Perturbed Bernstein-type operators

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Perturbed Bernstein-type operators Ana-Maria Acu1

· Heiner Gonska2

Received: 2 November 2018 / Revised: 4 September 2020 / Accepted: 4 September 2020 © Springer Nature Switzerland AG 2020

Abstract The present paper deals with modifications of Bernstein, Kantorovich, Durrmeyer and genuine Bernstein–Durrmeyer operators. Some previous results are improved in this study. Direct estimates for these operators by means of the first and second modulus of continuity are given. Also the asymptotic formulas for the new operators are proved. Keywords Approximation by polynomials · Bernstein operators · Kantorovich operators · Durrmeyer operators · Voronovskaya type theorem · First and second order moduli Mathematics Subject Classification 41A25 · 41A36

1 Introduction In 2018 Khosravian-Arab, Dehghan and Eslahchi introduced three modifications of the classical Bernstein operator. In this note we follow their approach, explain it and discuss further relevant, but truly different Bernstein-type operators which have been attracting attention in the past. Thus we will discuss the modifications of the classical Bernstein operators (pointwise defined, preserve linear functions, but not commutative), classical Kantorovich operators (defined on L 1 , do not preserve linear functions), Durrmeyer operators (globally defined, commutative, do not preserve linear functions) and genuine Bernstein–Durrmeyer operators (globally defined, also commutative, preserve linear functions). Only in the Bernstein case we will go one step further and add remarks on a second perturbation created by modifying the classical recursion twice.

B

Ana-Maria Acu [email protected] Heiner Gonska [email protected]

1

Department of Mathematics and Informatics, Lucian Blaga University of Sibiu, Str. Dr. I. Ratiu, No.5-7, 550012 Sibiu, Romania

2

Faculty of Mathematics, University of Duisburg-Essen, Bismarckstr. 90, 47057 Duisburg, Germany 0123456789().: V,-vol

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Page 2 of 26

A.-M. Acu, H. Gonska

The organization of this note follows the lines given above. Before we will give estimates we add two short sections on the recursion for the fundamental functions of the Bernstein operator and on the use of ω2 .

2 On the recursion for the fundamental functions of Bernstein operators For f ∈ C[0, 1] the Bernstein operator Bn : C[0, 1] → Bn ( f ; x) =

n  k=0

 n

is given by

  k , x ∈ [0, 1], pn,k (x) f n

where the fundamental functions are defined by ⎧  ⎨ n k x (1 − x)n−k , 0 ≤ k ≤ n, x ∈ [0, 1], pn,k (x) := k ⎩ 0, k < 0 or n < k. It is well-known that these functions satisfy the recursion pn,k (x) = (1 − x) pn−1,k (x) + x pn−1,k−1 (x), 0 ≤ k ≤ n.

(2.1)

In particular, pn,0 (x) =(1 − x) pn−1,0 (x) = (1 − x)n , pn,n (x) =x pn−1,n−1 (x) = x n . This recursion is closely related to the so-called de Casteljau algorithm and other methods to compute a value Bn ( f ; x), x fixed. See [12] for details. In [19] the recursion form (2.1) is perturbed by replacing it in the first modification BnM,1 by M,1 (x) =a(x, n) pn−1,k (x) + a(1 − x, n) pn−1,k−