Approximation Properties of Modified Srivastava-Gupta Operators Based on Certain Parameter

In the present article, we give a modified form of generalized SrivastavaGupta operators based on certain parameter which preserve the constant as well as linear functions. First, we estimate moments of the operators and then prove Voronovskaja type theorem. Next, direct approximation theorem, rate of convergence and weighted approximation by these operators in terms of modulus of continuity are studied. Then, we obtain point-wise estimate using the Lipschitz type maximal function. Finaly, we study the A-statistical convergence of these operators.


Introduction
The approximation of functions by linear positive operators is an important research topic in general mathematics and it also provides powerful tools to application areas such as computer-aided geometric design, numerical analysis, and solutions of differential equations.In order to approximate Lebesgue integrable functions on [0, ∞), Srivastava and Gupta [31] introduced a general family of summation-integral type operators which includes some well-known operators as special cases.They obtained the rate of convergence for functions of bounded variation.After that several researchers studied different approximation properties of these operators (see [1], [3], [12], [14], [22], [34], [35]).Verma [33] define the following generalization of Srivastava-Gupta operators based on certain parameter ρ > 0 as: where For the properties of φ n,c (x), we refer the readers to [31].For ρ = 1 the operators (1.1) reduced to the Srivastava-Gupta operators [31].In [33], Verma studied some results in simultaneous approximation by the operators L n,ρ .It is observed that the operators (1.1) reproduce only constant functions.So here we modify the operators (1.1) so that they may be capable to reproduce constant as well as linear function.King [20] gave an approach for modification of the classical Bernstein polynomials and he achieved better approximation.Here we give some alternate approach and we propose the modification of the operators (1.1) as follows: In the present paper, we study the basic convergence theorem, Voronovskaja type asymptotic formula, local approximation, rate of convergence, weighted approximation, pointwise estimation and A-statistical convergence of the operators (1.3).

Preliminaries
In this section we collect some results about the operators L * n,ρ useful in the sequel.
Proof: For x ∈ [0, ∞), in view of Lemma 2.1, we have Next, for f (t) = t, we get Proceeding similarly, we have ✷ Remark 2.3.For every x ∈ [0, ∞) and nρ > 2c, we have Lemma 2.4.For f ∈ C B [0, ∞) (space of all real valued bounded and uniformly continuous functions on [0, ∞) endowed with norm Proof: In view of (1.3) and Lemma 2.2, the proof of this lemma easily follows.✷ For C B [0, ∞), let us define the following Peetre's K-functional: where δ > 0 and By, p. 177, Theorem 2.4 in [4], there exists an absolute constant M > 0 such that where is the second order modulus of smoothness of f .By we denote the usual modulus of continuity of f ∈ C B [0, ∞).

Main results
In this section we establish some approximation properties in several settings.
Proof: Using Taylor's theorem, we have where r(t, x) is the remainder term and lim t→x r(t, x) = 0.
Applying L * n,ρ (f, x) to (3.1), we get In view of Remark 2.3, we have Applying the Cauchy-Schwarz inequality, we have We observe that r 2 (x, x) = 0 and r 2 (., Proof: Let g ∈ W 2 and x, t ∈ [0, ∞).Using Taylor's series, we have Applying L * n,ρ on both sides and using Lemma 2.2, we get Obviously, we have Finally, taking the infimum over all g ∈ W 2 and using (2.1) we obtain We observe that for a function f ∈ C B [0, ∞), the modulus of continuity ω b (f, δ) tends to zero.
Now, we give a rate of convergence theorem for the operators L * n,ρ .Theorem 3.4.Let f ∈ C B [0, ∞) and ω b+1 (f, δ) be its modulus of continuity on the finite interval [0, b + 1] ⊂ [0, ∞), where b > 0.Then, we have where ξ n,ρ (b) is defined in Remark 2.3 and M f is a constant depending only on f .
From the above, we have for x ∈ [0, b] and t ≥ 0. Applying Cauchy-Schwarz inequality, we have on choosing δ = ξ n,ρ (b).This completes the proof of the theorem.✷ Next, we obtain the Korovkin type weighted approximation by the operators defined in (1.3).The weighted Korovkin-type theorems were proved by Gadzhiev [5].
Let B ν (R) denote the weighted space of real-valued functions f defined on R with the property |f (x)| ≤ M f ν(x) for all x ∈ R, where M f is a constant depending on the function f .We also consider the weighted subspace exists finitely.
Theorem 3.6.For each f ∈ C * ν [0, ∞), we have Proof: From [5], we know that it is sufficient to verify the following three conditions lim Since L * n,ρ (1; x) = 1, the condition in (3.7) holds for k = 0.By Lemma 2.2, we have which implies that the condition in (3.7) holds for k = 1.Similarly, we can write for nρ > 2c which implies that lim n→∞ L * n,ρ (t 2 ; x)−x 2 ν = 0, the equation (3.7) holds for k = 2.This completes the proof of theorem.✷ Now we give the following theorem to approximate all functions in C * ν .Such type of results are given in [6] for locally integrable functions.Theorem 3.7.For each f ∈ C * ν and α > 0, we have Proof: For any fixed The first term of the above inequality tends to zero from Theorem 3.4.By Lemma 2.2, for any fixed x 0 > 0, it is easily prove that as n → ∞.We can choose x 0 > 0 so large that the last part of the above inequality can be small.Hence the proof is completed.
where M is a constant depending only on η and f .Now, we obtain some pointwise estimates of the operators L * n,ρ .
Then, we have where M is a constant depending on η and f and d(x, E) is the distance between x and E defined as Proof: Let E be the closure of E in [0, ∞).Then, there exists at least one point By our hypothesis and the monotonicity of L * n,ρ , we get Now, applying Hölder's inequality with p = 2 η and q = 2 2 − η , we obtain from which the desired result immediate.✷ Next, we obtain a local direct estimate of the operators defined in (1.3), using the Lipschitz-type maximal function of order η introduced by B. Lenze [21] as Proof: From the equation (3.8), we have Now, using the Hölder's inequality with p = 2 η and 1 q = 1 − 1 p , we obtain Thus, the proof is completed.✷ For a, b > 0, Özarslan and Aktuglu [30] consider the Lipschitz-type space with two parameters: where M is any positive constant and 0 < η ≤ 1.
Proof: First we prove the theorem for η = 1.Then, for f ∈ Lip This completes the proof of the theorem.✷