A Complete Asymptotic Expansion for Bernstein\u2013Chlodovsky Polynomials for Functions on mathbb {R}

Ulrich Abel,Harun Karsli

doi:10.1007/s00009-020-01632-1

Abstract

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].

Highlights

Let f be a real function on R which is bounded on each finite interval
The Bernstein–Chlodovsky operators applied to the function f described above are defined by x−a (Cn,a,bf ) (x) = (Bnfa,b) b − a, a ≤ x ≤ b, where Bn denote the Bernstein operators defined by n (Bnf ) (x) = pn,ν (x) f ν=0 ν n
Explicit expressions of the coefficients c[kbn] (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

Summary

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 fa,b on [0, 1] by fa,b (t) = f (a + (b − a) t). He showed that under the condition (1.3), if a function f satisfies lim exp − σn n→∞ He derived the asymptotic expansion if the function f satisfies condition (1.1) for every σ > 0. The purpose of this note is a pointwise complete asymptotic expansion for the sequence of Bernstein–Chlodovsky operators in the form:

Results

Conclusion