Polynomial approximation in Lp (0<p<1)

Ronald A. DeVore; Dany Leviatan; Xiang Ming Yu

doi:10.1007/BF01238268

Polynomial approximation in L_p (0<p<1)

Ronald A. DeVore^*, Dany Leviatan, Xiang Ming Yu

^*Corresponding author for this work

School of Mathematical Sciences

University of South Carolina

Research output: Contribution to journal › Article › peer-review

43 Scopus citations

Abstract

We prove that for f∈L_p, 0<p<1, and k a positive integer, there exists an algebraic polynomial P_n of degree ≤n such that {Mathematical expression} where ω_k^φ{symbol}(f, t)p is the Ditzian-Totik modulus of smoothness of f in L_p, and C is a constant depending only on k and p. Moreover, if f is nondecreasing and k≤2, then the polynomial P_n can also be taken to be nondecreasing.

Original language	English
Pages (from-to)	187-201
Number of pages	15
Journal	Constructive Approximation
Volume	8
Issue number	2
DOIs	https://doi.org/10.1007/BF01238268
State	Published - Jun 1992

Keywords

AMS classification: 41A25, 41A20
Degree of approximation
Monotone approximation
Polynomials

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abstract = "We prove that for f∈Lp, 0n of degree ≤n such that {Mathematical expression} where ωkφ{symbol}(f, t)p is the Ditzian-Totik modulus of smoothness of f in Lp, and C is a constant depending only on k and p. Moreover, if f is nondecreasing and k≤2, then the polynomial Pn can also be taken to be nondecreasing.",

keywords = "AMS classification: 41A25, 41A20, Degree of approximation, Monotone approximation, Polynomials",

author = "DeVore, {Ronald A.} and Dany Leviatan and Yu, {Xiang Ming}",

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AU - DeVore, Ronald A.

AU - Leviatan, Dany

AU - Yu, Xiang Ming

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N2 - We prove that for f∈Lp, 0n of degree ≤n such that {Mathematical expression} where ωkφ{symbol}(f, t)p is the Ditzian-Totik modulus of smoothness of f in Lp, and C is a constant depending only on k and p. Moreover, if f is nondecreasing and k≤2, then the polynomial Pn can also be taken to be nondecreasing.

AB - We prove that for f∈Lp, 0n of degree ≤n such that {Mathematical expression} where ωkφ{symbol}(f, t)p is the Ditzian-Totik modulus of smoothness of f in Lp, and C is a constant depending only on k and p. Moreover, if f is nondecreasing and k≤2, then the polynomial Pn can also be taken to be nondecreasing.

KW - AMS classification: 41A25, 41A20

KW - Degree of approximation

KW - Monotone approximation

KW - Polynomials

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JO - Constructive Approximation

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IS - 2

ER -

Polynomial approximation in L_p (0<p<1)

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Keywords

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