Research Article | Open Access

Md. Nasiruzzaman, A. F. Aljohani, "Approximation by Szász-Jakimovski-Leviatan-Type Operators via Aid of Appell Polynomials", *Journal of Function Spaces*, vol. 2020, Article ID 9657489, 11 pages, 2020. https://doi.org/10.1155/2020/9657489

# Approximation by Szász-Jakimovski-Leviatan-Type Operators via Aid of Appell Polynomials

**Academic Editor:**Syed Abdul Mohiuddine

#### Abstract

The main purpose of the present article is to construct a newly Szász-Jakimovski-Leviatan-type positive linear operators in the Dunkl analogue by the aid of Appell polynomials. In order to investigate the approximation properties of these operators, first we estimate the moments and obtain the basic results. Further, we study the approximation by the use of modulus of continuity in the spaces of the Lipschitz functions, Peetres K-functional, and weighted modulus of continuity. Moreover, we study -statistical convergence of operators and approximation properties of the bivariate case.

#### 1. Introduction

In 1969, Jakimovski and Leviatan introduced a sequence of positive linear operators [1], by using Appell polynomials [2] and defined as where , , and . For all and , the positive linear operators are defined on given by Wood in [3]. If we take , then an analogue of Szász operators was proved by Jakimovski and Leviatan, where denotes the set of functions on } such that , where , are positive constants. They established is uniformly on each compact subset of (see [1, 4]). Precisely, for in (1), the well-known classical Szász operators [5] were obtained defined in 1950 such that

Recently, Szász-Mirakyan operators have been obtained by researchers via the Dunkl generalization in approximation process; for instance, we refer the readers to [6–12]. For more details, related results relevant to the present article in different functional spaces are seen in [13–19] and [20–23]. Sucu [24] introduced Szász-Mirakyan operators by using the new exponential function given in [25] as

For a recursion of is given as

These types of generalizations gave rise to exponential function and generalization of Hermite-type polynomials, expressed in the form of the confluent hypergeometric function (see [25]).

#### 2. Construction of Operators and Estimation of Moments

For every , and all , , , , , we define

Lemma 1. *For all , , , and , if we define
**Then for all , we have
*

Lemma 2. *Let , and take for .**Then, for operators by (5), we have the following estimates:
*

*Proof. *(1)Take , then(2)For (3)For Similarly, we can prove easily (4) and (5).

Lemma 3. *Let be the central moments, then
*

#### 3. Global Approximation

In the present section, we follow Gadžiev [11] and recall the weighted spaces of the functions on , as well as additional conditions under which the analogous theorem of P.P. Korovkin holds for such a kind of functions. Take be continuous and strictly increasing function with and . Let be a set of functions defined on [), verifying the results where is a constant and depending only on function and is space of all continuous as well as bounded functions on . Let the set of all continuous functions on will be denoted by and equipped with the norm .

Let us denote

It is well known that (see [26]) the sequence of linear positive operators maps into if and only if where is a positive constant.

*Definition 4. *For all , the modulus of continuity for a uniformly continuous function defined by
For every and uniformly continuous function , we suppose

Theorem 5. *For all , operators defined in (5) satisfy on each compact subset of , with stands for uniform convergence.*

*Proof. *From the well-known Korovkin’s theorem (see [27]), for all , it is sufficient to see that
In the view of Lemma 2, it is obvious that as , , which completes Theorem 5.

Theorem 6. *Let . Then for every , we have
*

*Proof. *We prove this theorem by applying Korovkin’s theorem so it is sufficient to show that
From Lemma 2, we easily see that
Similarly, for
which imply that as . For which clearly shows that , whenever

Theorem 7. *For all operators given by (5) satisfy
where and stand for space of all continuous and bounded functions defined on .*

*Proof. *We prove Theorem 7 by using the well-known Cauchy-Schwarz inequality and modulus of continuity. Thus, we see that
If we take , we get the required result asserted by Theorem 7.

#### 4. Some Direct Results of

The present section gives some direct approximation results in the space of -functional and in the Lipschitz spaces. We suppose the following.

*Definition 8. *For every and , we define
where is defined by
Now, there exists an absolute constant such that
where is the second-order modulus of continuity given by
Moreover, the modulus of continuity of order one is

Theorem 9. *Let we define an auxiliary operators such that
**Then, for every , operators satisfy
where and are defined in Theorem 7.*

*Proof. *Take ; then, we easily conclude that and
We also know easily
Therefore,
From the Taylor series we see
Applying , we have
Since we know
Therefore, we get
This gives the complete proof.

Theorem 10. *Let and any . Then, there exists a constant such that
where is defined by Theorem 9.*

*Proof. *We prove the result asserted by Theorem 10 in the light of Theorem 9. Therefore, for all and , we get
Taking infimum over all and using (26), we get
Here, we obtain some local approximation results of in the Lipschitz spaces. For all the Lipschitz maximal function and , we recall that

Theorem 11. *Let , then for all , operators satisfy
where is the Lipschitz maximal function defined by (43) and by Theorem 7.*

*Proof. *To prove Theorem 11, we use the well-known Hölder inequality by applying (43)
The proof is complete.

From [28] for an arbitrary , the weighted modulus of continuity is introduced such that

The two main properties of this modulus of continuity are lim_{δ→0} and
where .

Theorem 12. *Let the operators be defined by (5); then for every , there exists a constant such that
where and with .*

*Proof. *In light of (46), (47), and Cauchy-Schwarz inequality, we prove this theorem. Thus, we see
From Lemma 3, we easily conclude that for any positive and Therefore,
Hence, in light of (49), (50), (51), (52), (53) and (54) and choosing , if we take the supremum , we get the result.

#### 5. -Statistical Convergence

Here, we obtain the -statistical convergence for the operators by (5). From [29], we recall the needed notations and notions for -statistical convergence. Take be a nonnegative infinite summability matrix. For a given sequence , the -transform of is denoted by where the series converges for each and defined by

The matrix is said to be regular if whenever and are said to be a -statistically convergent to , i.e., if for each . For the recent work on statistical convergence and statistical approximation, we refer to [30–37].

Theorem 13. *Let operators be defined by 1 and a nonnegative regular summability matrix be ; then, for every *

*Proof. *It is enough to show that
From Lemma 2, we conclude that
which implies that