Abstract
In this paper, we introduce a new type λ-Bernstein operators with parameter lambdain[-1,1], 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 B_{n,lambda }(f;x) to f(x), and we see that in some cases the errors are smaller than B_{n}(f) to f.
Highlights
In 1912, Bernstein [1] proposed the famous polynomials called nowadays Bernstein polynomials to prove the Weierstrass approximation theorem as follows: nkBn(f ; x) = f n bn,k(x), (1)k=0 where x ∈ [0, 1], n = 1, 2, . . . , and Bernstein basis functions bn,k(x) are defined as: bn,k(x) =n k xk(1 – x)n–k. (2)Based on this, there are many papers about Bernstein type operators [2,3,4,5,6,7,8,9]
We introduce the new λ-Bernstein operators, n
Proof By the Korovkin theorem it suffices to show that lim n→∞
Summary
And Bernstein basis functions bn,k(x) are defined as: bn,k(x) = There are many papers about Bernstein type operators [2,3,4,5,6,7,8,9]. We introduce the new λ-Bernstein operators, n 3, we investigate a Korovkin approximation theorem, establish a local approximation theorem, give a convergence theorem for the Lipschitz continuous functions, and obtain a Voronovskaja-type asymptotic formula.
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