A Voronovskaja type formula for Soardi’s operators

In 1991 Soardi introduced a sequence of positive linear operators βn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{n}$$\end{document} associating to each function f∈C0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C\left[ 0,1\right] $$\end{document} a polynomial function which is closely related to the Bernstein polynomials on -1,+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ -1,+1\right] $$\end{document}. One of the authors already studied the operators βn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{n}$$\end{document} in several papers. This paper is devoted to other properties of Soardi’s operators. We introduce a version β~n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{\beta }}_{n}$$\end{document} which can be expressed in terms of the classical Bernstein operators and present the relations between βn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{n}$$\end{document} and β~n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{\beta }}_{n}$$\end{document}. We derive Voronovskaja-type results for both βn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{n}$$\end{document} and β~n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{\beta }}_{n}$$\end{document}. Furthermore, rates of convergence for β~n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{\beta }}_{n}$$\end{document}, respectively βn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{n}$$\end{document}, are estimated. Finally, we study the first and second moments of βn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{n}$$\end{document}.


Introduction
In 1991 Soardi [8] introduced the sequence of positive linear operators β n associating to each function f ∈ C [0, 1] the polynomial function Usually, the operators β n are given in the form where m = n/2 and w n,k (x) =w n,m−k (x) are the fundamental polynomials. The definition and the proofs in [8] are based on properties of random walks on hypergroups. Soardi proved that, for each f ∈ C [0, 1], the sequence (β n f ) is uniformly convergent to f . Furthermore, by an intensive use of probabilistic tools, Soardi [8,Theorem 2] estimated the rate of convergence of (β n f ) in terms of the usual modulus of continuity: Shape preserving properties of the operators β n were investigated in [5][6][7]. In particular, if f ∈ C [0, 1] is increasing, then β n f is increasing (see [6,Th. 2.1]; this fact will be used in Sect. 3). Moreover, if f ∈ C [0, 1] is increasing and convex, then β n f ≥ f (see [6,Th. 3.1]; this inequality will be instrumental in Sect. 5). For x ∈ (0, 1) and bounded functions f on [0, 1], Raşa [6,Theorem 4.1] proved the Voronovskaja-type formula This paper is devoted to other properties of Soardi's operators. In Sect. 2 we introduce a versionβ n which can be expressed in terms of the classical Bernstein operators. The relations between β n andβ n are presented in Sect. 3. Section 4 contains Voronovskaja-type results for both β n andβ n . Rates of convergence forβ n , respectively β n , are estimated in Sects. 5 and 2. The last two sections are devoted to the first and second moments of β n .

The variant˜n and its relation to Bernstein polynomials
In this section we introduce a variantβ n of Soardi's operator which seems to be more natural. Replacing f n−2k n with f n+1−2k n+1 leads to the definition where m = n/2 . The index manipulation k → n + 1 − k yields For even values of n we have This representation is valid also in the case of odd integers n since the term Writing we obtain the following relation to the classical Bernstein polynomials.

Lemma 1
For a function f on [0, 1], we have the relation and B n g denotes the classical Bernstein polynomial on [0, 1].

Voronovskaja-type results for the operatorsˇn and˜n
In 2000, Raşa [6, Theorem 4.1] proved the following Voronovskaja-type formula for the operators β n .

Theorem 1 Let x ∈ (0, 1) and f be a bounded function on
If x = 0, i.e., t = 1/2, you can insert the well-known asymptotic formulas for B n . One obtains as n → ∞. In the special case x = 0, we can use in order to obtain The asymptotic behaviour can easily be derived if f is an even function which is smooth in x = 0. If f is not an even function, B n+1ĝ 1 2 is an unpleasant expression. The link to Soardi's original operator is given by with u n (x) = ((n + 1) t − 1) /n. Therefore, as n → ∞. A look into the proof of asymptotic formulas for Bernstein polynomials reveals that the latter formula is valid if f is only locally smooth. We have where a k, j (x) are certain polynomials involving Stirling numbers of the first and the second kind. More precisely, we have provided that f is bounded on [0, 1] and admits a derivative of order 2q at x ∈ [0, 1] (see [1,Remark 2]). and Then

An estimate of the rate of convergence for the operators˜n
In this section we derive an estimate for the rate of convergence for the operatorsβ n in terms of the ordinary modulus of continuity ω ( f , δ).
For functions of the form we haveg = −g. Hence, Lemma 2 For all n ∈ N, Proof With the notations of Sect. 3 we havẽ Since β n preserves constant functions and β n f ≥ f , for all increasing and convex functions f ∈ C [0, 1], we obtaiñ β n e 1 ≥ n n + 1 e 1 + 1 n + 1 e 0 = e 1 + e 0 − e 1 n + 1 ≥ e 1 .
Proof We have

An estimate of rate of convergence for the Soardi operator
As already mentioned in the introduction Soardi [8,Theorem 2] estimated the rate of convergence of the operators β n in terms of the ordinary modulus of continuity: In this section we improve this estimate considerably by diminishing the absolute constant.

Remark 2
In particular, we have The essential ingredient of the proof is the following estimate of the second central moment of the operators β n .
Lemma 4 For all n ∈ N, the second central moment of β n satisfies the estimate .
Furthermore, for each ε > 0, there is an index n 0 such that for each n > n 0 ,

Proof of Lemma 4 Using the relation
By Lemma 2, which is the desired estimate.

The second moment ofˇn
We have Since It follows So, we need a good control on β n e 1 (0). This is our aim in what follows.

The value (ˇne 1 ) (0) of the first moment
By Eq. ( 3) , we infer that In particular, it follows that In the next section we derive closed expressions for (β n e 1 ) (0) and study its asymptotic behaviour as n tends to infinity. We prove that the exact asymptotic rate of convergence is Note that 2 √ 3 ≈ 3.4641 and 2 √ 2/π ≈ 1.59577.