Szász-Durrmeyer operators involving Boas-Buck polynomials of blending type

The present paper introduces the Szász-Durrmeyer type operators based on Boas-Buck type polynomials which include Brenke type polynomials, Sheffer polynomials and Appell polynomials considered by Sucu et al. (Abstr. Appl. Anal. 2012:680340, 2012). We establish the moments of the operator and a Voronvskaja type asymptotic theorem and then proceed to studying the convergence of the operators with the help of Lipschitz type space and weighted modulus of continuity. Next, we obtain a direct approximation theorem with the aid of unified Ditzian-Totik modulus of smoothness. Furthermore, we study the approximation of functions whose derivatives are locally of bounded variation.


In [], Sucu et al. introduced the Szász operators involving
Boas-Buck type polynomials as follows: where a generating function of the Boas-Buck type polynomials is given by and A(t), G(t) and H(t) are analytic functions described as Motivated by the above work, in the present paper we define Szász-Durrmeyer type operators based on Boas-Buck type polynomials as follows.
where B(k, n + ) is the beta function and x ≥ , n ∈ N. Alternatively, we may write operator (.) as and δ(t) is the Dirac-delta function.
We study the approximation properties of the operators M n for functions belonging to different function spaces.

Lemma  ([]) For the operators B n , one has
Proof Since identities (i)-(iii) are proved in [], we give below the proof of only (iv). Identity (v) follows similarly. It is easily seen that Now, by simple calculations, we obtain identities (iii) and (iv). Hence the details are omitted.
In the following lemma, we obtain the moments for the operators defined by (.) utilizing Lemma .

Lemma 
Hence, as a consequence of Lemma , we find the following.
Lemma  For operator (.), we have the following results: Now, in order to study the approximation properties of the considered operators (.), we make the following assumptions on the analytic functions A(t), H(t) and G(t). It is to be noted that the following assumptions are valid pointwise. These assumptions will be needed to prove Theorems ,  and  of this paper which are pointwise results.
As a result of the above assumptions, applying Lemma , we reach the following important result.

Results and discussion
Throughout the paper, we assume In the following theorem, we show that the operators defined by (.) are an approximation process for f ∈ C γ [, ∞), using the Bohman-Korovkin theorem.
Theorem  Let f ∈ C γ [, ∞). Then where M f is a constant which depends on f .
In the following theorem, we find the rate of convergence of the operators M n for functions in Lip * M ξ . We observe that due to the presence of x, in the denominator on the righthand side, we get only pointwise approximation. In the case of Szász operators [], this x gets canceled leading to the uniform convergence.
Theorem  Let f ∈ Lip * M (ξ ) and ξ ∈ (, ]. Then, for all x ∈ (, ∞), we have Proof By the linearity and positivity of the operators M n , from (.) we obtain Applying Hölder's inequality with p =  ξ and q =  -ξ and Lemma , we have Thus, we reach the desired result.
In our next result, we establish a Voronovskaja type approximation theorem.
Theorem  Let f ∈ C γ [, ∞), admitting a derivative of second order at a point x ∈ [, ∞), then there holds

If f is continuous on [, ∞), then the limit in (.) holds uniformly in x ∈ [, a] ⊂ [, ∞), a > .
Proof By Taylor's theorem Applying the operator M n (·, x) on both sides of (.), we have Using the Cauchy-Schwarz inequality in the last term of the right-hand side of (.), we get Since ε(t, x) → , as t → x, applying Theorem , for every x ∈ [, ∞), we obtain lim n→∞ M n (ε  (t, x); x) = ε  (x, x) = . Next applying Lemma , for sufficiently large n and every x ∈ [, ∞), we have In our next theorem, we obtain the degree of approximation of the M n operators for functions in the space C  [, ∞) in terms of the classical modulus of continuity.
Theorem  For f ∈ C  [, ∞), we have the following inequality: where ω(f ; δ n (x)) is the modulus of continuity of f on [, b + ].
The next section is devoted to the weighted approximation properties of the operators M n .

Weighted approximation
Next, we study the approximation of functions in the subspace

Such type of function spaces has been considered by several researchers (cf. [, ]).
It is well known that the classical modulus of continuity of first order ω(f ; δ), δ >  does not tend to zero, as δ → , on an infinite interval. A weighted modulus of continuity (f ; δ) which tends to zero as δ →  on [, ∞) was defined in []. For f ∈ C   [, ∞), the weighted modulus of continuity defined by Yüksel and Ispir [] is given as follows: Some properties of (f ; δ) are collected in the following lemma.
. Then the following results hold: () (f ; δ) is a monotonically increasing function of δ; Firstly, we establish the following basic approximation theorem for functions in the weighted space of continuous functions C   [, ∞) by the operators M n .
Theorem  For f ∈ C   [, ∞) and a > , we have Proof Let x  ∈ [, ∞) be an arbitrary but fixed point. Then Let >  be arbitrary. We choose x  to be so large that Hence, Applying Theorem , we can find n  ∈ N such that Let n  = max(n  , n  ). Combining (.)-(.), we obtain Hence the required result is obtained.
In our next theorem, we determine the order of approximation for functions in a weighted space of continuous functions on [, ∞) by M n operators.
Theorem  Let f ∈ C   [, ∞). Then, for sufficiently large n, we have

constants independent of x and n and η(x), ν(x) are as given in Lemma .
Proof For x ∈ (, ∞) and δ > , using (.) and Lemma , we have Applying M n (·; x) on both sides, we can write From Lemma , for sufficiently large n, it follows Now, applying the Cauchy-Schwarz inequality in the last term of (.), we obtain Combining the estimates (.)-(.) and taking we reach the required result.

Unified modulus theorem
We investigate a direct approximation theorem by utilizing the unified Ditzian-Totik modulus of smoothness ω φ τ (f , t),  ≤ τ ≤ . Guo and the appropriate K -functional is given by where C is independent of f and n.
Proof By the definition of K φ τ (f , t), for fixed n, x, τ , we can choose g = g n,x,τ ∈ W τ such that We may write Since g ∈ W τ , we have By applying Hölder's inequality, we get Hence, on using the inequality |a + b| r ≤ |a| r + |b| r ,  ≤ r ≤ .
Proof Using Lemma  and (.), we have when n is large enough. Similarly, we can prove (ii).
Theorem  Let f ∈ DBV [, ∞). Then, for every x ∈ (, ∞) and sufficiently large n, we have where C  is a positive constant and b a f denotes the total variation of f on [a, b] and f x is defined by (.)

Conclusion
We introduce Szász-Durremeyer type operators involving Boas-Buck type polynomials. Brenke type polynomials, Sheffer polynomials and Appell polynomials turn out to be the special cases of Boas-Buck type polynomials. We obtain the rate of convergence for functions belonging to a Lipschitz type space and also establish a Voronovskaja type theorem for twice continuously differentiable functions. We study the approximation properties of the considered operators for continuous functions in weighted spaces. Lastly, we discuss the rate of approximation of functions having derivatives of bounded variations.