Convex polynomial and spline approximation in C[-1, 1]

Yingkang Hu; Dany Leviatan; Xiang Ming Yu

doi:10.1007/BF01205165

Convex polynomial and spline approximation in C[-1, 1]

Yingkang Hu, Dany Leviatan, Xiang Ming Yu

Emeriti, Retired & In Memoriam (COSM)

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16 Scopus citations

Abstract

We prove that a convex function f ∈ C[-1, 1] can be approximated by convex polynomials p_n of degree n at the rate of ω₃(f, 1/n). We show this by proving that the error in approximating f by C² convex cubic splines with n knots is bounded by ω₃(f, 1/n) and that such a spline approximant has an L^∞ third derivative which is bounded by n³ω₃(f, 1/n). Also we prove that if f ∈ C²[-1, 1], then it is approximable at the rate of n^-2 ω(f″, 1/n) and the two estimates yield the desired result.

Original language	English
Pages (from-to)	31-64
Number of pages	34
Journal	Constructive Approximation
Volume	10
Issue number	1
DOIs	https://doi.org/10.1007/BF01205165
State	Published - Mar 1994

Scopus Subject Areas

Analysis
General Mathematics
Computational Mathematics

Keywords

AMS classification: 41A10, 41A15, 41A25, 41A29
Degree of convex approximation
Polynomial and spline approximation

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10.1007/BF01205165

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title = "Convex polynomial and spline approximation in C[-1, 1]",

abstract = "We prove that a convex function f ∈ C[-1, 1] can be approximated by convex polynomials pn of degree n at the rate of ω3(f, 1/n). We show this by proving that the error in approximating f by C2 convex cubic splines with n knots is bounded by ω3(f, 1/n) and that such a spline approximant has an L∞ third derivative which is bounded by n3ω3(f, 1/n). Also we prove that if f ∈ C2[-1, 1], then it is approximable at the rate of n-2 ω(f″, 1/n) and the two estimates yield the desired result.",

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author = "Yingkang Hu and Dany Leviatan and Yu, {Xiang Ming}",

year = "1994",

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T1 - Convex polynomial and spline approximation in C[-1, 1]

AU - Hu, Yingkang

AU - Leviatan, Dany

AU - Yu, Xiang Ming

PY - 1994/3

Y1 - 1994/3

N2 - We prove that a convex function f ∈ C[-1, 1] can be approximated by convex polynomials pn of degree n at the rate of ω3(f, 1/n). We show this by proving that the error in approximating f by C2 convex cubic splines with n knots is bounded by ω3(f, 1/n) and that such a spline approximant has an L∞ third derivative which is bounded by n3ω3(f, 1/n). Also we prove that if f ∈ C2[-1, 1], then it is approximable at the rate of n-2 ω(f″, 1/n) and the two estimates yield the desired result.

AB - We prove that a convex function f ∈ C[-1, 1] can be approximated by convex polynomials pn of degree n at the rate of ω3(f, 1/n). We show this by proving that the error in approximating f by C2 convex cubic splines with n knots is bounded by ω3(f, 1/n) and that such a spline approximant has an L∞ third derivative which is bounded by n3ω3(f, 1/n). Also we prove that if f ∈ C2[-1, 1], then it is approximable at the rate of n-2 ω(f″, 1/n) and the two estimates yield the desired result.

KW - AMS classification: 41A10, 41A15, 41A25, 41A29

KW - Degree of convex approximation

KW - Polynomial and spline approximation

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DO - 10.1007/BF01205165

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VL - 10

SP - 31

EP - 64

JO - Constructive Approximation

JF - Constructive Approximation

IS - 1

ER -

Convex polynomial and spline approximation in C[-1, 1]

Abstract

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Keywords

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