The Lower Estimate for Linear Positive Operators (II)

Knoop, Hans-Bernd; Zhou, Xiaoying

doi:10.1007/bf03323413

Cited by 48 publications

(15 citation statements)

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“…Hence (3.4) fails for g(x). established in 1994 in [7]. Similar strong converse inequalities are valid for many other linear positive operators.…”

Section: Upper and Lower Pointwise Boundssupporting

confidence: 58%

Pointwise Approximation by Bernstein Polynomials

Tachev¹

2012

Bull. Aust. Math. Soc.

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We improve the degree of pointwise approximation of continuous functions f (x) by Bernstein operators, when x is close to the endpoints of [0,1]. We apply the new estimate to establish upper and lower pointwise estimates for the test function g(x) = x log(x) + (1 − x) log(1 − x). At the end we prove a general statement for pointwise approximation by Bernstein operators.2010 Mathematics subject classification: primary 41A10; secondary 41A15, 41A25, 41A36.

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“…Hence (3.4) fails for g(x). established in 1994 in [7]. Similar strong converse inequalities are valid for many other linear positive operators.…”

Section: Upper and Lower Pointwise Boundssupporting

confidence: 58%

Pointwise Approximation by Bernstein Polynomials

Tachev¹

2012

Bull. Aust. Math. Soc.

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“…For f ∈ C[0, 1], n ∈ IN and constants c 1 , c 2 independent of f and n one has (the crucial result of Knoop & Zhou [33], Totik [47])…”

Section: Vol 54 (2009)mentioning

confidence: 99%

Estimates for Iterates of Positive Linear Operators Preserving Linear Functions

Kacsó¹

2009

Results. Math.

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We consider iterates of certain (general) positive linear operators preserving linear functions and derive quantitative upper estimates in terms of weighted and non-weighted moduli of smoothness and related K-functionals. We show that corresponding lower estimates in terms of the classical moduli are not possible, while for the Ditzian-Totik modulus the situation can be different. The results can be applied to several well-known operators; we present here the Bernstein and the genuine Bernstein-Durrmeyer operators. Classification (2000). 41A36, 41A25, 41A10. Mathematics SubjectKeywords. Iterates of operators, upper and lower inequalities, degree of approximation, K-functionals, moduli of smothness, Bernstein operators, genuine Bernstein-Durrmeyer operators. Main toolsLet f ∈ L ∞ [0, 1] be the space of essentially bounded measurable functions, f ∞ = vrai sup x∈ [0,1] |f (x)|, ϕ(x) = x(1 − x), x ∈ [0, 1] and r a natural number. Consider the K-functionalwhere the infimum is taken over all g such that g (r−1) ∈ AC loc (0, 1) (i.e., g (r−1) is absolutely continuous in every closed finite subinterval of (0, 1) and ϕ r g (r) ∞ < ∞).

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“…A crucial tool used in [12] is the so-called Ditzian-Totik modulus [8]. In addition to this important work, we note that several strong inverse inequalities for univariate Bernstein polynomials have been established by Ditzian and Ivanov [7], Totik [19] and Knoop and Zhou [13].…”

Section: Introductionmentioning

confidence: 98%