The rate of convergence of a generalization of Post–Widder operators and Rathore operators

In this paper, we study local approximation properties of certain gamma-type operators. They generalize the Post–Widder operators and the Rathore operators, and approximate locally integrable functions satisfying a certain growth condition on the infinite interval [0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,\infty )$$\end{document}. We derive the complete asymptotic expansion for these operators and prove a localization result. Also, we estimate the rate of convergence for functions of bounded variation.

The Post-Widder operators P n were intensively studied by several authors [3,4,9]. In recent years, several authors defined and studied variants of the Post-Widder operator which preserve several test functions [5-7, 13, 16]. In order to include the similar operator by Rathore [12] (see below), we study in this paper a more general gamma type operator depending on a positive parameter, which includes both, the Post-Widder operators and the Rathore operators as special cases.
Let E be the class of all locally integrable functions of exponential type on [0, +∞) with the property | f (t)| ≤ Me At (t ≥ 0) for some finite constants M, A > 0. The gamma-type operators P n,c (cf. [10,Eq. (3.3) ]) associate to each f ∈ E the function where c is a positive parameter. We emphasize the fact that c may depend on the variable x. Note that the integral exists if nc > A. The definition can be rewritten in the form with the kernel function In the special case c = 1 these operators reduce to the Rathore operators R n ≡ P n,1 , given by [10,Eq. (3.6)] If we substitute c = 1/x, we obtain the Post-Widder operators (1.1).
In this paper we derive the complete asymptotic expansion for the sequence of operators P n,c in the form The coefficients a k ( f , c, x), which are independent of n, will be given in an explicit form. It turns out that associated Stirling numbers of the first kind play an important role. As a special case we obtain the complete asymptotic expansion for the Rathore operators R n and for the Post-Widder operators P n . Secondly, we study the rate of convergence of the sequence P n,c f (x) as n → ∞ for functions of bounded variation. More precisely, we present an estimate of the difference P n,c f (x) − ( f (x+) + f (x−)) /2.

Main results
For q ∈ N and x ∈ (0, ∞), let K [q; x] be the class of all functions f ∈ E which are q times differentiable at x. The following theorem presents as our main result the complete asymptotic expansion for the operators P n,c . Theorem 2.1 Let q ∈ N and x ∈ (0, ∞). For each function f ∈ K [2q; x], the operators P n,c possess the asymptotic expansion as n → ∞, where s 2 ( j, i) denote the associated Stirling numbers of the first kind.
The associated Stirling numbers of the first kind can be defined by their double generating function ∞ i, j=0 as n → ∞. In particular, we obtain the Voronovskaja-type formula In the special case c = 1 we have the complete asymptotic expansion for the Rathore operators, In the special case c = 1/x we have the complete asymptotic expansion for the Post-Widder operators Our second main result is an estimate of the rate of convergence for functions f ∈ E, which are of bounded variation (BV) on each finite subinterval of (0, ∞).

Theorem 2.2
Let f ∈ E be a function of bounded variation on each finite subinterval of (0, ∞). Then, for each x > 0, we have the estimate For the proofs of Theorems 2.1 and 2.2 we need a localization result for the operators P n,c . Since it is interesting in itself we state it as a theorem.
The constant β can be chosen to be

Auxiliary results and proofs
Firstly, we study the moments of the operators P n,c . Throughout the paper, let e r denote the monomials, given by e r (x) = x r (r = 0, 1, 2, . . .). Furthermore, define In the following, the quantities m j denote the unsigned Stirling numbers of the first kind defined by We recall some known facts about Stirling numbers which will be useful in the sequel. The Stirling numbers of the first kind possess the representation In particular, we have P n,c e 0 = e 0 , P n,c e 1 = e 1 and P n,c e 2 (x) = x 2 + x nc , Application of formula ( 3.1) yields P n,c e r (x) = 1 (nc) r r j=0 r j (ncx) j and the index transform j = r − k completes the proof.

Lemma 3.2
The central moments of the operators P n,c are given by In particular, we have P n,c ψ 0 x (x) = 1, P n,c ψ 1 x (x) = 0 and P n,c ψ 2 x (x) = x/ (nc). Proof Application of the binomial formula yields for the central moments and an index shift r → r + k yields the desired representation.
Proof Taking advantage of the formula (3.2) we obtain Note that r +k i+k = 0, for i > r . Using the binomial identity j r +k r +k The inner sum is to be read as zero if i > j − k. Since which completes the proof.
In order to derive Theorem 2.1, a general approximation theorem due to Sikkema [14, Theorem 3] (see also [15]) will be applied. For j ∈ N and x > 0, let H ( j) (x) denote the class of all locally bounded real functions f : [0, ∞) → R, which are j times differentiable at x, and satisfy the additional condition f (t) = O t − j as t → +∞. An inspection of the proof of Sikkema's result reveals that it can be stated in the following form which is more appropriate for our purposes.
In the application used in the proof of Theorem 2.1, we restrict H ( j) (x) to consist only of locally integrable functions. We proceed with the proof of the localization result (Theorem 2.3), which will be applied in the proofs of Theorems 2.1 and 2.2.

Proof of Theorem 2.3 Let
say, where s = nc > 0 and denote the lower and the upper incomplete gamma function, respectively. We use the well-known asymptotic behaviour of the incomplete gamma function for large parameters z and b. It holds [17, Eq. (7.3.18)], as z, b → ∞ such that the ratio λ = b/z is bounded away from unity, i.e., λ ≤ λ 0 < 1, where λ 0 is a fixed number in (0, 1). In a similar kind it holds [17, Eq. (7.4.43)], as z, b → ∞ such that the ratio α = z/b is bounded away from unity, i.e., α ≤ α 0 < 1. If δ = x the integral I 1 vanishes. Let us consider the case δ < x. Since 3) implies that Application of Stirling's formula, Application of Stirling's formula leads to The latter inequality is equivalent to the obvious inequality 2t ≤ log 1+t 1−t = 2 t + t 3 /3 + t 5 /5 + t 7 /7 + · · · , for t = δ/x ∈ [0, 1). Combining the above results we obtain the desired estimate with the constant β = β 2 .

Proof of Theorem 2.1 Let x > 0 and put
Interchanging the order of summation, we obtain this implies the desired expansion (1.4) with the associated Stirling numbers of the first kind s 2 (i, j) as defined in Eq. (3.2).
Now we turn to the estimate of the rate of convergence for BV functions. For the proof of Theorem 2.2 we apply the following properties of the kernel function φ n,c (x, t) as defined in (1.3).

Lemma 3.5
The kernel function φ n,c (x, t) satisfies the following estimates: The second estimate is obtained in an analogous manner.

Lemma 3.6
For fixed x > 0, Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.