigl\Vert L_{m}^{u_{m},v_{m}}(h;y)-h \bigr\Vert _{\varrho }=0. $$

Proof

In view of Theorem 3.2, suppose \({\phi(y)}=y\), then \(\varrho(y)=1+y^{2}\). Since

$$ \bigl\Vert L_{m}^{u_{m},v_{m}}(e_{i};y)-y^{i-1} \bigr\Vert _{\varrho}=\sup_{y\geq 0}\frac{| L_{m}^{u_{m},v_{m}}(e_{i};y)-y^{i-1}|}{1+y^{2}}. $$

For i = 1, from Lemma 2.1 we get

$$ \lim_{{m\rightarrow\infty} } \bigl\Vert L_{m}^{u_{m},v_{m}}(e_{1};y)-1 \bigr\Vert _{\varrho}=0. $$

For \(i=2\),

$$\begin{aligned} \sup_{y\geq0}\frac{| L_{m}^{u_{m},v_{m}}(e_{2};y)-y|}{1+y^{2}} \leq & \biggl\vert \frac{u_{m}}{v_{m}}\frac{m}{(m-1)}-1 \biggr\vert \sup _{y\geq0}\frac {y}{1+y^{2}} \\ &{}+ \biggl\vert \frac{m}{v_{m}(m-1)} \biggl( 1+\frac{P^{\prime }(1)}{P(1)} \biggr) \biggr\vert \sup_{y\geq0}\frac{1}{1+y^{2}}. \end{aligned}$$

Therefore

$$ \lim_{{m\rightarrow\infty} } \bigl\Vert L_{m}^{u_{m},v_{m}}(e_{2};y)-y \bigr\Vert _{\varrho}=0. $$

For \(i=3\),

$$\begin{aligned} \sup_{y\geq0}\frac{| L_{m}^{u_{m},v_{m}}(e_{2};y)-y^{2}|}{1+y^{2}} \leq& \biggl\vert \biggl( \frac{u_{m}}{v_{m}} \biggr) ^{2}\frac {m^{2}}{(m-2)(m-1)}-1 \biggr\vert \sup_{y\geq0}\frac{y^{2}}{1+y^{2}} \\ &{}+ \biggl\vert \frac{u_{m}}{v_{m}^{2}}\frac{m^{2}}{(m-2)(m-1)} \biggl( 2 \frac{P^{\prime}(1)}{P(1)}+4 \biggr) \biggr\vert \sup_{y\geq0} \frac{y}{1+y^{2}} \\ &{}+ \biggl\vert \frac{m^{2}}{v_{m}^{2}(m-2)(m-1)} \biggl( \frac{P^{\prime\prime }(1)}{P(1)}+4 \frac{P^{\prime}(1)}{P(1)}+2 \biggr) \biggr\vert \sup_{y\geq0} \frac {1}{1+y^{2}}. \end{aligned}$$

Therefore,

$$ \lim_{{m\rightarrow\infty}} \bigl\Vert L_{m}^{u_{m},v_{m}}(e_{3};y)-y^{2} \bigr\Vert _{\varrho}=0. $$

Lemma 2.1 implies that

$$ \lim_{m\rightarrow\infty} \bigl\Vert L_{m}^{u_{m},v_{m}}(h;y)-h \bigr\Vert _{\varrho }=0. $$

 □

Rate of convergence

We write \({C^{*}_{B}}[0,\infty)\) for the set of all uniformly continuous and bounded functions on \([0,\infty)\) with \(\Vert f\Vert _{C_{B}[0,\infty)}=\sup_{y\in{}[0,\infty)}| f(y)|\). For

$$ \omega^{\circ}\bigl(h,\delta^{\circ}\bigr)=\sup _{ \vert t-y \vert \leq\delta } \bigl\vert h(t)-h(y) \bigr\vert ,\quad \delta^{\circ}>0, h\in C_{B}[0,\infty), $$
(4.1)

we have

$$ \bigl\vert h(t)-h(y) \bigr\vert \leq \biggl( \frac{\delta^{\circ}+ \vert t-y \vert }{\delta^{\circ}} \biggr) \omega^{\circ}\bigl(h,\delta^{\circ}\bigr) $$
(4.2)

and \(\lim_{\delta^{\circ}\rightarrow0+}\omega^{\circ}(h,\delta ^{\circ })=0\).

For \(C>0\) and \(0<\zeta\leq1\), the Lipschitz class is defined by

$$ Lip_{C}(\zeta)= \bigl\{ h: \bigl\vert h(\eta_{1})-h( \eta_{2}) \bigr\vert \leq C \vert {\eta_{1}}-\eta_{2} \vert ^{\zeta}\ \bigl( \eta_{1},\eta_{2}\in{}[0,\infty)\bigr) \bigr\} . $$
(4.3)

Theorem 4.1

Let\(m\in\mathbb{N}\)and\(m>2\), then for all\(h\in{C^{*}_{B}}[0,\infty)\)

$$ \bigl\vert L_{m}^{u_{m},v_{m}}(h;y)-h(y) \bigr\vert \leq2 \omega^{\circ} ( h;\varPsi _{m} ) , $$
(4.4)

where\((\varPsi_{m})^{2}= \bigl{[} ( \frac{u_{m}}{v_{m}} ) ^{2}\frac{m^{2}}{(m-2)(m-1)}-2\frac{u_{m}}{v_{m}}\frac{m}{(m-1)}+1 \bigr{]} y^{2} + \bigl{[} \frac{u_{m}}{v_{m}^{2}}\frac{m^{2}}{(m-2)(m-1)} \bigl( 2\frac {P^{\prime }(1)}{P(1)}+4 \bigr) -\frac{1}{v_{m}}\frac{m}{(m-1)}\bigl( \frac{2P^{\prime }(1)}{P(1)}+2 \bigr) \bigr{]} y+\frac{1}{v_{m}^{2}}\frac{m^{2}}{(m-2)(m-1)} \bigl( \frac{P^{\prime \prime}(1)}{P(1)}+\frac{P^{\prime}(1)}{P(1)}+2 \bigr) \).

Proof

We have

$$\begin{aligned} \bigl\vert L_{m}^{u_{m},v_{m}}(h;y)-h(y) \bigr\vert =& \bigl| L_{m}^{u_{m},v_{m}}(h;y)-h(y)L_{m}^{u_{m},v_{m}}(e_{1};y) \bigr| \\ =& \bigl\vert L_{m}^{u_{m},v_{m}} \bigl( h(t)-h(y);y \bigr) \bigr\vert \\ \leq& {L_{m}^{u_{m},v_{m}} \bigl( \bigl\vert h(t)-h(y) \bigr\vert ;y \bigr)} . \end{aligned}$$

In the light of (4.1) and (4.2), we get

$$\begin{aligned} \bigl\vert L_{m}^{u_{m},v_{m}}(h;y)-h(y) \bigr\vert \leq&L_{m}^{u_{m},v_{m}} \biggl( 1+\frac {| t-y|}{\delta^{\circ}};y \biggr) \omega^{\circ}\bigl(h,\delta ^{\circ }\bigr) \\ =& \biggl( 1+\frac{1}{\delta^{\circ}}L_{m}^{u_{m},v_{m}}\bigl(| t-y| ;y\bigr) \biggr) \omega^{\circ}\bigl(h,\delta^{\circ}\bigr). \end{aligned}$$

Apply the Cauchy–Schwarz inequality

$$ L_{m}^{u_{m},v_{m}}\bigl(| t-y|;y\bigr)\leq \bigl[ L_{m}^{u_{m},v_{m}}(e_{1};y)L_{m}^{u_{m},v_{m}} \bigl( (t-y)^{2};y \bigr) \bigr] ^{\frac{1}{2}} $$

so that

$$ \bigl\vert L_{m}^{u_{m},v_{m}}(h;y)-h(y) \bigr\vert \leq \biggl( 1+\frac{1}{\delta ^{\circ }}L_{m}^{u_{m},v_{m}} ( \varUpsilon_{2};y ) ^{\frac{1}{2}} \biggr) \omega^{\circ} \bigl(h,\delta^{\circ}\bigr). $$
(4.5)

Choosing \(\delta^{\circ}=\varPsi_{m}=\sqrt{L_{m}^{u_{m},v_{m}} ( \varUpsilon_{2};y ) }\) yields the result. □

Remark 4.2

For \(u_{m}=v_{m}=1\), the above estimate is reduced to [22], i.e.,

$$ \bigl\vert L_{m}(h;y)-h(y) \bigr\vert \leq2 \omega^{\circ} ( h,\varPhi_{m} ) , $$
(4.6)

where \((\varPhi_{m})^{2}= \bigl( \frac{m^{2}}{(m-2)(m-1)}- \frac{2m}{(m-1)}+1\bigr) y^{2} +\bigl[\frac{2m}{(m-2)(m-1)}\bigl( \frac{P^{\prime}(1)}{P(1)} ) +2 ) - \frac{2}{(m-1)} \bigl( \frac{P^{\prime}(1)}{P(1)} +1\bigr)\bigr]y + \frac{1}{(m-2)(m-1)} \bigl( \frac{P^{\prime\prime}(1)}{P(1)}+\frac{P^{\prime}(1)}{P(1)}+2 \bigr) \).

Theorem 4.3

For every\(h\in \operatorname{Lip}_{C}(\zeta)\), we have

$$ \bigl\vert L_{m}^{u_{m},v_{m}}(h;y)-h(y) \bigr\vert \leq C ( \varPsi_{m} ) ^{ {\zeta}}, $$

where\(m>2\), \(m \in\mathbb{N} \), and\(\varPsi_{m}=\sqrt{L_{m}^{u_{m},v_{m}} (\varUpsilon_{2};y )}\)by Theorem4.1.

Proof

We use (4.3) and Hölder’s inequality to get

$$\begin{aligned} &\bigl\vert L_{m}^{u_{m},v_{m}}(h;y)-h(y) \bigr\vert \\ &\quad\leq \bigl\vert L_{m}^{u_{m},v_{m}}\bigl(h(t)-h(y);y\bigr) \bigr\vert \\ &\quad\leq L_{m}^{u_{m},v_{m}} \bigl( \bigl\vert h(t)-h(y) \bigr\vert ;y \bigr) \\ &\quad\leq CL_{m}^{u_{m},v_{m}} \bigl( \vert t-y \vert ^{\zeta};y \bigr) . \\ &\quad\leq C\frac{e^{-u_{m}y}}{v_{m}P(1)}\sum_{r=0}^{\infty }Q_{r}(u_{m}y) \frac{1}{B(r+1,m)} \int_{0}^{\infty}\frac{t^{r}}{(1+t)^{r+m+1}} \vert t-y \vert ^{\zeta}\,\mathrm{d}t \\ &\quad=\frac{e^{-u_{m}y}}{v_{m}P(1)} \Biggl( \sum_{r=0}^{\infty }Q_{r}(u_{m}y)\frac{1}{B(r+1,m)} \Biggr) ^{1-\frac{\zeta}{2}} \\ &\qquad{}\times \biggl( Q_{r}(u_{m}y)\frac{1}{B(r+1,m)} \biggr) ^{\frac{\zeta }{2}} \int_{0}^{\infty}\frac{t^{r}}{(1+t)^{r+m+1}} \vert t-y \vert ^{\zeta}\,\mathrm{d}t \\ &\quad \leq C \Biggl( \frac{e^{-u_{m}y}}{v_{m}P(1)}\sum_{r=0}^{\infty }Q_{r}(u_{m}y)\frac{1}{B(r+1,m)} \int_{0}^{\infty}\frac{t^{r}}{(1+t)^{r+m+1}}\,\mathrm {d}t \Biggr) ^{1-\frac{\zeta}{2}} \\ &\qquad{}\times \Biggl( \frac{e^{-u_{m}y}}{v_{m}P(1)}\sum_{r=0}^{\infty }Q_{r}(u_{m}y) \frac{1}{B(r+1,m)} \int_{0}^{\infty}\frac {t^{r}}{(1+t)^{r+m+1}} \vert t-y \vert ^{2}\,\mathrm{d}t \Biggr) ^{\frac{\zeta}{2}} \\ &\quad=CL_{m}^{u_{m},v_{m}} ( \varUpsilon_{2};y ) ^{\frac{\zeta}{2}}. \end{aligned}$$

This completes the proof. □

Direct theorems

Let

$$ C_{B}^{\kappa}[0,\infty)= \bigl\{ f\in C_{B}[0, \infty):f^{\prime },f^{\prime\prime}\in C_{B}[0,\infty) \bigr\} $$
(5.1)

with the norm

$$ \Vert f \Vert _{C_{B}^{\kappa}[0,\infty)}= \Vert f \Vert _{C_{B}[0,\infty )}+ \bigl\Vert f^{\prime} \bigr\Vert _{C_{B}[0,\infty)}+ \bigl\Vert f^{\prime\prime } \bigr\Vert _{C_{B}[0,\infty)}, $$
(5.2)

and let \(\varOmega^{\circ}= \{ f\in C_{B}[0,\infty ):f^{\prime},f^{\prime\prime}\in C_{B}[0,\infty) \}\). For \(h\in C_{B}[0,\infty)\), Peetre’s K-functional is defined by

$$ K_{2}^{\circ}\bigl(h,\delta^{\circ}\bigr)=\inf _{{f\in\varOmega^{\circ}}} \bigl\{ \bigl( \Vert h-f \Vert _{C_{B}[0,\infty)}+ \delta^{\circ } \bigl\Vert f^{\prime\prime} \bigr\Vert _{C_{B}^{2}[0,\infty)} \bigr) :f\in\varOmega ^{\circ } \bigr\} . $$
(5.3)

For a positive constant M, one has \(K_{2}^{\circ}(h,\delta^{\circ })\leq M\omega_{2}^{\circ}(h,\sqrt{\delta^{\circ}})\), where \({\delta^{\circ}}>0\) and the second order modulus of continuity \(\omega _{2}^{\circ}\) is defined by

$$ \omega_{2}^{\circ}\bigl(h,\sqrt{\delta^{\circ}} \bigr)=\sup_{0< u< \sqrt{\delta ^{\circ}}}\sup_{y\in{}[0,\infty)} \bigl\vert h(y+2u)-2h(y+u)+h(y) \bigr\vert . $$
(5.4)

Theorem 5.1

Let\(m>2\), \(m\in\mathbb{N}\). Then for all\(f\in C_{B}^{\kappa}[0,\infty )\)we have

$$ \bigl\vert L_{m}^{u_{m},v_{m}}(f;y)-f(y) \bigr\vert \leq \biggl( \varPsi_{m}+\frac{(\varPsi _{m})^{2}}{2} \biggr) \Vert f \Vert _{C_{B}^{\kappa}[0,\infty)}, $$

where\(\varPsi_{m}\)is defined by Theorem4.1.

Proof

By Taylor’s formula, one has

$$\begin{aligned}& f(t)=f(y)+f^{\prime}(y) (t-y)+f^{\prime\prime}(\chi) \frac {(t-y)^{2}}{2},\quad {\chi} \in(y,t), \\& \bigl\vert f(t)-f(y) \bigr\vert \leq W_{1} \vert t-y \vert +\frac{1}{2} W_{2}(t-y)^{2}, \end{aligned}$$

where

$$\begin{aligned}& W_{1}=\sup_{y \in{}[0,\infty)} \bigl\vert f^{\prime}(y) \bigr\vert = \bigl\Vert f^{\prime} \bigr\Vert _{C_{B}{}[0,\infty)}\leq \Vert f \Vert _{C_{B}^{\kappa}{}[0,\infty)}, \\& W_{2}=\sup_{y \in{}[0,\infty)} \bigl\vert f^{\prime\prime}(y) \bigr\vert = \bigl\Vert f^{\prime \prime} \bigr\Vert _{C_{B}{}[0,\infty)}\leq \Vert f \Vert _{C_{B}^{\kappa}{}[ 0,\infty)}. \end{aligned}$$

Therefore,

$$ \bigl\vert f(t)-f(y) \bigr\vert \leq \biggl( \vert t-y \vert + \frac{(t-y)^{2}}{2} \biggr) \Vert f \Vert _{C_{B}^{\kappa}{}[0,\infty)} $$

and

$$ \bigl\vert L_{m}^{u_{m},v_{m}}(f,y)-f(y) \bigr\vert = \bigl\vert L_{m}^{u_{m},v_{m}}\bigl(f(t)-f(y);y\bigr) \bigr\vert \leq L_{m}^{u_{m},v_{m}}\bigl( \bigl\vert f(t)-f(y) \bigr\vert ;y\bigr). $$

Thus, we get

$$\begin{aligned} \bigl\vert L_{m}^{u_{m},v_{m}}(f;y)-f(y) \bigr\vert \leq& \biggl(L_{m}^{u_{m},v_{m}}\bigl( \vert t-y \vert ;y\bigr)+ \frac {L_{m}^{u_{m},v_{m}}((t-y)^{2};y)}{2} \biggr) \Vert f \Vert _{C_{B}^{\kappa}[0,\infty)} \\ \leq& \biggl(\varPsi_{m}+\frac{(\varPsi_{m})^{2}}{2} \biggr) \Vert f \Vert _{C_{B}^{\kappa}[0,\infty)}, \end{aligned}$$

where

$$ L_{m}^{u_{m},v_{m}} \bigl( \vert t-y \vert ;y \bigr) \leq \sqrt{L_{m}^{u_{m},v_{m}} \bigl( (t-y)^{2};y \bigr) }=\sqrt{L_{m}^{u_{m},v_{m}} ( \varUpsilon _{2};y ) }. $$

Hence the result. □

Theorem 5.2

For every\(h\in C_{B}[0,\infty)\)and\(m>2\), \(m\in\mathbb{N}\), we have

$$\begin{gathered} \bigl\vert L_{m}^{u_{m},v_{m}}(h;y)-h(y) \bigr\vert \\\quad \leq2M \biggl\{ \omega_{2}^{\circ} \biggl( h;\sqrt{2 \varPsi_{m}+\frac{(\varPsi_{m})^{2}}{4}} \biggr)+\min \biggl(1, 2 \varPsi_{m}+\frac{(\varPsi_{m})^{2}}{4} \biggr) \Vert h \Vert _{C_{B}{}[0,\infty)} \biggr\} .\end{gathered} $$

Proof

As previously, we easily conclude that

$$\begin{aligned} { \bigl\vert L_{m}^{u_{m},v_{m}}(h;y)-h(y) \bigr\vert } \leq& \bigl| L_{m}^{u_{m},v_{m}}(h-f;y) \bigr|+ \bigl| L_{m}^{u_{m},v_{m}}(f;y)-f(y) \bigr| + \bigl\vert h(y)-f(y) \bigr\vert \\ \leq&2 \Vert h-f \Vert _{C_{B}(\mathbb{R}^{+})}+ \biggl( \varPsi_{m}+ \frac {(\varPsi _{m})^{2}}{2} \biggr) \Vert f \Vert _{C_{B}^{\kappa}[0,\infty)} \\ \leq&2 \biggl( \Vert h-f \Vert _{C_{B}[0,\infty)}+ \biggl( 2\varPsi _{m}+\frac{(\varPsi_{m})^{2}}{4} \biggr) \Vert f \Vert _{C_{B}^{\kappa}[0,\infty )} \biggr). \end{aligned}$$

By taking infimum and using (5.3), we get

$$ \bigl\vert L_{m}^{u_{m},v_{m}}(h;y)-h(y) \bigr\vert \leq2K_{2} \biggl( h;2\varPsi_{m}+\frac {(\varPsi _{m})^{2}}{4} \biggr) . $$

Now, for an absolute constant \(M>0\) [10], we use the relation

$$ K_{2}^{\circ}\bigl(h;\delta^{\circ}\bigr)\leq M \bigl\{ \omega_{2}^{\circ}\bigl(h;\sqrt{\delta^{\circ}}\bigr)+\min\bigl(1,\delta^{\circ}\bigr) \Vert h \Vert \bigr\} . $$

This completes the proof. □

Conclusion

We have constructed an integral type modification of Jakimovski–Leviatan operators by using beta function and two sequences of unbounded and increasing functions \(\{u_{m}\}\), \(\{v_{m}\}\) such that \(\lim_{m\rightarrow\infty}\frac{u_{m}}{v_{m}}=1+O ( \frac{1}{v_{m}} ) \) and \(\lim_{m\rightarrow\infty}\frac{1}{v_{m}}{=0}\). We derived some uniform convergence results of these operators via Korovkin’s theorem and obtained the rate of convergence by using the modulus of continuity and Lipschitz class. Furthermore, we obtained some direct theorems with the help of Peetre’s K-functional.

References

  1. 1.

    Acar, E., İzgi, A., Serenbay, S.K.: Note on Jakimovski–Leviatan operators preserving \(e^{-x}\). Appl. Math. Nonlinear Sci. 4(2), 543–549 (2019)

    MathSciNet  Google Scholar 

  2. 2.

    Al-Salam, W.A.: q-Appell polynomials. Ann. Mat. Pura Appl. 4, 31–45 (1967)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Alotaibi, A., Mursaleen, M.: Approximation of Jakimovski–Leviatan-beta type integral operators via q-calculus. AIMS Math. 5(4), 3019–3034 (2020)

    Article  Google Scholar 

  4. 4.

    Ansari, K.J., Mursaleen, M., Rahman, S.: Approximation by Jakimovski–Leviatan operators of Durrmeyer type involving multiple Appell polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(2), 1007–1024 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Appell, P.: Une classe de polynômes. Ann. Sci. Éc. Norm. Supér. 9, 119–144 (1880)

    MATH  Article  Google Scholar 

  6. 6.

    Ari, D.A., Serenbay, S.K.: Approximation by a generalization of the Jakimovski–Leviatan operators. Filomat 33(8), 2345–2353 (2019)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Atakut, C., Büyükyazici, I.: Approximation by modified integral type Jakimovski–Leviatan operators. Filomat 30, 29–39 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Bernstein, S.N.: Démonstation du théorème de Weierstrass fondée sur le calcul de probabilités. Commun. Soc. Math. Kharkow 13, 1–2 (1912–1913)

  9. 9.

    Büyükyazıcı, İ., Tanberkan, H., Serenbay, S., Atakut, C.: Approximation by Chlodowsky type Jakimovski–Leviatan operators. J. Comput. Appl. Math. 259, 153–163 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Ciupa, A.: A class of integral Favard–Szász type operators. Stud. Univ. Babeş–Bolyai, Math. 40, 39–47 (1995)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Gadžiev, A.D.: A problem on the convergence of a sequence of positive linear operators on unbounded sets, and theorems that are analogous to P. P. Korovkin’s theorem. Dokl. Akad. Nauk SSSR 218, 1001–1004 (1974) (Russian)

    MathSciNet  Google Scholar 

  12. 12.

    Gadžiev, A.D.: Weighted approximation of continuous functions by positive linear operators on the whole real axis. Izv. Akad. Nauk Azerb. SSR, Ser. Fiz.-Teh. Mat. Nauk 5, 41–45 (1975) (Russian)

    MathSciNet  Google Scholar 

  13. 13.

    Jakimovski, A., Leviatan, D.: Generalized Szász operators for the approximation in the infinite interval. Mathematica 11(34), 97–103 (1969)

    MATH  Google Scholar 

  14. 14.

    Kac, V., De Sole, A.: On integral representations of q-gamma and q-beta functions. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. 9, 11–29 (2005)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Keleshteri, M.E., Mahmudov, N.I.: A study on q-Appell polynomials from determinantal point of view. Appl. Math. Comput. 260, 351–369 (2015)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Korovkin, P.P.: Linear Operators and Approximation Theory. Hindustan Publ. Co., Delhi (1960)

    Google Scholar 

  17. 17.

    Milovanović, G.V., Mursaleen, M., Nasiruzzaman, M.: Modified Stancu type Dunkl generalization of Szász–Kantorovich operators. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112(1), 135–151 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Mohiuddine, S.A., Acar, T., Alghamdi, M.A.: Genuine modified Bernstein–Durrmeyer operators. J. Inequal. Appl. 2018, Article ID 104 (2018)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Mursaleen, M., Ansari, K.J., Nasiruzzaman, M.: Approximation by q-analogue of Jakimovski–Leviatan operators involving q-Appell polynomials. Iran. J. Sci. Technol. Trans. A, Sci. 41, 891–900 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Mursaleen, M., Karakaya, V., Ertürk, M., Gürsoy, F.: Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl. Math. Comput. 218, 9132–9137 (2012)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Mursaleen, M., Khan, T.: On approximation by Stancu type Jakimovski–Leviatan–Durrmeyer operators. Azerb. J. Math. 7(1), 16–26 (2017)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Mursaleen, M., Nasiruzzaman, M.: Approximation of modified Jakimovski–Leviatan-beta type operators. Constr. Math. Anal. 1(2), 88–98 (2018)

    Google Scholar 

  23. 23.

    Mursaleen, M., Nasiruzzaman, M., Alotaibi, A.: On modified Dunkl generalization of Szász operators via q-calculus. J. Inequal. Appl. 2017, Article ID 38 (2017)

    MATH  Article  Google Scholar 

  24. 24.

    Mursaleen, M., Nasiruzzaman, M., Srivastava, H.M.: Approximation by bicomplex beta operators in compact BC-disks. Math. Methods Appl. Sci. 39, 2916–2929 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Nasiruzzaman, Md., Mukheimer, A., Mursaleen, M.: Approximation results on Dunkl generalization of Phillips operators via q-calculus. Adv. Differ. Equ. 2019, Article ID 244 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Srivastava, H.M., Mursaleen, M., Nasiruzzaman, M.: Approximation by a class of q-beta operators of the second kind via the Dunkl-type generalization on weighted spaces. Complex Anal. Oper. Theory 13(3), 1537–1556 (2019)

    MathSciNet  MATH  Article  Google Scholar 

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Nasiruzzaman, M., Mursaleen, M. Approximation by Jakimovski–Leviatan-beta operators in weighted space. Adv Differ Equ 2020, 393 (2020). https://doi.org/10.1186/s13662-020-02848-x

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MSC

  • 41A10
  • 41A25
  • 41A36

Keywords

  • Appell polynomials
  • Jakimovski–Leviatan operators
  • Korovkin’s theorem
  • Modulus of continuity
  • Lipschitz functions
  • Peetre’s K-functional