Quantitative-Voronovskaya and Grüss-Voronovskaya type theorems for Szász-Durrmeyer type operators blended with multiple Appell polynomials

In this paper, we establish a link between the Szász-Durrmeyer type operators and multiple Appell polynomials. We study a quantitative-Voronovskaya type theorem in terms of weighted modulus of smoothness using sixth order central moment and Grüss-Voronovskaya type theorem. We also establish a local approximation theorem by means of the Steklov means in terms of the first and the second order modulus of continuity and Voronovskaya type asymtotic theorem. Further, we discuss the degree of approximation by means of the weighted spaces. Lastly, we find the rate of approximation of functions having a derivative of bounded variation.


Introduction
For f ∈ C(R +  ) and x ∈ R +  (R +  = [, ∞)), Szász [] introduced the well-known operators such that S n (|f |; x) < ∞. Several generalizations of Szász operators have been introduced in the literature and authors have studied their approximation properties. In [], the author considered Baskakov-Szász type operators and studied the rate of convergence for absolutely continuous functions having a derivative equivalent with a function of bounded variation. In [], the authors introduced the q-Baskakov-Durrmeyer type operators and studied the rate of convergence and the weighted approximation properties. In [] the authors proposed the β-operators based on q-integers and established some direct theorems by means of modulus of continuity and also studied the weighted approximation and better approximation using King type approach. For exhaustive literature on approximation by linear positive operators one can refer to [-] and the references therein. Now let us recall some results on multiple Appell polynomials []. Let g(z) = ∞ n= a n z n , g() = , be an analytic function in the disc |z| ≤ r, r >  and p k (x) be the Appell polynomials having the generating function g(u)e ux = ∞ k= p k (x)u k , with g() =  and p k (x) ≥ , ∀x ∈ R +  . Jakimovski and Leviatan [] proposed a generalization of Szász-Mirakjan operators by means of the Appell polynomials as follows: For g(u) = , these operators reduce to Szász-Mirakjan operators (). A set of polynomials {p k  ,k  (x)} ∞ k  ,k  = with degree k  + k  for k  , k  ≥  is called multiple polynomial system (multiple PS) and a multiple PS is called multiple Appell if it is generated by the relation where A is given by with A(, ) = a , = .
Theorem . For multiple PS, {p k  ,k  (x)} ∞ k  ,k  = , the following statements are equivalent: (c) For every k  + k  ≥ , we have Varma [] defined a sequence of linear positive operators for any f ∈ C(R +  ), by For α > , ρ >  and f : R +  → R, being integrable function, Pǎltǎnea [] defined a modification of the Szász operators by Motivated by [], for f ∈ C E (R +  ), the space of all continuous functions satisfying |f (t)| ≤ Ke at (t ≥ ) for some positive constant K and a, we propose an approximation method by linking the operators () and the multiple Appell polynomials by and establish a quantitative Voronovskaya type theorem, a Grüss Voronovskaya type theorem, a local approximation theorem by means of the Steklov mean, a Voronovskaya type asymptotic theorem and error estimates for several weighted spaces. Lastly, we study the rate of convergence of functions having a derivative of bounded variation.

Basic results
In order to prove the main results of the paper, we shall need the following auxiliary results.
The values of the moments K n (t i ; x) for i = , ,  are given in [] while the values of Consequently, The expression for L ρ n ((tx)  ; x) has not been included in Lemma . because it is very lengthy and complicated. It will be required to prove the quantitative Voronovskaya type theorem.
Applying the Bohman Korovkin theorem, we obtain the desired result.
For f ∈ C B (R +  ), the space of bounded and continuous functions on R +  endowed with the norm f = sup x∈R +  |f (x)|, the first and second order modulus of continuity are, respectively, defined as Further, for f ∈ C B (R +  ), the Steklov mean of second order [] is defined as Thus, it follows that where δ n,ρ (x) is defined by equation ().
Proof Using the Steklov mean f h defined by (), we may write Applying Lemma ., we have Using inequality () and equation (), we have Now by Taylor's expansion and applying the Cauchy-Schwarz inequality, we have Applying Lemma ., equations (), () and choosing h as δ n,ρ (x), we get the required result.
Proof By Taylor's expansion of f for some fixed x ∈ [, a], we obtain where ξ (t, x) ∈ C E (R +  ) and lim t→x ξ (t, x) = . Hence by linearity of the operators L ρ n , from equation (), we get Applying the Cauchy-Schwarz inequality in the last term of equation (), we have x) ∈ C E (R +  ) and hence from Theorem ., we get Hence from equation (), we obtain uniformly in x ∈ [, a]. Now taking the limit n → ∞ in () and using Remark ., we get the desired result. This completes the proof.

Weighted approximation
Let θ (x) ≥  be a weight function on R +  . We consider the weighted space defined on R +  : Further, let We define the norm in the space B θ (R +  ) as The usual modulus of continuity of the function f on [, p] is defined as Proof Let x ∈ [, c] and t > c +  then tx > . Then, for f ∈ C θ (R +  ), we have From equations () and (), for x ∈ [, c] and t ≥ , we have Applying the Cauchy-Schwarz inequality and choosing δ = √ η n,ρ , we get This completes the proof.
where η is some positive constant.
Proof Since |f (x)| ≤ f θ ( + x  ), for fixed y > , we may write Using Theorem ., for a given > , there exists k ∈ N such that Hence, Therefore, Let us choose y so large that Also, in view of Theorem ., for >  there exists a n ≥ l such that Taking m = max(k, l) and combining equations ()-(), we get This completes the proof.
Following [], the weighted modulus of continuity ω(g; δ) for g ∈ C θ (R +  ) is defined as Also, for g ∈ C * θ (R +  ), the weighted modulus of continuity has the following properties: For g ∈ C θ (R +  ), from equations () and () where C is a positive constant independent of n.
Proof By the linearity and positivity of the operators L ρ n , we get Using equation () and the Cauchy-Schwarz inequality, we get Using Lemma ., we obtain for some positive constants C  and C  dependent on ρ and A(t  , t  ). Now combining equations ()-() and taking δ =  n , we have Hence, we get . This completes the proof.

Quantitative Voronovskaya theorems
In the following result, we discuss a quantitative Voronovskaja type theorem by using the weighted modulus of smoothness ω(f ; δ). Recently, many researchers [-] have made remarkable contributions in this area.
and any x ∈ R +  , we have Proof Let x, t ∈ R +  , then, by Taylor's expansion, we have ! (tx)  and ϕ lies between t and x. Now, we get Multiplying by n on both sides of the above inequality and using Lemma ., we obtain Using the property of weighted modulus of smoothness given by (), we get Therefore, we get Now by the linearity and positivity of the operator L ρ n and using Remark ., for any x ∈ R +  , we obtain Choosing δ =  √ n , we obtain Hence combining () and (), we reach the required result.

Grüss-Voronovskaya-type theorem
For the first time Gal and Gonska [], studied the Grüss Voronovskaya type theorem for the Bernstein, Păltănea and Bernstein-Faber operators by means of the Grüss inequality which concerns the non-multiplicavity of these operators. For more papers in this direction we refer the reader to (cf. [-] etc.) Next, we study the non-multiplicativity of the positive linear operator L ρ n .
, we have the following equality: Proof We have By making an appropriate arrangement, we get Therefore, using Remark ., we get the desired result.

Rate of approximation of functions having derivative of bounded variation
In the last decade, the degree of approximation for the functions having a derivative of where g ∈ BV (R +  ), i.e., g is a function of bounded variation on every finite subinterval of R +  . In order to discuss the approximation of functions with derivatives of bounded variation, we express the operators L ρ n in an integral form as follows: where the kernel K ρ n (x, t) is given by δ(t) being the Dirac-delta function.
Lemma . For a fixed x ∈ R +  and sufficiently large n, we have Proof (i) Using Lemma ., we get The proof of (ii) is similar; hence the details are omitted.
Theorem . Let f ∈ DBV(R +  ). Then, for every x ∈ R +  and sufficiently large n, we have where b a f (x) denotes the total variation of f (x) on [a, b] and f x is defined by Collecting the estimates ()-(), we get the required result.

Conclusion
A link between Szász-Durrmeyer type operators and multiple Appell polynomials has been established. The quantitative Voronovskaya type theorem and the Grüss-Voronovskaya type theorem have been proved. A local approximation result and the weighted approximation theorem have been discussed besides the approximation of functions whose derivatives are locally of bounded variation.