- Research
- Open access
- Published:
Ibragimov–Gadjiev operators preserving exponential functions
Journal of Inequalities and Applications volume 2024, Article number: 72 (2024)
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
where
\(x,t\in \mathbb{R} ^{+}\) 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:
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:
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 [6–11].
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
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 [15–32].
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:
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
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.,
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
Since \(\varphi _{n} ( 0 ) =0\) and \(K_{n} ( \lambda _{n} ( x ) ,0,0 ) =1\), by taking \(t=0\) the above equation turns into
Using Eq. (4), we get
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
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
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
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
uniformly in \([ 0,\infty ) \) then for \(f\in [ 0,\infty ) \),
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
Then, for every function of \(f\in C^{\ast } [ 0,\infty ) \),
where
is the modulus of continuity. Further,
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
where
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
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:
where \(M_{f}\) and \(k_{f}\) are constants depending on f. All three spaces are normed spaces with the norm
It is obvious that for any \(f\in C_{\varphi }^{k} ( \mathbb{R} ^{+} ) \), the inequality
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 \(f\in C_{\varphi }^{k} ( \mathbb{R} ^{+} ) \),
Proof
Using the general result established in [25], it is sufficient to verify that following three conditions
For \(\upsilon =0\), from Lemma 3, one has
By passing to limit condition, using equality (4), we have
We now prove for \(\upsilon =2\), similarly from Lemma 3 and equality (4), we get
which leads to
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
Additionally, if \(f\in C^{2} [ 0,\infty ) \) ∩K and \(f^{{\prime \prime }}\in Lip ( c,B ) \), \(0< c\leq 1\), then for \(x\in [ 0,\infty ) \),
Proof
By considering the Taylor expansion of the function \(f\in C^{2} [ 0,\infty ) \) at \(x\in [ 0,\infty ) \), we obtain
where
Applying the operators \(G_{n}^{\ast }\) to equality (6), we have
Additionally,
It was proved by Tachev et al. [35] that, for each \(h>0\) and \(k\in \mathbb{N} \),
With the help of the above inequality, we obtain
Thus,
Applying the operators \(G_{n}^{\ast }\) to both sides of the above inequality, we have
Using some computations, we get
For sufficiently large n, it is obvious that
Making use of the Cauchy–Schwarz inequality, we have the following inequalities
Thus, using inequalities (8), (9), and (10) in (7), we obtain
Finally, when \(h=\sqrt{ \frac{\mu _{n,4} ( x ) }{\mu _{n,2} ( x ) }}\) is chosen and substituted in the above equality, we get
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
1. In case
the operator (11) turns into the form
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
where
defined by Aral et al. [14]. Use of L’Hospital’s rule gives
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
with
defined by Özsaraç et al. [26].
3. By choosing
conditions \(( 1^{\ast } )\)–\(( 4^{\ast } ) \) are fulfilled, and \(m=0\). The operator (11) becomes
If we choose \(\alpha _{n}=n\), \(\mathit{\psi }_{n} ( 0 ) = \frac{1}{n}\), we get modified Szász–Mirakjan operators
where
introduced by Acu et al. [15].
Besides, with the choice of \(\lambda _{n} ( x ) \) in (13) as
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
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
all conditions are fulfilled and \(m=1\). Thus, the operator defined by (11) turns into
Choosing \(\alpha _{n}=n\), \(\mathit{\psi }_{n} ( 0 ) = \frac{1}{n}\), we get Baskakov operators
where
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
Ibragimov, I.I., Gadjiev, A.D.: On a sequence of linear positive operators. Sov. Math. Dokl. 11, 1092–1095 (1970)
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)
Bozma, G., Bars, E.: Approximation with a Kantorovich type Ibragimov–Gadjiev operator. Euroasia J. Math. Eng. Nat. Med. Sci. 9(20), 74–83 (2022)
Herdem, S., Buyukyazıcı, İ.: Ibragimov–Gadjiev operators based on q-integers. Adv. Differ. Equ. 2008, 304 (2018)
Herdem, S., Buyukyazıcı, İ.: Weighted approximation by q-ibragimov-Gadjiev operators. Math. Commun. 25(2), 201–212 (2020)
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)
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)
Gonul, N., Coskun, E.: Approximation with modified Gadjiev–Ibragimov operators in \(C[0,A]\). J. Comput. Anal. Appl. 15(1), 868–879 (2013)
Bilgin, N.G., Coskun, N.: Comparison result of some Gadjiev–Ibragimov type operators. Karaelmas J. Sci. Eng. 8(1), 188–196 (2018)
Bilgin, N.G., Ozgur, N.: Approximation by two dimensional Gadjiev–Ibragimov type operators. Ikon. J. Math. 1(1), 1–10 (2019)
Deniz, E., Aral, A.: Convergence properties of Ibragimov–Gadjiev–Durrmeyer operators. Creative Math. Inform. 24(1), 17–26 (2015)
Aldaz, J.M., Render, H.: Optimality of generalized Bernstein operators. J. Approx. Theory 162, 1407–1416 (2010)
Acar, T., Aral, A., Gonska, H.: On Szász–Mirakyan operators preserving e\(^{2ax}\), \(a>0\). Mediterr. J. Math. 14(6), 1–14 (2017)
Aral, A., C’ardenas-Morales, D., Garrancho, P.: Bernstein-type operators that reproduce exponential functions. J. Math. Inequal. 12(3), 861–872 (2018)
Acu, A.M., Tachev, G.: Yet another new variant of Szász–Mirakyan operator. Symmetry 13(11), Article ID 2018 (2021)
Aral, A., Inoan, D., Raşa, I.: Approximation properties of Szász-Mirakyan operators preserving exponential functions. Positivity 23, 233–246 (2019)
Bodur, M., Yılmaz Ö, G., Aral, A.: Approximation by Baskakov–Szász-Stancu operators preserving exponential functions. Constr. Math. Anal. 1(1), 1–8 (2018)
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)
Deo, N., Pratap, R.: α-Bernstein–Kantorovich operators. Afr. Math. 31, 609–618 (2020)
Gupta, V.: A new type of exponential operator. Filomat 37(14), 4629–4638 (2023)
Gupta, V., Milovanović, G.V.: A solution to exponential operators. Results Math. 77, 207 (2022)
Gupta, V., Tachev, G.: On approximation properties of Phillips operators preserving exponential functions. Mediterr. J. Math. 14(4), Article ID 177 (2017)
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)
Goyal, M.: Reconstruction of Szasz–Mirakyan operators preserving exponential type functions. Filomat 37(2), 427–434 (2023)
Özsaraç, F., Acar, T.: Reconstruction of Baskakov operators preserving some exponential functions. Math. Meth. Appl. Sci., spl. 42, 5124–5132 (2019)
Özsaraç, F., Aral, A., Karslı, H.: On Bernstein–Chlodovsky Type Operators Preserving Exponential Functions. Springer Proceedings in Mathematics & Statistics, vol. 306. Springer, Singapore (2018)
Pratap, R.: The family of λ-Bernstein–Durrmeyer operators based on certain parameters. Math. Found. Comput. 6(3), 546–557 (2023)
Sofyalıoğlu Aksoy, M.: New modification of the post Widder operators preserving exponential functions. Arab J. Basic Appl. Sci. 31(1), 93–103 (2024)
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)
Sofyalıoğlu, M., Kanat, K.: Approximation properties of generalized Baskakov–Schurer–Szasz–Stancu operators preserving \(e ^{-2ax},a>0\). J. Inequal. Appl. 1, 112 (2019)
Usta, F., Mursaleen, M., Çakır, İ.: Approximation properties of Bernstein–Stancu operators preserving \(e^{-2x}\). Filomat 37(5), 1523–1534 (2023)
Yılmaz, Ö.G., Gupta, V., Aral, A.: On Baskakov operators preserving the exponential function. J. Numer. Anal. Approx. Theory 46(2), 150–161 (2017)
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)
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)
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
Funding
There was no funding for this research article.
Author information
Authors and Affiliations
Contributions
No competing interest.
Corresponding author
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/.
About this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-024-03147-9