Semi Post–Widder Operators and Difference Estimates

We consider the Post–Widder operators of semi-exponential type, which are a generalization of the exponential operators connected with x2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^2$$\end{document}. This modification has the beauty to find difference with other operators, while the original Post–Widder operators do not have such property. We estimate quantitative difference of these operators with Baskakov type operators and Szász–Kantorovich operators, along with some composition of operators. Finally, we further consider a form preserving linear functions and estimate some direct results.

Alternatively, (1.1) can be written as where the kernel e −λt/x t λ+k−1 satisfies the partial differential equation (1.2) which is the required condition for P β λ to be of semi-exponential type operator. Also, for specific value β = 0, we get the Post-Widder operators [12, (3.9)] defined by Abel et al. [2] and Gupta and Milovanović [9] introduced all remaining semiexponential operators from available exponential-type operators. In a very recent paper [6], some more general form of exponential-type operators was introduced and discussed. Also, we refer the readers to the recent related work [10,13].
In this paper, we shall investigate the difference between two operators which is an active area of research in the recent years. For example, in papers [3,7,8], the differences amongst operators having the same/different basis under summation are estimated. It is pointed out here that in case of the Post-Widder operators, the difference with other operators is not analogous due to the purely integral term of the operators, but for the semi-exponential Post-Widder operators, one can find the difference with other operators. We also consider composition of such operators with some other operators to capture some other operators. Also, some modification of P β λ is proposed here which preserves the linear functions.

Difference and Composition
In this section, we deal with the difference between P β λ to the general Baskakov operators and some Kantorovich variants. We shall apply some estimates from paper [7] and [8]. Also, we indicate some composition estimates.
First, we write the operators (1.1) in an alternative form as

Remark 2.1 By simple computation, we have
In particular Also, using above, we have The general Baskakov type operators for x ∈ I are defined as In particular if c = 1, φ 1,λ (x) = (1 + x) −λ in this case, we get the Baskakov operators, and if c = 0, φ 0,λ (x) = e −λx is a limit for c → 0 + and we obtain the Szász-Mirakyan operators.

Difference with Discrete Operator
In the following two theorems, we find the difference between semi-exponential Post-Widder operators and the generalized Baskakov type operators.
for c = 0 and c = 1, we obtain the difference of semi Post-Widder operators with Szász-Mirakyan and Baskakov operators, respectively.
Proof Applying Remarks 2.1 and 2.2 to the aforementioned theorem, we have This completes the proof.

Theorem 2.5 If D(I ) be the set of all functions in C(I ) for which the two operators P
for c = 0 and c = 1, we obtain the difference of semi-exponential Post-Widder operators with the Szász-Mirakyan and the Baskakov operators, respectively.
Proof Following [8, Theorem 2] and using Remarks 2.1 and 2.2, we have The estimates of δ 1 and δ 2 can be obtained as in Theorem 2.4. Collecting the above estimates, the result follows.

Difference with Integral Operator
In the following two theorems, we provide the difference of semi-exponential Post-Widder operators with the generalized Szász-Kantorovich operators.

Theorem 2.6 If D(I ) be the set of all functions in C(I ) for which the two operators P
Proof Applying Remarks 2.3 and 2.1 to the aforementioned theorem, we have This completes the proof.

Theorem 2.7 If D(I ) be the set of all functions in C(I ) for which the two operators P
Proof Following [8, Theorem 2] and using Remarks 2.1 and 2.2, we have: Next The estimates of δ 1 and δ 2 can be obtained as in Theorem 2.6. Collecting the above estimates, the result follows.

Proposition 2.8 Composition of semi-exponential Post-Widder and the Szász-Mirakyan operators provides the new operator
which may be considered as representation of semi-exponential Baskakov operator, slightly different from [2]. If β = 0, then m = s and we get the Baskakov operators.
Proof By definition This completes the proof.

Proposition 2.9 The composition of Post-Widder operators and the Szász-Kantorovich operators provides the Baskakov-Kantorovich operators (see [1])
Proof We can write This completes the proof of the proposition.

Modified Semi-exponential Post-Widder Operators
Let us consider the following modified form of the semi-exponential Post-Widder operators: Also, we can calculate We observe that the operators ( P ) preserve constants and linear functions, but we do not capture the exact Post-Widder operators (1.3). Also, these operators are neither exponential nor semi-exponential type operators as, for these operators, condition (1.2) is not satisfied for β ≥ 0.

Lemma 3.1
The moment producing function of ( P β λ f )(x) for A in some neighborhood of zero is given as In particular with e s (x) = x s , we have the representation Proof By the definition of the operators P β λ , we have Using the following connection between moments and moment generating function: we may get the moments by simple computation.

Weighted Convergence
According to [5], we consider the following spaces: where the constant M f depends on f , and exists and it is finite .
If the space B e 2 (I ) is yielded with the norm . e 2 defined by then the same norm is considered in both of the spaces defined above. The aim of the section is to achieve approximating theorems including Voronovskaya-type result in the aforementioned spaces. Now, we apply the theorem to our operators.

Theorem 3.3
Let f ∈ C * e 2 (I ), then the following holds true: Proof As we mentioned above, we shall examine the assumptions of Theorem A, applying them to the operators P β λ . According to Lemma 3.1, for the operators P β λ on C e 2 (I ), the result holds true for v = 0, 1. Next, for v = 2, we obtain We have to prove that the above expression tends to zero as λ → ∞.
First, we notice that for z ∈ I and λ > 0, we have (z + λ) 2 ≥ (2z + λ)λ. Now, we substitute z = 2βx which is no-negative by our assumptions, and we get Multiplying by λ(4βx + λ) > 0 we have Due to monotonicity of the square root, we achieve the following estimation: which is equivalent to the inequality Now, we proceed to the estimation from above. For u ≥ −1 and r ∈ [0, 1], we have Bernoulli's inequality as follows: (1 + u) r ≤ 1 + ru.

Theorem 3.4
Let f and f belong to C * e 2 (I ), then, for x ∈ I and λ > 4βx, one has where ω is the classical modulus of continuity.
Proof By Taylor's expansion and applying the operator P β λ , we can write that , and ξ lying between x and t. Thus, using Lemma 3.2 for λ > 4βx and arguing as follows: Using the classical modulus of continuity, we get Considering δ = λ −1/2 , we obtain the required result.

Corollary 3.5
Let f and f ∈ C * e 2 (I ), then, for x ∈ I , we have While, for the original operators, we have Theorem 3.6 For f ∈ C B (I ), there exists a constant C 1 > 0, such that

Proof
Let h ∈ C 2 B (I ) and x, t ∈ I . By Taylor's expansion, we have Hence, arguing as in Theorem 3.4, we have Due to constants preservation of P β λ , we have Therefore Considering infimum over all h ∈ C 2 B (I ), and using the inequality between Kfunctional and second-order moduli property given in [4], we obtain the assertion.

Graphical Representation
In this section, we use the Mathematica software to visualize the convergence of our operators. For t ∈ [0, 20] ⊂ I , we deal with the function f (t) = t 2 + e −t which belongs to the space C * e 2 (I ). Figure 1performs six terms of the sequence of operators P β λ , for λ = 1, 5, 10, 20, 50, 100 and β = 1.
In Fig. 2, we enlarge the plots that we have above. In Fig. 3 , we can see the comparison between the convergence of the classical Post-Widder operators P λ , the semi-exponential Post-Widder operators P β λ , and the We propose the graphs of the following operators: P 5 , P 1 5 , P 1 5 and the function f for x ∈ [0, 20]. The last picture demonstrates the approximation error for the operators P 1 5 , P 1 50 , P 1 100 . Observe that the difference d λ ( f ) = P β λ ( f ) − f tends to 0 as λ → +∞. In Fig. 4, we have the difference for β = 1 and λ = 1, 5, 50.