A counterexample in comonotone approximation in L p space

Xiang Wu; Song Zhou

Colloquium Mathematicae (1993)

  • Volume: 64, Issue: 2, page 265-274
  • ISSN: 0010-1354

Abstract

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Refining the idea used in [24] and employing very careful computation, the present paper shows that for 0 < p ≤ ∞ and k ≥ 1, there exists a function f ∈ C [ - 1 , 1 ] k , with f ( k ) ( x ) ≥ 0 for x ∈ [0,1] and f ( k ) ( x ) ≤ 0 for x ∈ [-1,0], such that lim supn→∞ (en(k)(f)p) / (ωk+2+[1/p](f,n-1)p) = + ∞ where e n ( k ) ( f ) p is the best approximation of degree n to f in L p by polynomials which are comonotone with f, that is, polynomials P so that P ( k ) ( x ) f ( k ) ( x ) ≥ 0 for all x ∈ [-1,1]. This theorem, which is a particular case of a more general one, gives a complete solution to the converse result in comonotone approximation in L p space for 1 < p ≤ ∞.

How to cite

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Wu, Xiang, and Zhou, Song. "A counterexample in comonotone approximation in $L^p$ space." Colloquium Mathematicae 64.2 (1993): 265-274. <http://eudml.org/doc/210190>.

@article{Wu1993,
abstract = {Refining the idea used in [24] and employing very careful computation, the present paper shows that for 0 < p ≤ ∞ and k ≥ 1, there exists a function $f ∈ C_\{[-1,1]\}^k$, with $f^\{(k)\}(x)≥ 0$ for x ∈ [0,1] and $f^\{(k)\}(x) ≤ 0$ for x ∈ [-1,0], such that lim supn→∞ (en(k)(f)p) / (ωk+2+[1/p](f,n-1)p) = + ∞ where $e_n^\{(k)\}(f)_p$ is the best approximation of degree n to f in $L^p$ by polynomials which are comonotone with f, that is, polynomials P so that $P^\{(k)\}(x)f^\{(k)\}(x) ≥ 0$ for all x ∈ [-1,1]. This theorem, which is a particular case of a more general one, gives a complete solution to the converse result in comonotone approximation in $L^p$ space for 1 < p ≤ ∞.},
author = {Wu, Xiang, Zhou, Song},
journal = {Colloquium Mathematicae},
language = {eng},
number = {2},
pages = {265-274},
title = {A counterexample in comonotone approximation in $L^p$ space},
url = {http://eudml.org/doc/210190},
volume = {64},
year = {1993},
}

TY - JOUR
AU - Wu, Xiang
AU - Zhou, Song
TI - A counterexample in comonotone approximation in $L^p$ space
JO - Colloquium Mathematicae
PY - 1993
VL - 64
IS - 2
SP - 265
EP - 274
AB - Refining the idea used in [24] and employing very careful computation, the present paper shows that for 0 < p ≤ ∞ and k ≥ 1, there exists a function $f ∈ C_{[-1,1]}^k$, with $f^{(k)}(x)≥ 0$ for x ∈ [0,1] and $f^{(k)}(x) ≤ 0$ for x ∈ [-1,0], such that lim supn→∞ (en(k)(f)p) / (ωk+2+[1/p](f,n-1)p) = + ∞ where $e_n^{(k)}(f)_p$ is the best approximation of degree n to f in $L^p$ by polynomials which are comonotone with f, that is, polynomials P so that $P^{(k)}(x)f^{(k)}(x) ≥ 0$ for all x ∈ [-1,1]. This theorem, which is a particular case of a more general one, gives a complete solution to the converse result in comonotone approximation in $L^p$ space for 1 < p ≤ ∞.
LA - eng
UR - http://eudml.org/doc/210190
ER -

References

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