Approximation properties of λ-Bernstein operators

Abstract

In this paper, we introduce a new type λ-Bernstein operators with parameter , we investigate a Korovkin type approximation theorem, establish a local approximation theorem, give a convergence theorem for the Lipschitz continuous functions, we also obtain a Voronovskaja-type asymptotic formula. Finally, we give some graphs and numerical examples to show the convergence of to , and we see that in some cases the errors are smaller than to f.

Convergence of λ-Bernstein operators based on (p, q)-integers

“…In 2018, Cai et al [1] introduced the following new family of Bernstein operators with parameter λ ∈ [-1, 1]:…”

“…They call these operators (1) λ-Bernstein operators, they investigated some approximation theorems and also gave some numerical examples. In the same year, Acu et al [2] studied some approximation properties of Kantorovich type of operators (1). Later, Özger [3] gave some statistical approximation results of (1).…”

“…Obviously, when p → 1 -, B λ n,p,q (f ; x) reduces to q-analogue of λ-Bernstein operators in [4]; when p, q → 1 -, B λ n,p,q (f ; x) reduces to (1). Here we mention certain notations on (p, q)-calculus, details can be found in [23][24][25][26][27].…”

Approximation properties of λ‐Bernstein‐Kantorovich operators with shifted knots

“…Recently, Cai et al generalized Bernstein operators depending on the parameter λ ∈ [ − 1,1] and studied their approximation as well as convergence properties. Acu et al considered a Kantorovich variant of λ ‐Bernstein operators as where 1 ≤ j ≤ m − 1 and b m , j ( y ), j = 0,1,… m are given by .…”

Simultaneous Approximation of New Sequence of Integral Type Operators with Parameter δ_0

“…Many papers interested in the classical sequences of Bernstein and gave some modifications of them [5], [11]. In addition, the numerical application for this sequence are very limited [3], [13]. Szãsz in 1950, generalized the Bernstein sequence to approximate the space of continuous functions on the interval [0, ∞) as [16] S n (f ; x) = ∞ k=0 q n,k (x)f k n (1.2) where q n,k (x) = (nx) k k!e nx , x ∈ [0, ∞).…”