# A counterexample in comonotone approximation in ${L}^{p}$ space

Colloquium Mathematicae (1993)

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

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topWu, 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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