Approximation Properties and -Statistical Convergence of Stancu-Type Generalized Baskakov-Szász Operators

Abstract

In this article, we introduce Stancu-type modification of generalized Baskakov-Szász operators. We obtain recurrence relations to calculate moments for these new operators. We study several approximation properties and -statistical approximation for these operators.

1. Introduction

In 1912, Bernstein [1] proposed the famous polynomial known as the Bernstein polynomial to give a simple, short, and most elegant proof of the Weierstrass approximation theorem. Since then, several papers have appeared to study approximation properties in different settings and spaces. Many new operators were constructed, e.g., Szász [2], Mirakjan [3], Kantorovic [4], Durrmeyer [5], Stancu [6], and many more [7–9]. These operators provide the improvement of approximating functions of different classes and give better and better estimates. For example, the Baskakov operators were given in [10]: For , the space of all continuous functions on normed with standard sup-norm

Devore and Lorentz [11] introduced a generalization of operators (1) dependent on a constant and independent of as follows: where and

Recently, Agrawal et al. [12] studied the following operators (2): for , for some , where

Inspired by Stancu’s work [6], we have studied recently the Stancu-type generalization in [13]. Now, we propose the Stancu-type generalization of operators (4) as follows: for any bounded and integrable function defined on , where . For , the operators (5) reduce to operators (4).

We establish recurrence relations to find moments and central moments. We study some approximation properties and the Voronovskaja-type asymptotic formula. We also study weighted approximation.

2. Auxiliary Results

Our first result is the recurrence formula for moments.

Theorem 1. The order moment for (5) is defined by . Then, and

Proof. We use the identity Then, where where where Therefore, Substituting (17), (14), and (13) in (12), we get Further, substituting (18) in (10), we get the result.

Corollary 2. From the above theorem, we get (i)(ii)(iii)

Theorem 3. The order central moment is defined by . The following recurrence relation holds:

Corollary 4. From the above theorem, we get (a)(b)

Corollary 5. We further get (a)(b)

3. Main Results

Peetre’s -functional is defined as for , where is bounded on and Note that where is the second-order modulus of continuity [11].

The usual modulus of continuity of is defined as

Theorem 6. For , where and

Proof. Put Note that and Let Then, by using Taylor’s theorem, we may write which gives Hence, Since and we have Now, by Corollary 4 (b), we get By (27), we get Since we get From (33), we get Now, taking the infimum over all , we obtain Hence, by using (22), we get the result.

For our next result, we consider the functions belonging to the Lipschitz class:

where and ;

Theorem 7. For , we have where .

Proof. First, we prove for . For , we get Since , we get by the Cauchy-Schwarz inequality: For , applying Hölder’s inequality, we get Since , we have Therefore, we get (39).

Next, we obtain a Voronovskaja-type asymptotic formula.

Theorem 8. If exists at a point for , then