Abstract
We present a sort of Korovkin-type result that provides a tool to obtain asymptotic formulae for sequences of linear positive operators.
Highlights
This paper deals with the approximation of continuous functions by sequences of positive linear operators
5, the novelty here being that instead of using the Taylor formula of the function f to be approximated, we consider the Taylor formula of f ◦ φ−1 for a certain function φ. It is the intention of the authors that the paper offers a clear and quick procedure to obtain asymptotic expressions for a wide variety of sequences of linear positive operators
Notice that for i 0, identity 2.2 becomes limn → ∞λn Lne[0] x − 1 0, which is obviously fulfilled if the operators Ln preserve the constants
Summary
This paper deals with the approximation of continuous functions by sequences of positive linear operators. A classical key ingredient to prove an asymptotic formula for a sequence of positive linear operators Ln is Taylor’s theorem. The formula appears after some minor work if one is able to find easy-to-use expressions for the first moments of the operators, namely, Ln eix x Ln e1 − xe[0] i x , i 0, 1, 2, 4, 1.2 ei and eix denote the monomials ei t ti and eix t t − x i. 5 , the novelty here being that instead of using the Taylor formula of the function f to be approximated, we consider the Taylor formula of f ◦ φ−1 for a certain function φ It is the intention of the authors that the paper offers a clear and quick procedure to obtain asymptotic expressions for a wide variety of sequences of linear positive operators
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