Skip to main content

Ibragimov–Gadjiev operators preserving exponential functions

Abstract

In this paper, a modification of general linear positive operators introduced by Ibragimov and Gadjiev in 1970 is constructed. It is shown that this modification preserves exponential mappings and also contains modified Bernstein-, Szász- and Baskakov-type operators as special cases. The convergence properties of corresponding operators on \([ 0,\infty ) \) and in exponentially weighted spaces are investigated. Finally, the quantitative Voronovskaja theorem in terms of modulus of continuity for functions having exponential growth is examined.

1 Introduction

After the construction of a general sequence of positive operators by Ibragimov and Gadjiev [1] in 1970, some authors, inspired by this work, introduced various generalizations of these operators. This is because the results obtained for the generalized operators are also valid for the operators included in them. Thus, Aral and Acar [2] introduced a general class of Durrmeyer operators by modifying the Ibragimov–Gadjiev operators as

$$\begin{aligned} M_{n} ( f;x ) ={}& ( n-m ) \alpha _{n} \mathit{\psi }_{n} ( 0 ) \sum_{\nu =0}^{\infty }K_{n}^{ ( \nu ) } \bigl( x,0,\alpha _{n}\mathit{\psi }_{n} ( 0 ) \bigr) \frac{(-\alpha _{n}\mathit{\psi }_{n} ( 0 ) )^{\nu }}{\nu !} \\ &{}\times \int _{0}^{\infty }f ( y ) K_{n}^{ ( \nu ) } \bigl( y,0,\alpha _{n}\mathit{\psi }_{n} ( 0 ) \bigr) \frac{(-\alpha _{n}\mathit{\psi }_{n} ( 0 ) )^{\nu }}{\nu !}\,dy, \end{aligned}$$

where

$$\begin{aligned} K_{n}^{ ( \nu ) } \bigl( x,0,\alpha _{n}\mathit{\psi }_{n} ( 0 ) \bigr) = \frac{\partial ^{\nu }}{\partial u^{\nu }}K_{n} ( x,t,u ) |_{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}} \end{aligned}$$

\(x,t\in \mathbb{R} ^{+}\) and \(-\infty < u<\infty \) is a sequence of functions of three variables x, t, u such that \(K_{n\text{ }}\) is entire analytic function with respect to variable u for each \(x,t\in \mathbb{R} ^{+}\) and for each \(n\in \mathbb{N} \). Bozma and Bars [3] defined a Kantorovich-type generalization on a variable bounded interval:

$$\begin{aligned} R_{n} ( f;x ) = ( \beta _{n}+n+2 ) \sum _{\nu =0}^{\infty }K_{n}^{ ( \nu ) } ( x ) \frac{(-\alpha _{n})^{\nu }}{\nu !} \int _{ \frac{\nu +n+1}{\beta _{n}+n+2}}^{\frac{\nu +n+2}{\beta _{n}+n+2}}f ( p ) \,dp, \end{aligned}$$

where \(f:L_{1} [ 0,\frac{n+1}{n+2} ] \rightarrow C [ 0, \frac{n+1}{n+2} ] \). Herdem and Büyükyazıcı [4] constructed an extension in q-Calculus of these operators. The q-generalization of Ibragimov–Gadjiev operators have the following form:

$$\begin{aligned} L_{n} ( f;q;x ) =\sum_{\nu =0}^{\infty }q^{ \frac{\nu ( \nu -1 ) }{2}}f \biggl( \frac{ [ \nu ] }{ [ n ] ^{2}\mathit{\psi }_{n} ( 0 ) } \biggr) D_{q,u}K_{n,\nu }^{q} ( x,t,u ) |_{{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}}} \frac{(-\alpha _{n}\mathit{\psi }_{n} ( 0 ) )^{\nu }}{ [ \nu ] !} \end{aligned}$$

for \(x\in \mathbb{R} ^{+}\) and any function f defined on \(\mathbb{R} ^{+}\). Furthermore, Korovkin-type theorems for continuous and unbounded functions defined on \([ 0,\infty ) \) were established, and some representation formulas using q-derivatives were given in [5]. Many other investigations about Ibragimov–Gadjiev operators may also be found in [611].

Now, we recall the original construction. Let \(\{ \varphi _{n} ( t ) \} \) and \(\{ \mathit{\psi }_{n} ( t ) \} \) be the sequence of functions in \(C [ 0,A ] \) such that \(\varphi _{n} ( 0 ) =0\) and \(\mathit{\psi }_{n} ( t ) >0\) for all \(t\in [ 0,A ] \), \(A>0\). Let also \(\{ \alpha _{n} \} \) be a sequence of positive numbers having the following properties \(\lim_{n\rightarrow \infty }\frac{\alpha _{n}}{n}=1\), \(\lim_{n\rightarrow \infty } \frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) }=0\).

Assume that a sequence of functions of three variables \(\{ K_{n} ( x,t,u ) \} \) (\(x,t\in [ 0,A ] \), \(-\infty < u<\infty \)) satisfies the following conditions:

\(1^{o}\) Each function of this family is an entire analytic function with respect to u for fixed \(x,t\in [ 0,A ] \);

\(2^{o}\) \(K_{n} ( x,0,0 ) =1\) for any \(x\in [ 0,A ] \) and for all \(n\in \mathbb{N} \);

\(3^{o}\) \(\{ ( -1 ) ^{\nu } [ \frac {\partial ^{\nu }}{\partial u^{\nu }}K_{n} ( x,t,u ) ] _{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}} \} \geq 0\) \(( \nu ,n\in \{ 1,2,\ldots \} ;\text{ }x\in [ 0,A ] ) \);

\(4^{o}\) \(-\frac {\partial ^{\nu }}{\partial u^{\nu }}K_{n} ( x,t,u ) |_{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}}=nx [ \frac {\partial ^{\nu -1}}{\partial u^{\nu -1}}K_{n+m} ( x,t,u ) |_{{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}}} ] (\nu ,n\in \{ 1,2,\ldots \} ; x\in [ 0,A ] )\), where m is a number such that \(m+n\) is zero or a naturel number.

Under these conditions, Ibragimov–Gadjiev operators are defined as

$$\begin{aligned} G_{n} ( f;x ) =\sum_{\nu =0}^{\infty }f \biggl( \frac{\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) } \biggr) \biggl[ \frac {\partial ^{\nu }}{\partial u^{\nu }}K_{n} ( x,t,u ) |_{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}} \biggr] \frac{(-\alpha _{n}\mathit{\psi }_{n} ( 0 ) )^{\nu }}{\nu !} \end{aligned}$$
(1)

for \(x\in \mathbb{R} ^{+}\) and any function f defined on \(\mathbb{R} ^{+}\).

We should mention that Ibragimov–Gadjiev operators contain, as a particular case, a series of operators. By choosing \(K_{n} ( x,t,u ) = ( 1-\frac{ux}{1+t} ) ^{n},\alpha _{n}\mathit{=n,}\) \(\mathit{\psi }_{n} ( 0 ) =\frac{1}{n}\) the operators defined by (1) are transformed into Bernstein polynomials; for \(\alpha _{n}\mathit{=n,}\) \(\mathit{\psi }_{n} ( 0 ) =\frac{1}{nb_{n}}\) \(( \lim_{n\rightarrow \infty }b_{n}=\infty ,\text{ } \lim_{n\rightarrow \infty }\frac{b_{n}}{n}=0 ) \), we also get Bernstein–Chlodowsky polynomials. For \(K_{n} ( x,t,u ) =e^{-n ( t+ux ) }\), \(\alpha _{n}\mathit{=n,}\) \(\mathit{\psi }_{n} ( 0 ) =\frac{1}{n} \), we get Szász–Mirakyan operators. Moreover, if we choose \(K_{n} ( x,t,u ) =K_{n} ( t+ux ) \), \(\alpha _{n}\mathit{=n,}\) \(\mathit{\psi }_{n} ( 0 ) =\frac{1}{n}\), then we obtain Baskakov operators.

In recent years, by defining the linear positive operators, which preserve the exponential functions by Aldaz and Render [12], several researchers introduced linear positive operators that reproduce the exponential functions by conveniently modifying the well-known operators. In [13], Acar et al. presented a modification of Szász-Mirakyan operators that reproduces the functions 1 and e\(^{2ax}\), \(a>0\). They discussed approximation properties via a certain weighted modulus of continuity and a quantitative Voronovskaya-type theorem. In [14], recovered a generalization of the Bernstein operators that reproduce the exponential functions e\(^{ax}\) and e\(^{2ax}\), \(a>0\). The authors also showed that this way better approximates functions with modified operators than the classical one. After that, this method was studied and extended in numerous papers. We refer interested readers to [1532].

In parallel with these developments, this paper aims to construct a new generalization of Ibragimov–Gadjiev operators, \(G_{n}^{\ast }\), fixing the function e\(^{ax},a>0\). Then, for these operators, we provide some approximation properties and present special cases as examples.

The rest of this work is organized as follows: In Sect. 2, the technique to construct the modified Ibragimov–Gadjiev operators is discussed. In Sect. 3, moments, central moments, and a recurrence formula are calculated. In Sect. 4, convergence properties on \([ 0,\infty ) \) and in the light of weighted spaces are investigated. The rate of convergence using the exponential modulus of continuity is also examined. In Sect. 5, it is shown that the new operators, which will be constructed below, contain modified Bernstein-, Szász- and Baskakov-type operators that exist in the literature as a special case. Finally, in the last section, we summarize the main results and give some thoughts that can be applied to expand the scope of this study.

2 Construction of the operators

Let \(\{ \varphi _{n} ( t ) \} \) and \(\{ \mathit{\psi }_{n} ( t ) \} \) be the sequence of functions in \(C [ 0,A ] \) such that \(\varphi _{n} ( 0 ) =0\) and \(\mathit{\psi }_{n} ( t ) >0\) for all \(t\in [ 0,A ] \), \(A>0\). Let also \(\{ \alpha _{n} \} \) be a sequence of positive numbers such that \(\lim_{n\rightarrow \infty }\frac{\alpha _{n}}{n}=1 \), \(\lim_{n\rightarrow \infty } \frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) }=0\).

Assume that a sequence of functions of three variables \(\{ K_{n} ( \lambda _{n} ( x ) ,t,u ) \} \) (\(x,t\in [ 0,A ] \), \(-\infty < u<\infty \) and \(\lim_{n\rightarrow \infty }\lambda _{n} ( x ) =x\)) satisfies the following conditions:

1 Each function of this family is an entire analytic function with respect to u for fixed \(x,t\in [ 0,A ] \);

2 \(K_{n} ( \lambda _{n} ( x ) ,0,0 ) =1\) for any \(x\in [ 0,A ] \) and for all \(n\in \mathbb{N} \);

3 \(\{ ( -1 ) ^{\nu } [ \frac {\partial ^{\nu }}{\partial u^{\nu }}K_{n} ( \lambda _{n} ( x ) ,t,u ) ] _{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}} \} \geq 0\) \(( \nu ,n\in \{ 1,2,\ldots \} ;\text{ }x\in [ 0,A ] ) \);

4 \(-\frac {\partial ^{\nu }}{\partial u^{\nu }}K_{n} ( \lambda _{n} ( x ) ,t,u ) |_{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}}=n\lambda _{n} ( x ) [ \frac {\partial ^{\nu -1}}{\partial u^{\nu -1}}K_{n+m} ( \lambda _{n} ( x ) ,t,u ) |_{{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}}} ] \) (\(\nu ,n\in \{ 1,2,\ldots \} \); \(x\in [ 0,A ] \)), where m is a number such that \(m+n\) is zero or a natural number.

According to these conditions, modified Ibragimov–Gadjiev operators have the following form:

$$\begin{aligned} G_{n}^{\ast } ( f;x ) =\sum_{\nu =0}^{\infty }e^{- \frac{a\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) }}e^{ax}f \biggl( \frac{\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) } \biggr) \biggl[ \frac {\partial ^{\nu }}{\partial u^{\nu }}K_{n} \bigl( \lambda _{n} ( x ) ,t,u \bigr) \big| _{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}} \biggr] \frac{(-\alpha _{n}\mathit{\psi }_{n} ( 0 ) )^{\nu }}{\nu !} \end{aligned}$$
(2)

for \(x,a\in \mathbb{R} ^{+}\) and any function f defined on \(\mathbb{R} ^{+}\). We also assume that \(G_{n}^{\ast }\) satisfies the following condition

$$\begin{aligned} \sum_{\nu =0}^{\infty }e^{- \frac{ \mu \nu }{n^{2}\mathit{\psi }_{n} ( 0 ) }} \biggl[ \frac {\partial ^{\nu }}{\partial u^{\nu }}K_{n} \bigl( \lambda _{n} ( x ) ,t,u \bigr) \big|_{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}} \biggr] \frac{(-\alpha _{n}\mathit{\psi }_{n} ( 0 ) )^{\nu }}{\nu !}=e^{\beta _{n} ( \mu x ) } \end{aligned}$$
(3)

for all \(n\in \mathbb{N} \) and \(\mu \in \mathbb{Z} \). Here, \(\beta _{n}(x)\) is a sequence such that \(\lim_{n\rightarrow \infty }\beta _{n}(x)=x\).

It can be easily shown that the operators \(G_{n}^{\ast }\) preserve \(e^{ax}\), i.e.,

$$\begin{aligned} G_{n}^{\ast } \bigl( e^{at};x \bigr) =e^{ax}. \end{aligned}$$

We can use the Taylor expansion of \(\{ K_{n} ( \lambda _{n} ( x ) ,t,u ) \} \) due to condition \(( 1^{o} ) \). Setting \(u=\varphi _{n} ( t ) \) and \(u_{1}=\alpha _{n}\mathit{\psi }_{n} ( t ) \), we have

$$\begin{aligned} K_{n} \bigl( \lambda _{n} ( x ) ,t,\varphi _{n} ( t ) \bigr) =\sum_{\nu =0}^{\infty } \biggl[ \frac {\partial ^{\nu }}{\partial u^{\nu }}K_{n} \bigl( \lambda _{n} ( x ) ,t,u \bigr) \big|_{u=\alpha _{n}\mathit{\psi }_{n} ( t ) } \biggr] \frac{ ( \varphi _{n} ( t ) -\alpha _{n}\mathit{\psi }_{n} ( t ) ) ^{\nu }}{\nu !}. \end{aligned}$$

Since \(\varphi _{n} ( 0 ) =0\) and \(K_{n} ( \lambda _{n} ( x ) ,0,0 ) =1\), by taking \(t=0\) the above equation turns into

$$\begin{aligned} \sum_{\nu =0}^{\infty } \biggl[ \frac {\partial ^{\nu }}{\partial u^{\nu }}K_{n} \bigl( \lambda _{n} ( x ) ,t,u \bigr) \big|_{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}} \biggr] \frac{ ( -\alpha _{n}\mathit{\psi }_{n} ( 0 ) ) ^{\nu }}{\nu !}=1. \end{aligned}$$
(4)

Using Eq. (4), we get

$$\begin{aligned} G_{n}^{\ast } \bigl( e^{at};x \bigr) &=\sum _{\nu =0}^{ \infty }e^{-\frac{a\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) }}e^{ax}e^{ \frac{a\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) }} \biggl[ \frac {\partial ^{\nu }}{\partial u^{\nu }}K_{n} \bigl( \lambda _{n} ( x ) ,t,u \bigr) \big| _{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}} \biggr] \frac{(-\alpha _{n}\mathit{\psi }_{n} ( 0 ) )^{\nu }}{\nu !} \\ &=e^{ax}\sum_{\nu =0}^{\infty } \biggl[ \frac {\partial ^{\nu }}{\partial u^{\nu }}K_{n} \bigl( \lambda _{n} ( x ) ,t,u \bigr) \big| _{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}} \biggr] \frac{(-\alpha _{n}\mathit{\psi }_{n} ( 0 ) )^{\nu }}{\nu !} \\ &=e^{ax}. \end{aligned}$$

3 Auxilary results

In this part, we mention some obvious properties of the modified Ibragimov–Gadjiev operators.

Lemma 1

Let \(f ( t ) =e^{\theta at}\), \(\theta \in \mathbb{Z} \). Then, for the operators defined by (2), we have

$$\begin{aligned} G_{n}^{\ast } \bigl( e^{\theta at};x \bigr) =e^{ax+\beta _{n}( ( \theta -1 ) ax)} \end{aligned}$$

here \(\beta _{n}\) is the same as (3).

We give the following lemma without proof since it similarly follows from the work by the authors dealing with moments for the q-Ibragimov–Gadjiev operators [4].

Lemma 2

Let \(e_{i} ( t ) :=t^{i}\), \(i=0,1,2\). Then the operators \(G_{n}^{\ast }\) satisfies

$$\begin{aligned} &G_{n}^{\ast } ( e_{0};x ) =e^{ax+\beta _{n} ( -ax ) }, \\ &G_{n}^{\ast } ( e_{1};x ) =e^{ ( ax+\beta _{n+m} ( -ax ) - \frac{a}{n^{2}\mathit{\psi }_{n} ( 0 ) } ) } \frac{\alpha _{n}}{n}\lambda _{n} (x), \\ &G_{n}^{\ast } ( e_{2};x ) =e^{ ( ^{ax+e^{\beta _{n+2m}(-ax)}- \frac{2a}{n^{2}\mathit{\psi }_{n} ( 0 ) }} ) } \biggl( \frac{\alpha _{n}}{n} \biggr) ^{2}\frac{n+m}{n}\lambda _{n} ^{2}(x) \\ &\phantom{G_{n}^{\ast } ( e_{2};x ) =}{}+e^{ ( ^{ax+\beta _{n+m}(-ax)- \frac{2a}{n^{2}\mathit{\psi }_{n} ( 0 ) }} ) } \frac{\alpha _{n}}{n}\frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) }\lambda _{n} (x), \\ &G_{n}^{\ast } ( e_{3};x ) =e^{ ( ^{ax+\beta _{n+3m}(-ax)- \frac{3a}{n^{2}\mathit{\psi }_{n} ( 0 ) }} ) } \biggl( \frac{\alpha _{n}}{n} \biggr) ^{3} \frac{ ( n+m ) ( n+2m ) }{n^{2}}\lambda _{n} ^{3}(x) \\ &\phantom{G_{n}^{\ast } ( e_{3};x ) =}{}+e^{ ( ax+\beta _{n+2m}(-ax)- \frac{2a}{n^{2}\mathit{\psi }_{n} ( 0 ) } ) } \biggl( \frac{\alpha _{n}}{n} \biggr) ^{2} \frac{3 ( n+m ) }{n} \frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) }\lambda _{n} ^{2}(x) \\ &\phantom{G_{n}^{\ast } ( e_{3};x ) =}{}+e^{ ( ax+\beta _{n+m}(-ax)- \frac{a}{n^{2}\mathit{\psi }_{n} ( 0 ) } ) } \frac{\alpha _{n}}{n} \biggl( \frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) } \biggr) ^{2} \lambda _{n} (x), \\ &G_{n}^{\ast } ( e_{4};x ) =e^{ ( ^{ax+\beta _{n+4m}(-ax)- \frac{4a}{n^{2}\mathit{\psi }_{n} ( 0 ) }} ) } \biggl( \frac{\alpha _{n}}{n} \biggr) ^{4} \frac{ ( n+m ) ( n+2m ) ( n+3m ) }{n^{3}}\lambda _{n} ^{4}(x) \\ &\phantom{G_{n}^{\ast } ( e_{4};x ) =}{}+e^{ ( ax+\beta _{n+3m}(-ax)- \frac{3a}{n^{2}\mathit{\psi }_{n} ( 0 ) } ) } \biggl( \frac{\alpha _{n}}{n} \biggr) ^{3} \frac{6 ( n+m ) ( n+2m ) }{n^{2}} \frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) }\lambda _{n} ^{3}(x) \\ &\phantom{G_{n}^{\ast } ( e_{4};x ) =}{}+e^{ ( ax+\beta _{n+2m}(-ax)-\frac{2a}{n^{2}\mathit{\psi }_{n} ( 0 ) } ) } \biggl( \frac{\alpha _{n}}{n} \biggr) ^{2} \frac{7 ( n+m ) }{n} \biggl( \frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) } \biggr) ^{2}\lambda _{n} ^{2}(x) \\ &\phantom{G_{n}^{\ast } ( e_{4};x ) =}{}+e^{ ( ^{ax+\beta _{n+m}(-ax)-\frac{a}{n^{2}\mathit{\psi }_{n} ( 0 ) }} ) } \frac{\alpha _{n}}{n} \biggl( \frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) } \biggr) ^{3}\lambda _{n} (x). \end{aligned}$$

Lemma 3

Let \(\mu _{n,r} ( x ) =G_{n}^{\ast } ( ( t-x ) ^{r};x ) \), \(r=0,1,2,4\). Then by considering above Lemma, we have

$$\begin{aligned} &\mu _{n,0} ( x ) =e^{ax+\beta _{n} ( -ax ) }, \\ &\mu _{n,1} ( x ) =\frac{\alpha _{n}}{n}\lambda _{n} (x)e^{ ( ax+\beta _{n+m} ( -ax ) - \frac{a}{n^{2}\mathit{\psi }_{n} ( 0 ) } ) }-xe^{ax+\beta _{n} ( -ax ) }, \\ &\mu _{n,2} ( x ) = \biggl( \frac{\alpha _{n}}{n} \biggr) ^{2} \frac{n+m}{n}\lambda _{n} ^{2}(x)e^{ ( ^{ax+e^{\beta _{n+2m}(-ax)}- \frac{2a}{n^{2}\mathit{\psi }_{n} ( 0 ) }} ) } \\ &\phantom{\mu _{n,2} ( x ) =}{}-\frac{\alpha _{n}}{n} \frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) }\lambda _{n} (x)e^{ ( ^{ax+\beta _{n+m}(-ax)- \frac{2a}{n^{2}\mathit{\psi }_{n} ( 0 ) }} ) } \\ &\phantom{\mu _{n,2} ( x ) =}{}-2x\frac{\alpha _{n}}{n}\lambda _{n} (x)e^{ ( ax+\beta _{n+m} ( -ax ) - \frac{a}{n^{2}\mathit{\psi }_{n} ( 0 ) } ) }+x^{2}e^{ax+\beta _{n} ( -ax ) }, \\ &\mu _{n,4} ( x ) =e^{ ( ax+\beta _{n+4m}(-ax)- \frac{4a}{n^{2}\mathit{\psi }_{n} ( 0 ) } ) } \biggl( \frac{\alpha _{n}}{n} \biggr) ^{4} \frac{ ( n+m ) ( n+2m ) ( n+3m ) }{n^{3}}\lambda _{n} ^{4}(x) \\ &\phantom{\mu _{n,4} ( x ) =}{}+e^{ ( ^{ax+\beta _{n+3m}(-ax)- \frac{3a}{n^{2}\mathit{\psi }_{n} ( 0 ) }} ) } \biggl( \frac{\alpha _{n}}{n} \biggr) ^{3} \frac{6 ( n+m ) ( n+2m ) }{n^{2}} \frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) }\lambda _{n} ^{3}(x) \\ &\phantom{\mu _{n,4} ( x ) =}{}+e^{ ( ^{ax+\beta _{n+2m}(-ax)- \frac{2a}{n^{2}\mathit{\psi }_{n} ( 0 ) }} ) } \biggl( \frac{\alpha _{n}}{n} \biggr) ^{2} \frac{7 ( n+m ) }{n} \biggl( \frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) } \biggr) ^{2}\lambda _{n} ^{2}(x) \\ &\phantom{\mu _{n,4} ( x ) =}{}+e^{ ( ^{ax+\beta _{n+m}(-ax)-\frac{a}{n^{2}\mathit{\psi }_{n} ( 0 ) }} ) } \frac{\alpha _{n}}{n} \biggl( \frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) } \biggr) ^{3}\lambda _{n} (x) \\ &\phantom{\mu _{n,4} ( x ) =}{}-4x \biggl\{ e^{ ( ^{ax+\beta _{n+3m}(-ax)- \frac{3a}{n^{2}\mathit{\psi }_{n} ( 0 ) }} ) } \biggl( \frac{\alpha _{n}}{n} \biggr) ^{3} \frac{ ( n+m ) ( n+2m ) }{n^{2}}\lambda _{n} ^{3}(x) \\ &\phantom{\mu _{n,4} ( x ) =}{}+e^{ ( ax+\beta _{n+2m}(-ax)- \frac{2a}{n^{2}\mathit{\psi }_{n} ( 0 ) } ) } \biggl( \frac{\alpha _{n}}{n} \biggr) ^{2} \frac{3 ( n+m ) }{n} \frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) }\lambda _{n} ^{2}(x) \\ &\phantom{\mu _{n,4} ( x ) =}{} +e^{ ( ax+\beta _{n+m}(-ax)-\frac{a}{n^{2}\mathit{\psi }_{n} ( 0 ) } ) } \frac{\alpha _{n}}{n} \biggl( \frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) } \biggr) ^{2} \lambda _{n} (x) \biggr\} \\ & \phantom{\mu _{n,4} ( x ) =}{}+6x^{2} \biggl\{ e^{ ( ^{ax+e^{ \beta _{n+2m}(-ax)}- \frac{2a}{n^{2}\mathit{\psi }_{n} ( 0 ) }} ) } \biggl( \frac{\alpha _{n}}{n} \biggr) ^{2}\frac{n+m}{n}\lambda _{n} ^{2}(x) \textit{ }\\ &\phantom{\mu _{n,4} ( x ) =}{} +e^{ ( ^{ax+\beta _{n+m}(-ax)- \frac{2a}{n^{2}\mathit{\psi }_{n} ( 0 ) }} ) } \frac{\alpha _{n}}{n}\frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) }\lambda _{n} (x) \biggr\} \\ &\phantom{\mu _{n,4} ( x ) =}{}-4x^{3}e^{ ( ax+\beta _{n+m} ( -ax ) -\frac{a}{n^{2}\mathit{\psi }_{n} ( 0 ) } ) } \frac{\alpha _{n}}{n}\lambda _{n} (x)+x^{4}. \end{aligned}$$

4 Main results

In this section, first, we analyze the uniform convergence of \(G_{n}^{\ast }\) on \([ 0,\infty ) \) by means of the modulus of continuity. In 1970, Boyanov and Vaselinov [33] gave approximation properties of a function in an infinite interval.

Now, suppose that \(C^{\ast } [ 0,\infty ) \) denotes the Banach space of all real-valued continuous functions on \([ 0,\infty ) \) with the property \(\lim_{x\rightarrow \infty }f ( x ) \) existing and finite, given with uniform norm \(\Vert . \Vert _{C^{\ast } [ 0,\infty ) }\).

Theorem A

[33] If the sequence \(A_{n}:C^{\ast } [ 0,\infty ) \rightarrow C^{\ast } [ 0,\infty ) \) of linear positive operators satisfies the conditions

$$\begin{aligned} \lim_{n\rightarrow \infty }A_{n} \bigl( e^{-kt};x \bigr) =e^{-kx},\quad k=0,1,2, \end{aligned}$$

uniformly in \([ 0,\infty ) \) then for \(f\in [ 0,\infty ) \),

$$\begin{aligned} \lim_{n\rightarrow \infty }A_{n} ( f;x ) =f ( x ) . \end{aligned}$$

Later, Holhoş [34] expanded Theorem A to find the rate of uniform convergence.

Theorem B

[34] Let \(A_{n}:C^{\ast } [ 0,\infty ) \rightarrow C^{\ast } [ 0,\infty ) \) be a sequence of positive linear operators satisfying that

$$\begin{aligned} &\bigl\Vert A_{n} ( 1;x ) -1 \bigr\Vert _{ [ 0, \infty ) } =a_{n}, \\ &\bigl\Vert A_{n} \bigl( e^{-t};x \bigr) -e^{-x} \bigr\Vert _{ [ 0,\infty ) } =b_{n}, \\ &\bigl\Vert A_{n} \bigl( e^{-2t};x \bigr) -e^{-2x} \bigr\Vert _{ [ 0,\infty ) } =c_{n}. \end{aligned}$$

Then, for every function of \(f\in C^{\ast } [ 0,\infty ) \),

$$\begin{aligned} \bigl\Vert A_{n} ( f;x ) -f ( x ) \bigr\Vert _{ [ 0,\infty ) }\leq \Vert f \Vert _{ [ 0,\infty ) }a_{n}+ ( 2+a_{n} ) \omega ^{ \ast } ( f,\sqrt{a_{n}+2b_{n}+c_{n}} ) \end{aligned}$$

where

$$\begin{aligned} \omega ^{\ast } ( f,\delta ) =\max_{ \substack{ \vert e^{-x}-e^{-t} \vert \leq \delta \\ x,t>0}} \bigl\vert f ( t ) -f ( x ) \bigr\vert , \quad \delta >0 \end{aligned}$$

is the modulus of continuity. Further,

$$\begin{aligned} \bigl\vert f ( t ) -f ( x ) \bigr\vert \leq \biggl( 1+\frac{ ( e^{-x}-e^{-t} ) ^{2}}{\delta ^{2}} \biggr) \omega ^{\ast } ( f,\delta ) . \end{aligned}$$

Now, we are ready to prove our main theorem about uniform convergence of \(G_{n}^{\ast }\).

Theorem 1

For \(f\in C^{\ast } [ 0,\infty ) \), we have

$$\begin{aligned} \bigl\Vert G_{n}^{\ast }f-f \bigr\Vert _{ [ 0,\infty ) } \leq \Vert f \Vert _{ [ 0,\infty ) }a_{n}+ ( 2+a_{n} ) \omega ^{\ast } ( f,\sqrt{a_{n}+2b_{n}+c_{n}} ) \end{aligned}$$

where

$$\begin{aligned} &a_{n} = \bigl\Vert G_{n}^{\ast } ( 1;x ) -1 \bigr\Vert _{ [ 0,\infty ) } \\ &b_{n} = \bigl\Vert G_{n}^{\ast } \bigl( e^{-t};x \bigr) -e^{-x} \bigr\Vert _{ [ 0,\infty ) } \\ &c_{n} = \bigl\Vert G_{n}^{\ast } \bigl( e^{-2t};x \bigr) -e^{-2x} \bigr\Vert _{ [ 0,\infty ) }. \end{aligned}$$

Moreover, \(a_{n}\), \(b_{n}\) and \(c_{n}\) tend to zero as n goes to infinity so that \(G_{n}^{\ast }\) converges uniformly to f.

Proof

Considering Lemma 1 and the definition of the operators \(G_{n}^{\ast }\), we can write

$$\begin{aligned} &G_{n}^{\ast } ( 1;x ) =e^{ax+\beta _{n} ( -ax ) }, \\ &G_{n} \bigl( e^{-t};x \bigr) =\sum _{\nu =0}^{\infty }e^{- \frac{a\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) }}e^{ax}e^{- \frac{\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) }} \biggl[ \frac {\partial ^{\nu }}{\partial u^{\nu }}K_{n} \bigl( \lambda ( x ) ,t,u \bigr) \big|_{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}} \biggr] \frac{(-\alpha _{n}\mathit{\psi }_{n} ( 0 ) )^{\nu }}{\nu !} \\ &\phantom{G_{n} \bigl( e^{-t};x \bigr) }=e^{ax}\sum_{\nu =0}^{\infty } \bigl( e^{- \frac{\nu ( a+1 ) }{n^{2}\mathit{\psi }_{n} ( 0 ) }} \bigr) \biggl[ \frac {\partial ^{\nu }}{\partial u^{\nu }}K_{n} \bigl( \lambda ( x ) ,t,u \bigr) \big|_{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}} \biggr] \frac{(-\alpha _{n}\mathit{\psi }_{n} ( 0 ) )^{\nu }}{\nu !}\text{ } \\ &\phantom{G_{n} \bigl( e^{-t};x \bigr) }=e^{ax}e^{\beta _{n}(-(a+1)x)} \\ &G_{n} \bigl( e^{-2t};x \bigr) =\sum _{\nu =0}^{\infty }e^{- \frac{a\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) }}e^{ax}e^{- \frac{2\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) }} \biggl[ \frac {\partial ^{\nu }}{\partial u^{\nu }}K_{n} \bigl( \lambda ( x ) ,t,u \bigr) \big|_{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}} \biggr] \frac{(-\alpha _{n}\mathit{\psi }_{n} ( 0 ) )^{\nu }}{\nu !} \\ &\phantom{G_{n} \bigl( e^{-2t};x \bigr)}=e^{ax}\sum_{\nu =0}^{\infty }e^{- \frac{(a+2)\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) }} \biggl[ \frac {\partial ^{\nu }}{\partial u^{\nu }}K_{n} \bigl( \lambda ( x ) ,t,u \bigr) \big|_{ \substack{ u=\alpha _{n}\mathit{\psi }_{n} ( t ) \\ t=0}} \biggr] \frac{(-\alpha _{n}\mathit{\psi }_{n} ( 0 ) )^{\nu }}{\nu !} \\ &\phantom{G_{n} \bigl( e^{-2t};x \bigr)}=e^{ax}e^{\beta _{n}(-(a+2)x)}. \end{aligned}$$

Thus, the result follows immediately from (4) and Theorem B. □

Now, we examine the behavior of the operators \(G_{n}^{\ast }\) on some weighted spaces and then prove a quantitative Voronovskaja theorem in terms of modulus of continuity for functions having exponential growth.

Set \(\varphi ( x ) =1+e^{2ax}\), \(x\in \mathbb{R} ^{+}\), and consider the following weighted spaces:

$$\begin{aligned} &B_{\varphi } \bigl( \mathbb{R} ^{+} \bigr) = \bigl\{ f:\mathbb{R} ^{+}\rightarrow \mathbb{R} :\text{ } \bigl\vert f ( x ) \bigr\vert \text{ }\leq M_{f}\varphi ( x ) \bigr\} , \\ &C_{\varphi } \bigl( \mathbb{R} ^{+} \bigr) =C\bigl(\mathbb{R} ^{+}\bigr)\cap B_{\varphi } \bigl( \mathbb{R} ^{+} \bigr) , \\ &C_{\varphi }^{k} \bigl( \mathbb{R} ^{+} \bigr) = \biggl\{ f\in C\bigl(\mathbb{R} ^{+}\bigr):\lim _{x\rightarrow \infty } \frac{f ( x ) }{\varphi ( x ) }=k_{f} \biggr\} , \end{aligned}$$

where \(M_{f}\) and \(k_{f}\) are constants depending on f. All three spaces are normed spaces with the norm

$$\begin{aligned} \Vert f \Vert _{\varphi }=\sup_{x\in \mathbb{R} ^{+}} \frac{ \vert f ( x ) \vert }{\varphi ( x ) }. \end{aligned}$$

It is obvious that for any \(f\in C_{\varphi }^{k} ( \mathbb{R} ^{+} ) \), the inequality

$$\begin{aligned} \bigl\Vert G_{n}^{\ast } ( f ) \bigr\Vert _{\varphi } \leq \Vert f \Vert _{\varphi } \end{aligned}$$

holds, and we conclude that \(G_{n}^{\ast }\) maps \(C_{\varphi }^{k} ( \mathbb{R} ^{+} ) \) to \(C_{\varphi }^{k} ( \mathbb{R} ^{+} )\) [25].

Theorem 2

For each function \(f\in C_{\varphi }^{k} ( \mathbb{R} ^{+} ) \),

$$\begin{aligned} \lim_{n\rightarrow \infty } \bigl\Vert G_{n}^{\ast } ( f ) -f \bigr\Vert _{\varphi }=0. \end{aligned}$$

Proof

Using the general result established in [25], it is sufficient to verify that following three conditions

$$\begin{aligned} \lim_{n\rightarrow \infty } \bigl\Vert G_{n}^{\ast } \bigl( e^{ \upsilon at} \bigr) -e^{\upsilon at} \bigr\Vert _{\varphi }=0, \quad\upsilon =0,1,2. \end{aligned}$$
(5)

For \(\upsilon =0\), from Lemma 3, one has

$$\begin{aligned} \bigl\Vert G_{n}^{\ast } ( 1 ) -1 \bigr\Vert _{\varphi }= \sup_{x\in \mathbb{R} ^{+}} \frac{ \vert e^{ax+\beta _{n} ( -ax ) }-1 \vert }{1+e^{2ax}}. \end{aligned}$$

By passing to limit condition, using equality (4), we have

$$\begin{aligned} \lim_{n\rightarrow \infty } \bigl\Vert G_{n}^{\ast } ( 1 ) -1 \bigr\Vert _{\varphi }=0. \end{aligned}$$

We now prove for \(\upsilon =2\), similarly from Lemma 3 and equality (4), we get

$$\begin{aligned} \bigl\Vert G_{n}^{\ast } \bigl( e^{2at} \bigr) -e^{2ax} \bigr\Vert _{\varphi } &=\sup_{x\in \mathbb{R} ^{+}} \frac{ \vert e^{ax+\beta _{n}(ax)}-e^{2ax} \vert }{1+e^{2ax}} \\ &\leq \frac{e^{ax}}{1+e^{2ax}} \bigl\vert e^{\beta _{n}(ax)}-e^{ax} \bigr\vert \\ &\leq \bigl\vert e^{\beta _{n}(ax)}-e^{ax} \bigr\vert , \end{aligned}$$

which leads to

$$\begin{aligned} \lim_{n\rightarrow \infty } \bigl\Vert G_{n}^{\ast } \bigl( e^{2at} \bigr) -e^{2ax} \bigr\Vert _{\varphi }=0. \end{aligned}$$

Since \(G_{n}^{\ast } ( e^{at};x ) =e^{ax}\), condition (5) is implemented for \(\upsilon =1\). Hence, the proof is completed. □

Theorem 3

Let \(G_{n}^{\ast }\): \(K\rightarrow C [ 0,\infty ) \) be the sequence of linear positive operators preserving \(e^{ax}\), \(a>0\). We suppose that for each constant \(B>0\) and fixed \(x\in [ 0,\infty ) \), \(G_{n}^{\ast }\) satisfy

$$\begin{aligned} G_{n}^{\ast } \bigl( ( t-x ) ^{2}e^{Bt};x \bigr) \leq C_{a} ( B,x ) \mu _{n,2} ( x ) . \end{aligned}$$

Additionally, if \(f\in C^{2} [ 0,\infty ) \)K and \(f^{{\prime \prime }}\in Lip ( c,B ) \), \(0< c\leq 1\), then for \(x\in [ 0,\infty ) \),

$$\begin{aligned} & \biggl\vert G_{n}^{\ast } ( f;x ) -f ( x ) -f^{ \prime } ( x ) \mu _{n,1}-\frac{1}{2}f^{\prime \prime } ( x ) \mu _{n,2} ( x ) \biggr\vert \\ &\quad\leq \mu _{n,2} ( x ) \biggl( \frac{\sqrt{C_{a} ( 2B,x ) }}{2}+ \frac{C_{a} ( B,x ) }{2}+e^{2Bx} \biggr) \omega _{1} \biggl( f^{\prime \prime },\sqrt{ \frac{\mu _{n,4} ( x ) }{\mu _{n,2} ( x ) }},B \biggr) . \end{aligned}$$

Proof

By considering the Taylor expansion of the function \(f\in C^{2} [ 0,\infty ) \) at \(x\in [ 0,\infty ) \), we obtain

$$\begin{aligned} f ( t ) =f ( x ) +f^{\prime } ( x ) ( t-x ) +f^{\prime \prime } ( x ) \frac{ ( t-x ) ^{2}}{2}+h ( t,x ), \end{aligned}$$
(6)

where

$$\begin{aligned} h ( t,x ) =\frac{ ( t-x ) ^{2}}{2} \bigl( f^{ \prime \prime } ( \xi ) -f^{\prime \prime } ( x ) \bigr) , \quad x< \xi < t. \end{aligned}$$

Applying the operators \(G_{n}^{\ast }\) to equality (6), we have

$$\begin{aligned} \biggl\vert G_{n}^{\ast } ( f;x ) -f ( x ) -f^{ \prime } ( x ) \mu _{n,1}-\frac{1}{2}f^{\prime \prime } ( x ) \mu _{n,2} ( x ) \biggr\vert &= \bigl\vert G_{n}^{\ast } \bigl( h ( t,x ) ;x \bigr) \bigr\vert \\ &\leq G_{n}^{\ast } \bigl( \bigl\vert h ( t,x ) \bigr\vert ;x \bigr) . \end{aligned}$$
(7)

Additionally,

$$\begin{aligned} h ( t,x ) =\frac{ ( t-x ) ^{2}}{2} \bigl( f^{ \prime \prime } ( \xi ) -f^{\prime \prime } ( x ) \bigr) \leq \frac{ ( t-x ) ^{2}}{2}\textstyle\begin{cases} e^{Bx}\omega _{1} ( f^{\prime \prime },h,B ) , & \vert t-x \vert \leq h, \\ e^{Bx}\omega _{1} ( f^{\prime \prime },kh,B ) , & h\leq \vert t-x \vert \leq kh,\end{cases}\displaystyle \end{aligned}$$

It was proved by Tachev et al. [35] that, for each \(h>0\) and \(k\in \mathbb{N} \),

$$\begin{aligned} \omega _{1} ( f,kh,B ) \leq ke^{B ( k-1 ) h} \omega _{1} ( f,h,B ) . \end{aligned}$$

With the help of the above inequality, we obtain

$$\begin{aligned} \frac{ ( t-x ) ^{2}e^{Bx}}{2}\omega _{1} \bigl( f^{\prime \prime },kh,B \bigr) & \leq \frac{ ( t-x ) ^{2}e^{Bx}}{2}ke^{B ( k-1 ) h}\omega _{1} \bigl( f^{\prime \prime },h,B \bigr) \\ &\leq \frac{ ( t-x ) ^{2}}{2} \biggl( \frac{ \vert t-x \vert }{h}+1 \biggr) e^{Bx}e^{B \vert t-x \vert }\omega _{1} \bigl( f^{\prime \prime },h,B \bigr) \\ &\leq \frac{ ( t-x ) ^{2}}{2} \biggl( \frac{ \vert t-x \vert }{h}+1 \biggr) \bigl( e^{Bt}+e^{2Bx} \bigr) \omega _{1} \bigl( f^{\prime \prime },h,B \bigr) . \end{aligned}$$

Thus,

$$\begin{aligned} \bigl\vert h ( t,x ) \bigr\vert \leq \frac{ ( t-x ) ^{2}}{2} \biggl( \frac{ \vert t-x \vert }{h}+1 \biggr) \bigl( e^{Bt}+e^{2Bx} \bigr) \omega _{1} \bigl( f^{\prime \prime },h,B \bigr) . \end{aligned}$$

Applying the operators \(G_{n}^{\ast }\) to both sides of the above inequality, we have

$$\begin{aligned} G_{n}^{\ast } \bigl( \bigl\vert h ( t,x ) \bigr\vert ;x \bigr) \leq {}&\frac{1}{2}G_{n}^{\ast } \biggl( \biggl( \frac{ \vert t-x \vert ^{3}}{h}+ \vert t-x \vert ^{2} \biggr) \bigl( e^{Bt}+e^{2Bx} \bigr) ;x \biggr) \omega _{1} \bigl( f^{\prime \prime },h,B \bigr) \\ ={}& \biggl( \frac{1}{2h}G_{n}^{\ast } \bigl( \vert t-x \vert ^{3}e^{Bt};x \bigr) +\frac{1}{2}G_{n}^{\ast } \bigl( \vert t-x \vert ^{2}e^{Bt};x \bigr) \biggr) \\ &{}+\frac{e^{2Bx}}{2h}G_{n}^{\ast } \bigl( \vert t-x \vert ^{3};x \bigr) +\frac{e^{2Bx}}{2}G_{n}^{\ast } \bigl( \vert t-x \vert ^{2};x \bigr) \omega _{1} \bigl( f^{\prime \prime },h,B \bigr) . \end{aligned}$$

Using some computations, we get

$$\begin{aligned} G_{n}^{\ast } \bigl( \vert t-x \vert ^{2}e^{Bt};x \bigr) ={}&G_{n}^{ \ast } \bigl( t^{2}e^{Bt};x \bigr) -2xG_{n}^{\ast } \bigl( te^{Bt};x \bigr) +x^{2}G_{n}^{\ast } \bigl( e^{Bt};x \bigr) \\ ={}&e^{ ( \beta _{n+2m} ( ( B-a ) x ) +ax+ \frac{ ( B-a ) 2}{n^{2}\mathit{\psi }_{n} ( 0 ) } ) } \biggl( \frac{\alpha _{n}}{n} \biggr) ^{2} \frac{ ( n+m ) }{n}\lambda ^{2}(x) \\ &{}+e^{ ( \beta _{n+m} ( ( B-a ) x ) +ax+ \frac{ ( B-a ) }{n^{2}\mathit{\psi }_{n} ( 0 ) } ) } \frac{\alpha _{n}}{n} \frac{1}{n^{2}\mathit{\psi }_{n} ( 0 ) }\lambda (x) \\ &{}-2xe^{ ( ^{\beta _{n+m} ( ( B-a ) x ) +ax+ \frac{ ( B-a ) }{n^{2}\mathit{\psi }_{n} ( 0 ) }} ) }\frac{\alpha _{n}}{n}\lambda (x)+x^{2}e^{ax+\beta _{n} ( ( B-a ) x ) }, \end{aligned}$$

For sufficiently large n, it is obvious that

$$\begin{aligned} G_{n}^{\ast } \bigl( \vert t-x \vert ^{2}e^{Bt};x \bigr) \leq C_{a} ( B,x ) \mu _{n,2} ( x ) . \end{aligned}$$
(8)

Making use of the Cauchy–Schwarz inequality, we have the following inequalities

$$\begin{aligned} &G_{n}^{\ast } \bigl( \vert t-x \vert ^{3}e^{Bt};x \bigr) \leq \sqrt{G_{n}^{\ast } \bigl( \vert t-x \vert ^{2}e^{2Bt};x \bigr) }\sqrt {G_{n}^{\ast } \bigl( \vert t-x \vert ^{4};x \bigr) } \\ &\phantom{G_{n}^{\ast } \bigl( \vert t-x \vert ^{3}e^{Bt};x \bigr) }\leq \sqrt{C_{a} ( 2B,x ) \mu _{n,2} ( x ) }\sqrt{\mu _{n,4} ( x ) }, \end{aligned}$$
(9)
$$\begin{aligned} & G_{n}^{\ast } \bigl( \vert t-x \vert ^{3};x \bigr) \leq \sqrt{G_{n}^{\ast } \bigl( \vert t-x \vert ^{4};x \bigr) } \sqrt{G_{n}^{\ast } \bigl( \vert t-x \vert ^{2};x \bigr) } \\ &\phantom{G_{n}^{\ast } \bigl( \vert t-x \vert ^{3};x \bigr) }\leq \sqrt{\mu _{n,4} ( x ) }\sqrt{\mu _{n,2} ( x ) }. \end{aligned}$$
(10)

Thus, using inequalities (8), (9), and (10) in (7), we obtain

$$\begin{aligned} & \biggl\vert G_{n}^{\ast } ( f;x ) -f ( x ) -f^{ \prime } ( x ) \mu _{n,1}-\frac{1}{2}f^{\prime \prime } ( x ) \mu _{n,2} ( x ) \biggr\vert \\ &\quad\leq \biggl( \frac{1}{2h}\sqrt{C_{a} ( 2B,x ) \mu _{n,2} ( x ) }\sqrt{\mu _{n,4} ( x ) }+\frac{1}{2}C_{a} ( B,x ) \mu _{n,2} ( x ) \\ &\qquad{} +\frac{e^{2Bx}}{2h}\sqrt{\mu _{n,4} ( x ) }\sqrt{ \mu _{n,2} ( x ) }+\frac{e^{2Bx}}{2}\mu _{n,2} ( x ) \biggr) \omega _{1} \bigl( f^{\prime \prime },h,B \bigr) . \end{aligned}$$

Finally, when \(h=\sqrt{ \frac{\mu _{n,4} ( x ) }{\mu _{n,2} ( x ) }}\) is chosen and substituted in the above equality, we get

$$\begin{aligned} & \biggl\vert G_{n}^{\ast } ( f;x ) -f ( x ) -f^{ \prime } ( x ) \mu _{n,1}-\frac{1}{2}f^{\prime \prime } ( x ) \mu _{n,2} ( x ) \biggr\vert \\ &\quad\leq \mu _{n,2} ( x ) \biggl( \frac{\sqrt{C_{a} ( 2B,x ) }}{2}+ \frac{C_{a} ( B,x ) }{2}+e^{2Bx} \biggr) \omega _{1} \biggl( f^{\prime \prime },\sqrt{ \frac{\mu _{n,4} ( x ) }{\mu _{n,2} ( x ) }},B \biggr) . \end{aligned}$$

Note that, for fixed \(x\in [ 0,\infty ) \), \(\frac{\mu _{n,4} ( x ) }{\mu _{n,2} ( x ) } \rightarrow 0\) as \(n\rightarrow \infty \), guarantees the convergence of Theorem 2. □

5 An application of modified Ibragimov–Gadjiev operators

Just like classical Ibragimov–Gadjiev operators, modified Ibragimov–Gadjiev operators also contain some modified operators preserving exponential functions under appropriate selection of \(K_{n} ( \lambda _{n} ( x ) ,t,u ) ,\alpha _{n}\) and \(\mathit{\psi }_{n} ( 0 ) \).

If property 4 is applied ν-times to the \(K_{n} ( \lambda _{n} ( x ) ,t,u ) \), the operators defined by (1) can be reduced to the form

$$\begin{aligned} G_{n}^{\ast } ( f;x ) ={}&\sum_{\nu =0}^{\infty }e^{- \frac{a\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) }}e^{ax}f \biggl( \frac{\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) } \biggr) \frac{n ( n+m ) ... ( n+ ( \nu -1 ) m ) }{\nu !} \\ &{}\times \bigl(\lambda _{n} ( x ) \alpha _{n}\mathit{\psi }_{n} ( 0 ) \bigr)^{\nu }K_{n+\nu m} \bigl( \lambda _{n} ( x ) ,0,\alpha _{n}\mathit{\psi }_{n} ( 0 ) \bigr) . \end{aligned}$$
(11)

1. In case

$$\begin{aligned} K_{n} \bigl( \lambda _{n} ( x ) ,t,u \bigr) = \biggl( 1- \frac{u\lambda _{n} ( x ) }{1+t} \biggr) ^{n} , \end{aligned}$$

the operator (11) turns into the form

$$\begin{aligned} G_{n}^{\ast } ( f;x ) ={}&\sum_{\nu =0}^{\infty }e^{- \frac{a\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) }}e^{ax}f \biggl( \frac{\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) } \biggr) \binom{n}{\nu } \\ &{}\times\bigl( \lambda _{n} ( x ) \alpha _{n}\mathit{\psi }_{n} ( 0 ) \bigr)^{\nu } \bigl( 1-\alpha _{n}\mathit{\psi }_{n} ( 0 ) \lambda _{n} ( x ) \bigr) ^{n-\nu }. \end{aligned}$$
(12)

Conditions \(( 1^{\ast } )\)\(( 4^{\ast } ) \) are fulfilled, and \(m=-1\). For \(\alpha _{n}=n\), \(\mathit{\psi }_{n} ( 0 ) =\frac{1}{n}\), we have modified Bernstein operators

$$\begin{aligned} G_{n}^{\ast } ( f;x ) =\sum_{\nu =0}^{\infty }e^{- \frac{a\nu }{n}}e^{ax}f \biggl( \frac{\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) } \biggr) \binom{n}{\nu }\bigl(\lambda _{n} ( x ) \bigr)^{\nu } \bigl( 1- \lambda _{n} ( x ) \bigr) ^{n-\nu }, \end{aligned}$$

where

$$\begin{aligned} \lambda _{n} ( x ) =\frac{e^{-ax/n}-1}{e^{a/n}-1} \end{aligned}$$

defined by Aral et al. [14]. Use of L’Hospital’s rule gives

$$\begin{aligned} \lim_{n\rightarrow \infty }\lambda _{n} ( x ) =x \end{aligned}$$

as claimed.

2. In (12), for \(\alpha _{n}=n\), \(\mathit{\psi }_{n} ( 0 ) =\frac{1}{nb_{n}}\) \(( \lim b_{n}= \infty ,\text{ lim}\frac{b_{n}}{n}=0 ) \), the operator (11) becomes modified Bernstein–Chlodowsky operators

$$\begin{aligned} G_{n}^{\ast } ( f;x ) =\sum_{\nu =0}^{\infty }e^{ \frac{-a\nu b_{n}}{n}}e^{ax}f \biggl( \frac{\nu b_{n}}{n} \biggr) \binom{n}{\nu }\biggl( \frac{\lambda _{n} ( x ) }{b_{n}} \biggr)^{\nu } \biggl( 1- \frac{\lambda _{n} ( x ) }{b_{n}} \biggr) ^{n-\nu } \end{aligned}$$

with

$$\begin{aligned} \lambda _{n} ( x ) =b_{n}\frac{e^{ax/n}-1}{e^{ab_{n}/n}-1},\qquad \lim_{n\rightarrow \infty }\lambda _{n} ( x ) =x \end{aligned}$$

defined by Özsaraç et al. [26].

3. By choosing

$$\begin{aligned} K_{n} \bigl( \lambda _{n} ( x ) ,t,u \bigr) =e^{-n ( t+u\lambda _{n} ( x ) ) }, \end{aligned}$$

conditions \(( 1^{\ast } )\)\(( 4^{\ast } ) \) are fulfilled, and \(m=0\). The operator (11) becomes

$$\begin{aligned} G_{n}^{\ast } ( f;x ) =\sum_{\nu =0}^{\infty }e^{- \frac{a\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) }}e^{ax}f \biggl( \frac{\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) } \biggr) \frac{ ( n\lambda _{n} ( x ) ) ^{\nu }}{\nu !}\bigl(\alpha _{n}\mathit{\psi }_{n} ( 0 ) \bigr)^{\nu }e^{-n ( \alpha _{n}\mathit{\psi }_{n} ( 0 ) \lambda _{n} ( x ) ) }. \end{aligned}$$
(13)

If we choose \(\alpha _{n}=n\), \(\mathit{\psi }_{n} ( 0 ) = \frac{1}{n}\), we get modified Szász–Mirakjan operators

$$\begin{aligned} G_{n}^{\ast } ( f;x ) =e^{-n\lambda _{n} ( x ) } \sum _{\nu =0}^{\infty }e^{-\frac{a\nu }{n}}e^{ax}f \biggl( \frac{\nu }{n} \biggr) \frac{ ( n\lambda _{n} ( x ) ) ^{\nu }}{\nu !}, \end{aligned}$$

where

$$\begin{aligned} \lambda _{n} ( x ) = \frac{-ax}{n ( e^{-a/n}-1 ) } \end{aligned}$$

introduced by Acu et al. [15].

Besides, with the choice of \(\lambda _{n} ( x ) \) in (13) as

$$\begin{aligned} \lambda _{n} ( x ) =\frac{ax}{n ( e^{a/n}-1 ) } \end{aligned}$$

another variant of Szász–Mirakjan operators is obtained, which was presented by Goyal [24]. As can be seen easily, using L’Hospital’s rule, the limit of both \(\lambda _{n} ( x ) \) yields

$$\begin{aligned} \lim_{n\rightarrow \infty }\lambda _{n} ( x ) =x. \end{aligned}$$

It must be noted that the new variants of Szász–Mirakjan operators obtained by different selection of \(\lambda _{n} ( x ) \) differ in terms of the functions they preserve as well as their structural features.

4. In addition, if we choose

$$\begin{aligned} K_{n} \bigl( \lambda _{n} ( x ) ,t,u \bigr) = \bigl( 1+t+u \lambda _{n} ( x ) \bigr) ^{-n} , \end{aligned}$$

all conditions are fulfilled and \(m=1\). Thus, the operator defined by (11) turns into

$$\begin{aligned} G_{n} ( f;x ) ={}&\sum_{\nu =0}^{\infty }e^{- \frac{a\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) }}e^{ax}f \biggl( \frac{\nu }{n^{2}\mathit{\psi }_{n} ( 0 ) } \biggr) \binom{n+\nu -1}{\nu } \\ &{}\times \bigl(\lambda _{n} ( x ) \alpha _{n}\mathit{\psi }_{n} ( 0 ) \bigr)^{\nu }K_{n+\nu } \bigl( \lambda _{n} ( x ) ,0,\alpha _{n}\mathit{\psi }_{n} ( 0 ) \bigr) . \end{aligned}$$

Choosing \(\alpha _{n}=n\), \(\mathit{\psi }_{n} ( 0 ) = \frac{1}{n}\), we get Baskakov operators

$$\begin{aligned} G_{n} ( f;x ) =\sum_{\nu =0}^{\infty }e^{- \frac{a\nu }{n}}e^{ax}f \biggl( \frac{\nu }{n} \biggr) \binom{n+\nu -1}{\nu } \frac{\lambda _{n}^{\nu } ( x ) }{ ( 1+\lambda _{n} ( x ) ) ^{n+\nu }}, \end{aligned}$$

where

$$\begin{aligned} \lambda _{n} ( x ) = \frac{e^{-\frac{ax}{n}}-1}{1-e^{\frac{a}{n}}},\qquad \lim _{n\rightarrow \infty }\lambda _{n} ( x ) =x \end{aligned}$$

defined by Özsaraç and Acar [25].

By choosing the appropriate sequences of \(K_{n} ( \lambda _{n} ( x ) ,t,u ) \), \(\alpha _{n}\) and \(\mathit{\psi }_{n} ( 0 ) \), one can obtain other new operators, and we leave it to readers.

6 Conclusions

Through this work, a new generalization of Ibragimov–Gadjiev operators, which fixes the function e\(^{ax},a>0\), has been constructed. Then, for these operators, some approximation properties have been provided, and it has been shown that the newly defined operators contain modified Bernstein-, Szász-, and Baskakov-type operators, which were studied by several authors, as special cases. The relationship between these operators obtained by different choices of \(\lambda _{n} ( x ) \) has also been revealed.

It is worth noting to readers that one can obtain new operators by taking different sequences of \(K_{n} ( \lambda _{n} ( x ) ,t,u ) \), \(\alpha _{n}\) and \(\mathit{\psi }_{n} ( 0 ) \). Moreover, the other approximation properties not covered in this study may also be investigated.

Data Availability

No datasets were generated or analysed during the current study.

References

  1. Ibragimov, I.I., Gadjiev, A.D.: On a sequence of linear positive operators. Sov. Math. Dokl. 11, 1092–1095 (1970)

    Google Scholar 

  2. Aral, A., Acar, T.: On approximation properties of generalized Durrmeyer operators, modern mathematical methods and high performance computing in science and technology. In: Springer Proc Math. Stat., vol. 171, pp. 1–15. Springer, Singapore (2016)

    Google Scholar 

  3. Bozma, G., Bars, E.: Approximation with a Kantorovich type Ibragimov–Gadjiev operator. Euroasia J. Math. Eng. Nat. Med. Sci. 9(20), 74–83 (2022)

    Google Scholar 

  4. Herdem, S., Buyukyazıcı, İ.: Ibragimov–Gadjiev operators based on q-integers. Adv. Differ. Equ. 2008, 304 (2018)

    Article  MathSciNet  Google Scholar 

  5. Herdem, S., Buyukyazıcı, İ.: Weighted approximation by q-ibragimov-Gadjiev operators. Math. Commun. 25(2), 201–212 (2020)

    MathSciNet  Google Scholar 

  6. Gadjiev, A.D., Ispir, N.: On a sequence of linear positive operators in weighted space. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 11, 45–56 (1999)

    MathSciNet  Google Scholar 

  7. Bozma, G., Bars, E.: On the approximation with an Ibragimov–Gadjiev type operator. Euroasia J. Math. Eng. Nat. Med. Sci. 9(20), 74–83 (2022)

    Google Scholar 

  8. Gonul, N., Coskun, E.: Approximation with modified Gadjiev–Ibragimov operators in \(C[0,A]\). J. Comput. Anal. Appl. 15(1), 868–879 (2013)

    MathSciNet  Google Scholar 

  9. Bilgin, N.G., Coskun, N.: Comparison result of some Gadjiev–Ibragimov type operators. Karaelmas J. Sci. Eng. 8(1), 188–196 (2018)

    Google Scholar 

  10. Bilgin, N.G., Ozgur, N.: Approximation by two dimensional Gadjiev–Ibragimov type operators. Ikon. J. Math. 1(1), 1–10 (2019)

    Google Scholar 

  11. Deniz, E., Aral, A.: Convergence properties of Ibragimov–Gadjiev–Durrmeyer operators. Creative Math. Inform. 24(1), 17–26 (2015)

    Article  MathSciNet  Google Scholar 

  12. Aldaz, J.M., Render, H.: Optimality of generalized Bernstein operators. J. Approx. Theory 162, 1407–1416 (2010)

    Article  MathSciNet  Google Scholar 

  13. Acar, T., Aral, A., Gonska, H.: On Szász–Mirakyan operators preserving e\(^{2ax}\), \(a>0\). Mediterr. J. Math. 14(6), 1–14 (2017)

    Google Scholar 

  14. Aral, A., C’ardenas-Morales, D., Garrancho, P.: Bernstein-type operators that reproduce exponential functions. J. Math. Inequal. 12(3), 861–872 (2018)

    Article  MathSciNet  Google Scholar 

  15. Acu, A.M., Tachev, G.: Yet another new variant of Szász–Mirakyan operator. Symmetry 13(11), Article ID 2018 (2021)

    Article  Google Scholar 

  16. Aral, A., Inoan, D., Raşa, I.: Approximation properties of Szász-Mirakyan operators preserving exponential functions. Positivity 23, 233–246 (2019)

    Article  MathSciNet  Google Scholar 

  17. Bodur, M., Yılmaz Ö, G., Aral, A.: Approximation by Baskakov–Szász-Stancu operators preserving exponential functions. Constr. Math. Anal. 1(1), 1–8 (2018)

    MathSciNet  Google Scholar 

  18. Deo, N., Pratap, R.: Approximation by integral form of Jain and Pethe operators. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 92, 31–38 (2022)

    Article  MathSciNet  Google Scholar 

  19. Deo, N., Pratap, R.: α-Bernstein–Kantorovich operators. Afr. Math. 31, 609–618 (2020)

    Article  MathSciNet  Google Scholar 

  20. Gupta, V.: A new type of exponential operator. Filomat 37(14), 4629–4638 (2023)

    Article  MathSciNet  Google Scholar 

  21. Gupta, V., Milovanović, G.V.: A solution to exponential operators. Results Math. 77, 207 (2022)

    Article  MathSciNet  Google Scholar 

  22. Gupta, V., Tachev, G.: On approximation properties of Phillips operators preserving exponential functions. Mediterr. J. Math. 14(4), Article ID 177 (2017)

    Article  MathSciNet  Google Scholar 

  23. Kanat, K., Sofyalioglu, M.: On Stancu type Szász–Mirakyan–Durrmeyer operators preserving e\(^{2ax}\), \(a>0\). Gazi Univ. J. Sci. 34(1), 196–209 (2021)

    Article  Google Scholar 

  24. Goyal, M.: Reconstruction of Szasz–Mirakyan operators preserving exponential type functions. Filomat 37(2), 427–434 (2023)

    Article  MathSciNet  Google Scholar 

  25. Özsaraç, F., Acar, T.: Reconstruction of Baskakov operators preserving some exponential functions. Math. Meth. Appl. Sci., spl. 42, 5124–5132 (2019)

    Article  MathSciNet  Google Scholar 

  26. Özsaraç, F., Aral, A., Karslı, H.: On Bernstein–Chlodovsky Type Operators Preserving Exponential Functions. Springer Proceedings in Mathematics & Statistics, vol. 306. Springer, Singapore (2018)

    Google Scholar 

  27. Pratap, R.: The family of λ-Bernstein–Durrmeyer operators based on certain parameters. Math. Found. Comput. 6(3), 546–557 (2023)

    Article  Google Scholar 

  28. Sofyalıoğlu Aksoy, M.: New modification of the post Widder operators preserving exponential functions. Arab J. Basic Appl. Sci. 31(1), 93–103 (2024)

    Article  Google Scholar 

  29. Sofyalıoğlu, M., Kanat, K.: Approximation properties of the post-Widder operators preserving \(e^{2ax},a>0\). Math. Methods Appl. Sci. 43(7), 4272–4285 (2020)

    MathSciNet  Google Scholar 

  30. Sofyalıoğlu, M., Kanat, K.: Approximation properties of generalized Baskakov–Schurer–Szasz–Stancu operators preserving \(e ^{-2ax},a>0\). J. Inequal. Appl. 1, 112 (2019)

    Article  MathSciNet  Google Scholar 

  31. Usta, F., Mursaleen, M., Çakır, İ.: Approximation properties of Bernstein–Stancu operators preserving \(e^{-2x}\). Filomat 37(5), 1523–1534 (2023)

    Article  MathSciNet  Google Scholar 

  32. Yılmaz, Ö.G., Gupta, V., Aral, A.: On Baskakov operators preserving the exponential function. J. Numer. Anal. Approx. Theory 46(2), 150–161 (2017)

    Article  MathSciNet  Google Scholar 

  33. Boyanov, B.D., Vaselinov, V.M.: A note on the approximation of functions in an infinite interval by linear positive operators. Bull. Math. Soc. Sci. Math. Roum. 14(62), 9–13 (1970)

    MathSciNet  Google Scholar 

  34. Holhoş, A.: The rate of approximation of functions in an infinite interval by positive linear operators. Stud. Univ. Babeş–Bolyai, Math. 55(2), 133–142 (2010)

    MathSciNet  Google Scholar 

  35. Tachev, G., Gupta, V., Aral, A.: Voronovskaja’s theorem for functions with exponential growth. Georgian Math. J. 27, 459–468 (2018). https://doi.org/10.1515/gmj-2018-0041

    Article  MathSciNet  Google Scholar 

Download references

Funding

There was no funding for this research article.

Author information

Authors and Affiliations

Authors

Contributions

No competing interest.

Corresponding author

Correspondence to Serap Herdem.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Herdem, S. Ibragimov–Gadjiev operators preserving exponential functions. J Inequal Appl 2024, 72 (2024). https://doi.org/10.1186/s13660-024-03147-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13660-024-03147-9

Keywords