1 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 [1–4, 6, 7, 11, 14, 15, 17–19, 21, 23–26].

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)

2 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∘:

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

3 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

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

 □

4 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. □

5 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. □

6 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.