Abstract
The current article deals with a modification of the Post-Widder operators which reproduce the exponential functions both and for The central moments, uniform convergence of the operators and the rate of convergence of these operators with the help of modulus of continuity are estimated. Also, a Voronovskaja-type asymptotic formula is established. After that, numerical results are obtained to confirm the theoretical results. Finally, new constructed operators are compared with modified Post-Widder operators.
1. Introduction
In 1976, May (May, Citation1976) introduces the Post-Widder operators for and as follows: where In 2020, for and the modified form of the Post-Widder operators (Sofyalıoğlu & Kanat, Citation2020) is defined by (1) (1)
Here, the operators given by Equation(1)(1) (1) reproduce constant and for fixed a > 0. For some recent studies on linear positive operators preserving exponential functions, we refer the readers to the literature (Acar, Aral, Cárdenas-Morales, & Garrancho, Citation2017; Acu, Aral, & Rasa, Citation2023; Aral, Cárdenas-Morales, & Garrancho, Citation2018; Aral, Limmam, and Ozsarac, Citation2019, Gupta & Agrawal, Citation2019; Gupta & Herzog, Citation2023; Gupta & López-Moreno, Citation2018; Gupta & Tachev, Citation2022a; Citation2022b, Kanat & Sofyalıoğlu, Citation2021; Sofyalıoğlu & Kanat, Citation2019 and Torun, Citation2022).
In the current paper, we construct a generalization of the Post-Widder operators, which preserve both and for The aim of this paper is to obtain better approximation results than the results in Equation(1)(1) (1) . Firstly, our starting point is the general sequence (2) (2) for and Here, and are positive functions to be calculated in such a way that the operators Equation(2)(2) (2) hold fixed the functions and for In order to define the sequences and explicitly, we consider the following identities for every and By choosing in Equation(2)(2) (2) , we achieve
Similarly, by taking in Equation(2)(2) (2) , we have
After simple calculations, we get
Attention that and tend to x as So, the operators return to Now, we substitute the values and in the operators Equation(2)(2) (2) . After some rearranging, we obtain (3) (3) where and (4) (4)
Henceforth, we will use the notation of our new construction for the Post-Widder operators throughout the article. The classical Post-Widder operators Equation(1)(1) (1) and the new construction of the Post-Widder operators Equation(3)(3) (3) are related by
Currently, we mention some auxilary lemmas.
Lemma 1.
Let , then we have (5) (5) where is given by Equation(4)(4) (4) .
Proof.
From Equation(3)(3) (3) , we have where is as given by Equation(4)(4) (4) .
Lemma 2.
For the rising factorial is denoted by . Then we have
Proof.
From Equation(3)(3) (3) , we have where is as given by Equation(4)(4) (4) .
Lemma 3.
For the operators Equation(3)(3) (3) we have the following moments where
Proof.
By choosing K = 0 and in Lemma 2, respectively, we obtain the desired results.
Lemma 4.
Let Then we write the following central moments
Proof.
By using the linearity of the operators Equation(3)(3) (3) , the proof is completed.
Furthermore, we obtain the limits of the central moments as follows:
1.1. Preliminaries
In the current part, we give the main definitions which will be used in the paper. Let the subspace of real-valued continuous functions on which possess finite limit at infinity be denoted by The space is equipped with the uniform norm. In the following remark, we will check that the operators belong to the space
In 2010, Holhoş (Holhoş, Citation2010) come to grips with the uniform convergence of the linear positive operators and obtained the following theorem with the help of modulus of continuity (6) (6)
Theorem
(Holhoş, Citation2010). Let be a sequence of linear positive operators, then for every function , where
As the variables and tend to zero.
Let the class of all bounded and uniform continuous functions f on be denoted by , equipped with the norm . The modulus of continuity of the function is defined by
Moreover, for , the second order modulus of continuity is described as where . Furthermore, Peetre’s -functionals are defined by indicates the space of the functions f, for which f, and be a member of . In 1993, DeVore (DeVore & Lorentz, Citation1993) represented the relation between the second order modulus of continuity and Peetre’s -functional as follows where M > 0.
The structure of this paper is as follows: In the next section, we give uniform convergence theorem as a main result. In Section 3, we give approximation results by using the modulus of continuity. In Section 3, we mention Voronovskaya-type theorem for asymptotic estimation. In last section, we present the comparison of the newly constructed operators with different operators theoretically and graphically.
2. Main results
In this part, we will briefly study the approximation properties of the new constructed operators
Remark 1.
Let and
Then we write and we already know from Lemma 3 that
We conclude from the exponential component that is bounded. So, we can write for fixed K and Let and For a fixed positive ϵ, such a positive L exists such that for all Let and then we write and hence
Thusly, for all we have scilicet, This result denotes that the operators belong to Finally, we write
In 1970, Boyanov and Veselinov (Boyanov & Veselinov, Citation1970) built a theorem in order to show uniform convergence of the linear positive operators. For the new constructed operators Equation(3)(3) (3) , we adapt this theorem in the next form.
Theorem 1.
Let the sequence of linear positive operators satisfy uniformly in . Then for each is satisfied uniformly in
Proof.
From Lemma 1 and Lemma 3, we achieve (7) (7) and (8) (8) where given in Equation(4)(4) (4) . By taking limit as λ goes to infinity, we complete the proof. Hereby, we obtain that uniformly in Then for each converges uniformly in
Now, we present a quantitive estimation of the new construction of the Post-Widder operators according to Holhoş’s (Holhoş, Citation2010) theorem as follows:
Theorem 2.
Let be a sequence of linear positive operators, then we write the next inequality for every where
Here, converges uniformly to f. Additionally, and tend to zero as
Proof.
From Holhoş’s theorem (Holhoş, Citation2010) and Lemma 3, we obtain that
In order to calculate we take into account the equality Equation(7)(7) (7) ,
By using the supremum values we achieve
We represent the convergence rate of in for and In the same manner,
By using we obtain the inequality
We investigate the convergence rate of in numerically.
Finally, we analyze the results of and and . Then we achieve that and tend to zero as
3. Modulus of continuity
In this part, we mention the rate of convergence with the help of usual modulus of continuity.
Lemma 5
. Let . Then we have
Proof.
From Equation(3)(3) (3) , we get
Theorem 3.
Let . Then for every , there is a constant M > 0, such that where
Proof.
For the proof, we describe the auxiliary operators (9) (9)
By using the Taylor expansion for a function we have (10) (10)
We apply operators to the EquationEqn. (10)(10) (10) . By using Lemma 4, we get (11) (11)
Further, (12) (12) and (13) (13)
Using the inequalities Equation(11)(11) (11) , Equation(12)(12) (12) and Equation(13)(13) (13) , we obtain (14) (14) where
From auxiliary operators Equation(9)(9) (9) and Lemma 5, we write (15) (15)
By using Equation(9)(9) (9) , Equation(14)(14) (14) , Equation(15)(15) (15) for every and by choosing we have
Remark 2.
One can check that and as This result guarantees the convergence of the Theorem 3.
4. Voronovskaya-type theorem
In this part, we mention some asymptotic estimation results of the pointwise convergence in the case of the functions with exponential growth.
Theorem 4.
For and we write where
Proof.
From Taylor’s formula, we can write (16) (16) where (17) (17) is the remainder term. ξ is a number between x and t. If we apply the operators to Equation(16)(16) (16) , we achieve
After that,
We briefly denote that and So,
Note that and tend to zero as from EquationEqn. (6)(6) (6) and EquationEqn. (6)(6) (6) . Right now, we concern about the part with the reminder term. (18) (18) where is given by Equation(6)(6) (6) . Using Equation(17)(17) (17) and the inequality Equation(18)(18) (18) , we write For if then so and if then we get Thusly, we have
So,
If we choose and we get
Remark 3.
We obtain the following result by direct calculations
Additionally, we get the following result
We present the next corollary as an important result of Theorem 4 and Remark 3.
Corollary 5.
Let and . Then we have (19) (19)
5. Comparison with Post-Widder operators preserving
Now, we give a theorem that the operators preserving both and approximate better than the Post-Widder operators preserving (Sofyalıoğlu & Kanat, Citation2020).
Corollary 6.
Let be a decreasing and convex function. Then there is a natural number m0 such that for , we write for all x > 0.
Theorem 7.
Let . Assume that there exists such that for all (20) (20)
Then (21) (21)
In particular,
Conversely, if Equation(21)(21) (21) holds with strict at a point , then there is a natural number m0 such that for
Proof.
From Equation(20)(20) (20) we have for all and that
By taking into consideration the Voronovskaya-type theorem given in (Sofyalıoğlu & Kanat, Citation2020), we write (22) (22)
From EquationEqn. (19)(19) (19) and EquationEqn. (22)(22) (22) , we have the following inequality
So, we directly obtain inequality Equation(21)(21) (21) .
On the contrary, if inequality Equation(21)(21) (21) holds with strict at a given point then
After that, by using EquationEqn. (19)(19) (19) and EquationEqn. (22)(22) (22) ve get the desired result.
Now, we illustrate various examples to verify our theoretical results. When specifying the functions for the following two examples, we take into account the inequality Equation(21)(21) (21) . Then we compare the operators with (Sofyalıoğlu & Kanat, Citation2020), and we present the attractive events.
Example 1.
Let we choose In , we illustrate the convergence of the new constructed operators to the function for different and values. We see that we obtain pleasant convergence results from these choices. In , we compare (red), (green) and the original function (blue).
From the and , one can easily arrive that, the operators converge more rapidly than to the function
Example 2.
Let In , we compare (red), (green) and the original function (blue) for
From the and , one can see that, the operators converge more rapidly than to the selected function
6. Conclusion
In this paper, we construct a new modification of Post-Widder operators which preserve both and for positive μ. After that, we submit the uniform convergence theorem, rate of convergence theorems and Voronovskaya-type theorem. Moreover, we give an comparison theorem by using the results of Voronovskaya-type theorem of the new operators and the reference operators According to the conditions Equation(21)(21) (21) of the comparison theorem, we determine the functions that are used in examples. Finally, the effectiveness of the newly constructed operators is shown with several graphs and error comparison tables. For the future work, Stancu-type generalization of the Post-Widder operators preserving two exponential functions can be investigated.
Disclosure statement
No potential conflict of interest was reported by the authors.
References
- Acar, T., Aral, A., Cárdenas-Morales, D., & Garrancho, P. (2017). Szasz-Mirakyan type operators which fix exponentials. Results in Mathematics, 72(3), 1393–1404. doi:10.1007/s00025-017-0665-9
- Acu, A. M., Aral, A., & Rasa, I. (2023). New properties of operators preserving exponentials. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 117(1). doi:10.1007/s13398-022-01332-3
- Aral, A., Cárdenas-Morales, D., & Garrancho, P. (2018). Bernstein-type operators that reproduce exponential functions. Journal of Mathematical Inequalities, 12(3), 861–872. doi:10.7153/jmi-2018-12-64
- Aral, A., Limmam, M. L., & Ozsarac, F. (2019). Approximation properties of Szasz-Mirakyan Kantorovich type operators. Mathematical Methods in the Applied Sciences, 42(16), 5233–5240. doi:10.1002/mma.5280
- Boyanov, B. D., & Veselinov, V. M. (1970). A note on the approximation of functions in an infinite interval by linear positive operators. Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie, 14(62), 9–13.
- DeVore, R. A., & Lorentz, G. G. (1993). Constructive approximation (pp. 177). Berlin: Springer.
- Gupta, V., & Agrawal, D. (2019). Convergence by modified Post-Widder operators. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 113(2), 1475–1486. doi:10.1007/s13398-018-0562-4
- Gupta, V., & Herzog, M. (2023). Semi Post-Widder operators and difference estimates. Bulletin of the Iranian Mathematical Society, 49(2), 18. doi:10.1007/s41980-023-00766-8
- Gupta, V., & López-Moreno, A.-J. (2018). Phillips operators preserving arbitrary exponential functions eat, ebt. Filomat, 32(14), 5071–5082. doi:10.2298/FIL1814071G
- Gupta, V., & Tachev, G. (2022b). Some results on Post-Widder operators preserving test function xr. Kragujevac Journal of Mathematics, 46(1), 149–165. doi:10.46793/KgJMat2201.149G
- Gupta, V., & Tachev, G. (2022a). A modified Post Widder operators preserving eAx. Studia Universitatis Babes-Bolyai Matematica, 67(3), 599–606. doi:10.24193/subbmath.2022.3.11
- Holhoş, A. (2010). The rate of approximation of functions in an infinite interval by positive linear operators. Studia Universitatis Babes-Bolyai, Mathematica, 2, 133–142.
- Kanat, K., & Sofyalioğlu, M. (2021). On Stancu type Szász-Mirakyan-Durrmeyer operators preserving e2ax, a > 0. Gazi University Journal of Science, 34(1), 196–209. doi:10.35378/gujs.691419
- May, C. P. (1976). Saturation and inverse theorems for combinations of a class of exponential type operators. Canadian Journal of Mathematics, 28(6), 1224–1250. doi:10.4153/CJM-1976-123-8
- Sofyalıoğlu, M., & Kanat, K. (2019). Approximation properties of generalized Baskakov-Schurer-Szasz-Stancu operators preserving e2ax, a > 0. J Inequal Appl, 112, https://doi.org/10.1186/s13660-019-2062-2
- Sofyalıoğlu, M., & Kanat, K. (2020). Approximation properties of the Post-Widder operators preserving e2ax, a > 0. Mathematical Methods in the Applied Sciences, 43(7), 4272–4285. doi:10.1002/mma.6192
- Torun, G. (2022). On approximation properties of Stancu type Post-Widder operators preserving exponential functions. Gazi University Journal of Science Part A: Engineering and Innovation, 9(2), 173–186. doi:10.54287/gujsa.1113567