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Approximation by a power series summability method of Kantorovich type Szász operators including Sheffer polynomials
Advances in Difference Equations volume 2021, Article number: 165 (2021)
Abstract
The main purpose of this paper is to use a power series summability method to study some approximation properties of Kantorovich type Szász–Mirakyan operators including Sheffer polynomials. We also establish Voronovskaya type result.
1 Introduction and background
Let \(\mathcal{K}_{m}=\{i\leq m:i\in \mathcal{K}\subseteq \mathbb{N}\}\). Then the natural density of \(\mathcal{K}\) is defined by \(\sigma (\mathcal{K})=\lim_{m}\frac{1}{m}{|\mathcal{K}_{m}|}\) provided the limit exists, where \(|\mathcal{K}_{m}|\) denotes the cardinality of \(\mathcal{K}_{m}\). A sequence \(\eta =(\eta _{i})\) is “statistically convergent” (see [9]) to \(\mathfrak{s}\) if for every \(\epsilon >0\)
and we write \(st-\lim_{m}\eta {_{m}}=\mathfrak{s}\).
Let \(\mathfrak{T}=(\mathfrak{d}_{ij})\) be an infinite matrix. It is said to be regular if it transforms a convergent sequence into a convergent one with the same limit.
Let \(\mathfrak{T}=(\mathfrak{d}_{ij})\) be regular matrix. A sequence \(\zeta =(\zeta _{j})\) is said to be \(\mathfrak{T}\)-statistically convergent (see [10]) to the number \(\mathfrak{s}\) if, for any \(\epsilon >0\), \(\lim_{i}{\sum_{j:|\eta _{j}-\mathfrak{s}|\geq \epsilon }} \mathfrak{d}_{ij}=0\), and denote \(st_{\mathfrak{T}}-\lim \eta =\mathfrak{s}\). If
Then it reduces to statistical convergence.
For a sequence of positive real numbers \((\mathfrak{p}_{j})\), denote the corresponding power series \(\mathfrak{p}(\mathfrak{y})=\sum_{j=1}^{\infty }\mathfrak{p}{_{j}}\mathfrak{y}^{j-1}\) which has radius of convergence \(R>0\). A sequence \(\eta =(\eta _{j})\) is convergent in the sense of power series method (see [12, 21]) if \(\lim_{\mathfrak{y}\rightarrow R^{-}}{ \frac{1}{\mathfrak{p}(\mathfrak{y})}\sum_{j=1}^{\infty }}\eta _{j}{{ \mathfrak{p}}}_{j}\mathfrak{y}{{^{j-1}}}=\mathcal{L}\) for all \(\mathfrak{y}\in (0,R)\). Moreover, the power series method is regular if and only if \(\lim_{\mathfrak{y}\rightarrow R^{-}}{ \frac{{{\mathfrak{p}}}_{j}\mathfrak{y}{{^{j-1}}}}{\mathfrak{p}(\mathfrak{y})}}=0\) holds for each \(j\in \{1,2,\ldots \}\) (see [2]). The power series method is more effective than the ordinary convergence (see [22, 23]). For more summability methods, see [3–5, 7, 13, 15–19].
We study a Korovkin type theorem for the Kantorovich type generalization of Szász operators involving Sheffer polynomials via power series method. We determine the rate of convergence for these operators. Furthermore, we give a Voronovskaya type theorem for \(\mathfrak{T}\)-statistical convergence.
The multiple Sheffer polynomials \(\{S_{k_{1},k_{2}}(x)\}_{k_{1},k_{2}=0}^{\infty }\) are defined as follows. The generating function is
where \(A(t_{1},t_{2})\) and \(H(t_{1},t_{2})\) have series expansions of the form
and
respectively, with the conditions
In [1], one defined the positive linear operators involving multiple Sheffer polynomials for \(x\in [ 0,\infty )\) as follows:
provided that the right-hand side of the above series converge, under conditions that:
-
(1)
\(S_{k_{1},k_{2}}(x)\geq 0,k_{1},k_{2}\in \mathbb{N}\),
-
(2)
\(A(1,1)\neq 0, H_{t_{1}}(1,1)=1,H_{t_{2}}(1,1)=1 \),
-
(3)
Series (1.2), (1.3) and (1.4) are convergent for \(|t_{1}|< R\), \(|t_{2}|< R\) and \((R_{1},R_{2})>1\).
In [6] one defined the Kantorovich variant of Szász operators induced by multiple Sheffer polynomials as follows:
provided that the right-hand side of the above relation exists.
Example 1.1 of [6], gives us the following expressions for moments of the Kantorovich variant of Szász operators induced by multiple Sheffer polynomials:
The central moments of the Kantorovich variant of Szász operators induced by multiple Sheffer polynomials are [6]
Similarly, there exist constants \(C_{di}\) (dependent only on \(\tilde{a}_{i,j}\) and \(\tilde{h}_{i,j}\)) such that
As a consequence of the above relations, we obtain
where \(E_{3},E_{4},E_{5},E_{6}\) are constant dependent on the derivatives of \(A(t_{1},t_{2})\) and \(H(t_{1},t_{2})\) up to order three at the point \((t_{1},t_{2})=(1,1)\).
2 Korovkin type results
The statistical form of Korovkin’s theorem was studied in [11] and the A-statistical version was considered in [8] (see also [13, 17] for other summability methods).
Let \(B[0,\infty )\) (\(C[0,\infty )\)) be “the space of all bounded (continuous) functions” on the interval \([0,\infty )\).
Theorem 2.1
Let \(\mathfrak{T}=(\mathfrak{d}_{ij})\) be regular matrix and \(K_{n}^{(S)}(f,x)\) be as in (1.6) on \([0,M]\), for any finite M. If
then
\(\mathfrak{h}\in C([0,M])\), where \(\Vert \mathfrak{h} \Vert =\sup_{t\in [ 0,M]}{ |\mathfrak{h}(t)|}\).
Proof
From Example 1.1 of [6], we have \(st_{\mathfrak{T}}-\lim_{n}{ \Vert K_{n}^{(S)}e_{0}-e_{0} \Vert }=0\). Now
Also \(\lim_{n\rightarrow \infty } \Vert K_{n}^{(S)}e_{1}-e_{1} \Vert =0\). Moreover,
as \(n\rightarrow \infty \). Now the proof follows directly from the statistical version of the Korovkin theorem [11]. □
Example 2.2
([14])
Under the conditions given in Theorem 2.1, set
where
then
If the matrix \(\mathfrak{T}\) is as in (1.1), then, by Theorem 2.1 we obtain \(st_{\mathfrak{T}}-\lim_{n}{ \Vert N_{n}h-h \Vert }=0\), but the operators \(N_{n}(h,x)\), do not satisfy the conditions of the theorem in [11].
In the following result we use a power series method as in [20, 24]; the Abel summability method was used.
Theorem 2.3
Let \((K_{n}^{(S)})\) be a sequence of positive linear operators from \(C[0,M]\) into \(B[0,M]\) \((0< M<\infty )\) such that
Then, for \(\mathfrak{h}\in C[0,M]\),
Proof
Clearly, from (2.2) follows (2.1). Now we show the converse that (2.1) implies (2.2). Let \(\mathfrak{h}\in C[0,M]\). Then there exists a constant \(K>0\) such that \(|\mathfrak{h}(u)|\leq K\) for all \(u\in [ 0,M]\). Therefore
For every given \(\epsilon >0\), there exists a \(\delta >0\) such that
whenever \(|u-x|<\delta \) for all \(u\in [ 0,M]\). Define \(\psi \equiv \psi (u,x)=(u-x)^{2}\). If \(|u-x|\geq \delta \), then
From (2.3)–(2.5), we have \(|\mathfrak{h}(u)-\mathfrak{h}(x)|<\epsilon +\frac{2K}{\delta ^{2}}\psi (u,x)\), namely,
By applying the operator \(K_{n}^{(S)}(1,x)\), \(K_{n}^{(S)}(1,x)\) is a monotone and linear operator, we obtain
which implies
On the other hand
Now we estimate the following expression:
By (2.8), we obtain
Therefore,
From the above relations and the linearity of \(K_{n}^{(S)}\), we obtain
3 Rate of convergence
The modulus of continuity is defined by
Note that
Theorem 3.1
Let \(\mathfrak{T}=(\mathfrak{a}_{ij})\) be regular and \(\mathfrak{h}\in C[0,M]\). If \((\alpha _{n})\) is a sequence of positive real numbers such that \(\omega (\mathfrak{h},\delta _{n})=st_{\mathfrak{T}}-0 ( {\alpha _{n}} ) \), then
where
for any positive integer n.
Proof
By (3.1), we see
By applying the Cauchy–Schwartz inequality, we have
From Example 1.1 of [6], we obtain
By taking
we get \(\Vert K_{n}^{(S)}\mathfrak{h}-\mathfrak{h} \Vert \leq 2\cdot \omega (\mathfrak{h},\delta _{n})\). Therefore, for every \(\epsilon >0\), we have
From the conditions that are given in the theorem, we have \(\Vert K_{n}^{(S)}\mathfrak{h}-\mathfrak{h} \Vert =st_{\mathfrak{T}}-0(\alpha _{i})\), as claimed. □
Now, we obtain the rate of convergence for our method.
Theorem 3.2
Let \(\mathfrak{h}\in C[0,M]\) and let ϕ be a positive real function defined on \((0,M)\). If \(\omega (\mathfrak{h},\psi (u))=O(\phi (u))\), as \(v\rightarrow R^{-}\), then we have
where the function \(\psi:(0,M)\rightarrow \mathbb{R}\) is defined by the relation
Proof
For any \(u\in (0,R)\), \(x\in (0,M)\) and \(\delta >0\), we have
which leads to
If we set \(\delta =\psi (u)\), then from the last inequality we have
as required. □
4 Voronovskaya type theorems
It is well known that there is a Voronovskaya type theorem for the Kantorovich type generalization of Szász operators involving Sheffer type polynomials and it is stated as follows.
Theorem 4.1
([6])
For \(f\in C_{B}[0,\infty )\),
for every \(x\in [0,M]\) and any finite M.
We extend the Voronovskaya type theorem for the \(\mathfrak{T}\)-statistical method for these operators. Let us consider the following operators.
Example 4.2
Define the operators
where
Lemma 4.3
Let \(\mathfrak{h}\in C[0,M]\) such that \(\mathfrak{h}^{{\prime }},\mathfrak{h}^{{\prime \prime }}\in C[0,M]\), \(x\in [ 0,M]\). Then we obtain
Proof
It follows directly from Remark 2.6 given in [6]. □
Theorem 4.4
Let \(\mathfrak{h}\in C[0,M]\) such that \(\mathfrak{h}^{{\prime }},\mathfrak{h}^{{\prime \prime }}\in C[0,M]\), \(x\in [ 0,M]\), for any finite M. Then
on \([ 0,M]\).
Proof
Taylor’s formula gives
where \(\psi (y-x)\rightarrow 0\), as \(y-x\rightarrow 0\). After applying \(NB_{n}^{(S)}\) on both sides of Eq. (4.1), we obtain
This yields
Therefore,
where \(K=\sup_{x\in [ 0,M]}{|\mathfrak{h}(x)|}\), \(K_{1}=\sup_{x\in [ 0,M]}{|\mathfrak{h}^{{\prime }}(x)|}\) and \(K_{2}=\sup_{x\in [ 0,M]}{|\mathfrak{h}^{{\prime \prime }}(x)|}\).
Now we have to prove that
By applying the Cauchy–Schwartz inequality, we obtain
Also, by setting \(\eta _{x}(y)=(\psi (y-x))^{2}\), we have \(\eta _{x}(x)=0\) and \(\eta _{x}(\cdot )\in C[0,M]\). So
Now from the previous relation, (4.2), (4.3), and Lemma 4.3, we obtain
From the construction of \((u_{n})\), it follows that \(nu_{n}\rightarrow 0(st_{\mathfrak{T}})\) on \([0,M]\).
For a given \(\epsilon >0\), we define the sets
From these relations we obtain \(A\leq A_{1}+A_{2}+A_{3}\). Hence the result follows. □
Theorem 4.5
Let \(\mathfrak{h},\mathfrak{h}^{{\prime }},\mathfrak{h}^{{\prime \prime }}\in C[0,\infty )\). Then
as \(n\rightarrow \infty \), and for every \(x\in [ 0,M]\), for any finite M.
Proof
From Taylor’s theorem, we have
where \(R(u,x)= \frac{\mathfrak{h}^{{\prime \prime }}(\theta )-\mathfrak{h}^{{\prime \prime }}(x)}{2}(u-x)^{2}\), for \(\theta \in (u,x)\). Now we obtain
From this we get
By the properties of the continuity modulus, we have
On the other hand
For \(0<\delta <1\), we obtain
which gives
By the linearity of \(K_{n}^{(S)}\) and the above relation we obtain
Taking into consideration Remark 2.6 in [6], for every \(x\in [0,M]\), we have
For \(\delta =n^{-\frac{1}{2}}\), we complete the proof. □
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References
Ali, M., Paris, R.B.: Generalization of Szász operators involving multiple Sheffer polynomials (2020). arXiv:2006.11131v1 [math.CA]
Boos, J.: Classical and Modern Methods in Summability. Oxford University Press, Oxford (2000)
Braha, N., Mansour, T., Mursaleen, M., Acar, T.: Convergence of λ-Bernstein operators via power series summability method. J. Appl. Math. Comput. https://doi.org/10.1007/s12190-020-01384-x
Braha, N.L.: Some properties of new modified Szász–Mirakyan operators in polynomial weight spaces via power summability method. Bull. Math. Anal. Appl. 10(3), 53–65 (2018)
Braha, N.L.: Some properties of Baskakov–Schurer–Szász operators via power summability method. Quaest. Math. 42(10), 1411–1426 (2019)
Braha, N.L., Mansour, T., Mursaleen, M., Acar, T.: Convergence of λ-Bernstein operators via power series summability method. J. Appl. Math. Comput. 65(1–2), 125–146 (2021)
Braha, N.L., Mansour, T., Mursaleen, M.: Some properties of Kantorovich–Stancu type generalization of Szász operators including Brenke type polynomials via power series summability method. J. Funct. Spaces 2020, Article ID 3480607 (2020)
Duman, O., Khan, M.K., Orhan, C.: A-statistical convergence of approximating operators. Math. Inequal. Appl. 6, 689–699 (2003)
Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)
Fridy, J.A., Miller, H.I.: A matrix characterization of statistical convergence. Analysis 11, 59–66 (1991)
Gadjiev, A.D., Orhan, C.: Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 32(1), 129–138 (2002)
Kratz, W., Stadtmuller, U.: Tauberian theorems for \(J_{p}\)-summability. J. Math. Anal. Appl. 139, 362–371 (1989)
Mohiuddine, S.A., Alotaibi, A., Mursaleen, M.: Statistical summability \((C,1)\) and a Korovkin type approximation theorem. J. Inequal. Appl. 2012, 172 (2012)
Moricz, F., Orhan, C.: Tauberian conditions under which statistical convergence follows from statistical summability by weighted means. Studia Sci. Math. Hung. 41(4), 391–403 (2004)
Mursaleen, M., Alotaibi, A.: Statistical summability and approximation by de la Vallée–Poussin mean. Appl. Math. Lett. 24, 320–324 (2011). [Erratum: Appl. Math. Lett. 25, 665 (2012)]
Mursaleen, M., Alotaibi, A.: Korovkin type approximation theorem for functions of two variables through statistical A-summability. Adv. Differ. Equ. 2012, 65 (2012)
Mursaleen, M., Karakaya, V., Erturk, M., Gursoy, F.: Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl. Math. Comput. 218, 9132–9137 (2012)
Mursaleen, M., Kiliçman, A.: Korovkin second theorem via B-statistical A-summability. Abstr. Appl. Anal. 2013, Article ID 598963 (2013). https://doi.org/10.1155/2013/598963
Mursaleen, M., Mohiuddine, S.A.: Korovkin type approximation theorem for functions of two variables via statistical summability \((C,1)\). Acta Sci., Technol. 37(2), 237–243 (2015)
Soylemez, D., Unver, M.: Korovkin type theorems for Cheney–Sharma operators via summability methods. Results Math. 72(3), 1601–1612 (2017)
Stadtmuller, U., Tali, A.: On certain families of generalized Nörlund methods and power series methods. J. Math. Anal. Appl. 238, 44–66 (1999)
Tas, E.: Some results concerning Mastroianni operators by power series method. Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 63(1), 187–195 (2016)
Tas, E., Yurdakadim, T.: Approximation by positive linear operators in modular spaces by power series method. Positivity 21(4), 1293–1306 (2017)
Unver, M.: Abel transforms of positive linear operators. In: ICNAAM 2013. AIP Conference Proceedings, vol. 1558, pp. 1148–1151 (2013)
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Loku, V., Braha, N.L., Mansour, T. et al. Approximation by a power series summability method of Kantorovich type Szász operators including Sheffer polynomials. Adv Differ Equ 2021, 165 (2021). https://doi.org/10.1186/s13662-021-03326-8
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DOI: https://doi.org/10.1186/s13662-021-03326-8
MSC
- 40G10
- 41A36
Keywords
- \(\mathfrak{T}\)-statistical convergence
- Vornovskaya type theorem
- Korovkin type theorem
- Power series summability method
- Kantorovich type generalization
- Szász operators
- Sheffer type polynomials