A counterexample in comonotone approximation in $L^p$ space
Colloquium Mathematicum, Tome 64 (1993) no. 2, pp. 265-274
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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 sup_{n→∞} (e_n^{(k)}(f)_p) / (ω_{k+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 ≤ ∞.
@article{10_4064_cm_64_2_265_274,
author = {Xiang Wu and Song Zhou},
title = {A counterexample in comonotone approximation in $L^p$ space},
journal = {Colloquium Mathematicum},
pages = {265--274},
publisher = {mathdoc},
volume = {64},
number = {2},
year = {1993},
doi = {10.4064/cm-64-2-265-274},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm-64-2-265-274/}
}
TY - JOUR AU - Xiang Wu AU - Song Zhou TI - A counterexample in comonotone approximation in $L^p$ space JO - Colloquium Mathematicum PY - 1993 SP - 265 EP - 274 VL - 64 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4064/cm-64-2-265-274/ DO - 10.4064/cm-64-2-265-274 LA - en ID - 10_4064_cm_64_2_265_274 ER -
Xiang Wu; Song Zhou. A counterexample in comonotone approximation in $L^p$ space. Colloquium Mathematicum, Tome 64 (1993) no. 2, pp. 265-274. doi: 10.4064/cm-64-2-265-274
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