A Complete Asymptotic Expansion for Bernstein–Chlodovsky Polynomials for Functions on R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document}

We consider a variant of the Bernstein–Chlodovsky polynomials approximating continuous functions on the entire real line and study its rate of convergence. The main result is a complete asymptotic expansion. As a special case we obtain a Voronovskaja-type formula previously derived by Karsli [11].


Introduction
Let f be a real function on R which is bounded on each finite interval. For a, b ∈ R with a < b, define the function f a,b on [0, 1] by f a,b (t) = f (a + (b − a) t). Furthermore, put Obviously, f 0,1 is the restriction of f to [0, 1] and we have f a,b = f a,b 0,1 .
The Bernstein-Chlodovsky operators applied to the function f described above are defined by where B n denote the Bernstein operators defined by See also [13].
In the following we suppose that the parameters a, b are coupled with n, i.e., a = a n and b = b n . Because the difference between two nodes of C n,a,b is at least (b − a) /n it is clear that the condition b n − a n = o (n) as n → ∞ is necessary for having convergence of (C n,an,bn f ) (x) to f (x).
In the special case a n = 0 < b n , for n ∈ N, these polynomials were introduced by I. Chlodovsky [7] in 1937 in order to approximate functions on infinite intervals. He showed that under the condition (1. for an arbitrary small η > 0. For more results on Chlodovsky operators see the survey article [9] by Karsli. Explicit expressions of the coefficients c [bn] k (f, x) in terms of Stirling numbers were given by Karsli [10]. He derived the asymptotic expansion if the function f satisfies condition (1.1) for every σ > 0.
Throughout the paper we assume that the sequences (a n ) and (b n ) satisfy b n − a n > 0, lim n→∞ (−a n ) = lim n→∞ b n = +∞, and lim n→∞ −a n b n n = 0. (1. 3) The purpose of this note is a pointwise complete asymptotic expansion for the sequence of Bernstein-Chlodovsky operators in the form: −a n b n n k + o −a n b n n q (1.4) as n → ∞, for sufficiently smooth functions f satisfying f (t) = O (exp (αt p )) as t → +∞, provided that the sequences (a n ) , (b n ) satisfy (−a n b n ) = , which depend on f and a n , b n , are bounded with respect to n.
The latter formula means that, for each fixed x > 0 and for all positive integers q:

Main Result
For real constants α ≥ 0 and p ≥ 0, let W α,p denote the class of functions f ∈ C (R) satisfying the growth condition: Note that in the special instance p = 0 the class W α,0 consists of the bounded continuous functions on R. Since W 0,p and W α,0 coincide we consider only the case α > 0.
Recall that the Stirling numbers s (n, k) and S (n, k) of first and second kind, respectively, are defined by the relations: where z 0 = 1 and z n = z (z − 1) · · · (z − n + 1), for n ∈ N, denote the falling factorials.
The following theorem is the main result.
Let (−a n ) and (b n ) be sequences of reals tending to infinity and satisfying the growth condition: Then, for any positive integer q, the Bernstein-Chlodovsky operators C n,an,bn possess the asymptotic expansion: The coefficients c (f, x) depend on n but are bounded with respect to n. Remark 2.3. Our assumption (2.1) on the sequence (−a n b n ) corresponds to Chlodovsky's condition (1.2). Furthermore, it is related to the assumption in the case a n = 0 (see, [3, Theorem 1, Eq. (4)]).
In the special case q = 1 Theorem 2.1 implies the following Voronovskajatype result.
Suppose that the function f ∈ W α,p admits a second derivative at the point x > 0. Let (−a n ) and (b n ) be sequences of reals tending to infinity and satisfying the growth condition (2.1) . Then, the Bernstein-Chlodovsky operators C n,an,bn satisfy the asymptotic relation: Remark 2.5. The expansion in Theorem 2.1 is completely different to the (pointwise) complete asymptotic expansion: for the classical Bernstein polynomials B n , which is valid for all bounded functions f : [0, 1] → R being sufficiently smooth in x ∈ [0, 1]. The Voronovskaja formula states that The same is true in the case of the classical Bernstein-Chlodovsky operators C n,0,bn . Their Voronovskaja-type formula: was derived in 1960 by Albrycht and Radecki [5]. For further history consult the survey article [9].

Auxiliary Results and Proof of the Main Theorem
Our starting-point is an explicit representation of the central moments of the Bernstein polynomials in terms of Stirling numbers of the first and second kind. In the following we write e m (x) = x m , m ∈ N 0 , for the m-th monomial and ψ x (t) = t − x for x ∈ R. For a proof see, e.g., [2].
where the coefficients A (k, s, j) are given by Eq. (2.3) .
Proof. We have x − a b − a and the lemma follows by Lemma 3.1.
As we have seen in the proof of Lemma 3.2 the central moments of the Bernstein-Chlodovsky operators can be expressed in terms of Bernstein polynomials: As a consequence from well-known properties of the Bernstein polynomials we obtain the following result: where d s,i are certain real numbers, we obtain This can be rewritten in the form as stated in the lemma.
For the sake of brevity, in the following, we write This immediately implies the following estimate for the central moment of the Bernstein-Chlodovsky operators. A crucial tool is the following estimate due to Bernstein (see [14, Theorem 1.5.3, p. 18f]).
The next lemma presents a form of Lemma 3.5 which is more useful for application to Chlodovsky operators on the real line. It follows the idea of Albrycht and Radecki [5] who proved a similar result for the classical Chlodovsky operators.
The latter inequality is equivalent to b − a which is a condition of the lemma.