A Parametric Generalization of the Baskakov-Schurer-Szász-Stancu Approximation Operators

In this paper, we introduce and investigate a new class of the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators, which considerably extends the well-known class of the classical Baskakov-Schurer-Szász-Stancu approximation operators. For this new class of approximation operators, we present a Korovkin type theorem and a Grüss-Voronovskaya type theorem, and also study the rate of its convergence. Moreover, we derive several results which are related to the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators in the weighted spaces. Finally, we prove some shape-preserving properties for the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators and, as a special case, we deduce the corresponding shape-preserving properties for the classical Baskakov-Schurer-Szász-Stancu approximation operators.


Introduction
One of the most powerful theorems in the approximation theory is known as the Weierstrass Approximation Theorem, which states that any continuous function f (x) defined on the closed interval [a, b] can be approximated by an algebraic polynomial P(x) with real coefficients for each x ∈ [a, b].
The idea of finding concrete algebraic functions for better approximation has been studied extensively, and a number of polynomial operators have been used directly. The first results are given for the Bernstein operators, which were generalized by Szász [1] as follows: for x ∈ [0, ∞). Baskakov [2] defined the following sequence of linear operators: L s ( f , x) = 1 (1 + x) s ∞ ∑ r=0 s + r − 1 r x r (1 + x) r f x s for s ∈ N and x ∈ [0, ∞), N being the set of positive integers. Subsequently, Schurer [3] generalized the Bernstein operators in the following form: Stancu [4] defined the following sequence of operators: for 0 α β. More recently, the following form of the Baskakov-Schurer-Szász-Stancu operators was introduced by Sofyalioglu and Kanat [5]: where s is a positive integer, p is a non-negative integer, and 0 α β.

Remark 1.
It is clearly seen that M 1,α,β The aims of this paper are to first study the Korovkin type theorem, the Grüss-Voronovskaya type theorem, and the rate of the convergence for the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators. We then present some results related to the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators in the weighted spaces. Finally, in the last section, we give some preserving properties of the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators such as convexity.

Preliminary Results
By simple applications of the principle of mathematical induction, one can obtain Lemmas 1 and 2 below: Proof. We proceed to the proof by the principle of mathematical induction on N given by as claimed in Lemma 1. We now assume that the claimed result holds true for some N given by We then prove the claimed result for Indeed, by partial integration, we have Thus, by the induction hypothesis, we have which shows that the claimed result also holds true for N + 1 = + k + 1. This evidently completes our proof of Lemma 1 by the principle of mathematical induction.

Lemma 2.
For all m 0, where f m , g m and h m are defined as follows: and they satisfy the following recurrence relations: Proof. By using similar arguments as in the proof of Lemma 1, we can establish the result asserted by Lemma 2. We choose to skip the details involved.
By means of Lemmas 1 and 2, and, by using the principle of mathematical induction on m, we are led to the following result.
Furthermore, for all m 0, Thus, by Lemma 1, we obtain which, by Lemma 2, implies that Hence, by applying the above Proposition, we can prove the following result.
where e i = x i . Then, for every f ∈ C[0, R], By means of the basic form of the Korovkin type theorem (see, for example, Ref. [25]), we complete the proof of Theorem 2.

Direct Estimates
With B[0, ∞), C[0, ∞), and C B ([0, ∞)), we will denote the space of all bounded functions, the space of all continuous functions, and the space of all continuous and bounded functions defined in the interval [0, ∞), respectively, endowed with the norm given by The modulus of continuity of the function f ∈ C[0, ∞) is defined by It is known that, for any value of the |x − y|, we have Then, the following inequality for the operators (1) holds true: Proof. We know that operators M γ,α,β n,p are linear and positive. Let f ∈ C B [0, ∞). In view of the modulus of continuity, we have Let us set Then, by the Cauchy-Schwarz inequality, we get By direct calculations, we see that and also that These last three equalities lead us to the following consequence: Hence, in view of the positivity of b k,α n,p (x), if we use the following expression: From (6) and (5), we find that Putting δ = √ n, we get the result asserted by Theorem 3.
In what follows, we will give an upper bound for the sequence of the parametric generalization of the Baskakov-Schurer-Szász operators.
Proof. From the definition of the parametric generalization of the Baskakov-Schurer-Szász- as asserted by Theorem 4.
For f ∈ C[0, ∞) and δ > 0, the second-order modulus of smoothness of f is defined as follows: where δ > 0 and It is known that there exists a positive constant C > 0 such that (see [26] (Theorem 3.1.2)), Theorem 5. Let f ∈ C[0, A] for any finite real number A. Then, Proof. Let f S be the Steklov function of the second order for the function Now, from the Lemmas in [27], we find that Knowing that f S ∈ C 2 [0, A], and from the Lemmas in [17], we obtain The following inequality is valid (see [27]): In light of (8) and (9), (7) takes the following form: From the relation (9) and the Landau inequality (see [28]), we get Using relations (9) and (10), and upon setting Now, from relation (8), we complete the proof of Theorem 5. Let and the Peetre's K-functional given by (see [29]) }.
Theorem 6. Let f ∈ C B [0, ∞). Then, the following inequality holds true: Proof. By using the Taylor formula and the linearity of the operators M γ,α,β where ϕ ∈ (x, t). In addition, from the above Example, we have which proves Theorem 6.
where M is a positive constant and A(n, p, α, β, γ, x) is defined as in Theorem 6.
Proof. From the linearity of the operator M γ,α,β n,p ( f ; x) and the following relation: Now, from Theorems 4 and 6, and, by considering that g ∈ C 2 B , we get It is known that where C is a positive constant, holds true for every δ > 0 (see [26]). From the last two relations, we get the result asserted by Theorem 7.
We will give the Voronovskaya type theorem for the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators.
Theorem 8. For f ∈ C B [0, ∞), the following limit relation: holds true for every x ∈ [0, M] and any finite M.
Proof. By Taylor's expansion theorem of the function f in C B [0, ∞), we obtain: where and the function ψ x (·) is the Peano form of the remainder, ψ x (·) ∈ C B [0, ∞) and ψ x (t) → 0 as t → x. Applying the operator M γ,α,β n,p on both sides of the above relation, we find that In addition, from the above Example, we get which, after applying the Cauchy-Schwarz inequality, yields We now observe that ψ 2 This completes the proof of Theorem 8.

Proof. After some calculations, we obtain
The proof of Theorem 9 now follows from Theorem 8 and the above Example.
The following results give light to the speed of the change between the difference of the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators and their derivatives, measured in terms of the modulus of continuity.
for every x ∈ [0, M] for any finite M.

Proof. From the Taylor's theorem, we have
for θ ∈ (u, x). We thus find that From this last relation, we get By the properties of the modulus of continuity, we have On the other hand, it is easily seen that For 0 < δ < 1, we obtain that By the linearity of M γ,α,β n,p and the above relation, we obtain Now, in view of the above Example, for every x ∈ [0, M], we have Thus, for we complete the proof of Theorem 10.
The next result gives an estimation of the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators in the special Lipschitz-type space Lip * M α( [1]), defined as follows: where M is a positive constant and α ∈ (0, 1].

I.
For for a positive constant M. If we apply the Cauchy-Schwarz inequality in the last expression, we get If we apply the Hölder inequality on the last relation under the conditions that for a positive constant M. Thus, after applying the Cauchy-Schwarz inequality, obtain the following estimate: which completes the proof of Theorem 11.
The Ditzian-Totik uniform modulus of smoothness of the first and the second orders are defined as follows (see [26]): respectively, where φ is an admissible step-weight function on [a, b], that is, . The corresponding K-functional is defined as follows: where δ > 0, means that g is absolutely continuous on [0, ∞). It is known that there exists an absolute constant C > 0 such that (see [26]) ) be a step-weight function of the Ditzian-Totik modulus of smoothness. Then, for any f ∈ C B [0, 1] and x ∈ [0, 1], n ∈ N and 2γ < β + 2, We then observe that Let g ∈ W 2 (φ). Then, by using Taylor's expansion, we write Therefore, we have M γ,α,β, * n,p · x+β 1 (n,p,α,β,x)

Weighted Approximation
Let ρ(x) = x 2 + 1 be the weight function and let M f be a positive constant. We define the weighted space of functions as follows: We note that the space B ρ [0, ∞) is a normed linear space with the norm given by .
In order to calculate the rate of convergence, we consider the weighted modulus of continuity Ω( f ; δ) defined on infinite interval [0, ∞) as For any µ ∈ [0, ∞), the weighted modulus of continuity Ω( f ; δ) verifies the following inequality: and, for every Using Theorem 1, the result for i = 0 is trivial. We now prove that the results are true for i = 1 and i = 2, respectively. Indeed, for By a similar consideration, we have We thus conclude that which completes the proof of Theorem 13.

Theorem 14.
Let f ∈ C * ρ [0, ∞). Then, the following inequality holds true: for a sufficiently large n, where C, D, E and F are positive constants dependent only on n, p, α, β and γ, and K is a positive constant.
Using the properties of the weighted modulus, we obtain Let us define Since s k n,p (t) > 0 for every t ∈ (0, ∞), we have which implies that Thus, clearly, we get Thus, by using the above Proposition, we have From the above relation, we obtain |M γ,α,β In addition, for δ n = n − 1 4 , we have where C, D, E , and F are positive constants depending only on n, p, α, β, and γ, and K is a positive constant. This proves Theorem 14.
Proof. Let us suppose that f (x) is convex and that x 0 and x 1 are distinct points in the interval [x, y], where x < x 0 < x 1 < y and x, y ∈ [a, b] ⊂ [0, ∞). Then, the Lagrangian interpolation polynomial through the points x 0 , f (x 0 ) and x 1 , f (x 1 ) is given by Then, based upon Theorem 1, we have On the other hand, we have From this last relation, we find that M γ,α,β n,p ( f ; x) = f (ξ t ) · (n + p)(n + 4γ + p − 3) (n + p + β) 2 > 0 under the given conditions. This completes the proof of Theorem 15.
Proof. We know that, for γ = 1, we are led to the Baskakov-Schurer-Szász-Stancu operators from the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators. Since n, p ∈ N, in the special case when γ = 1, we have n + p + 4γ > 3. The proof now follows from Theorem 15.

Concluding Remarks and Observations
In our present investigation, we have introduced, and systematically studied the properties and relations associated with, a new class of the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators. Our findings have considerably and significantly extended the well-known family of the classical Baskakov-Schurer-Szász-Stancu approximation operators. For our new class of the Baskakov-Schurer-Szász-Stancu approximation operators, we have established a Korovkin type theorem and a Grüss-Voronovskaya type theorem. We have also studied the rate of its convergence. Moreover, we have proved several results which are related to the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators in the weighted spaces. Finally, we have derived a number of shape-preserving properties for the parametric generalization of the Baskakov-Schurer-Szász-Stancu approximation operators. We have also appropriately specialized our results in order to deduce the corresponding shape-preserving properties for the classical Baskakov-Schurer-Szász-Stancu approximation operators.
The various results and their consequences, which we have presented in this article, will potentially motivate and encourage further researches on the subject dealing with the parametric generalization of the Baskakov-Schurer-Szász-Stancu approximation operators.