A counterexample in comonotone approximation in $L^p$ space

Wu, Xiang ; Zhou, Song

Colloquium Mathematicae, Tome 66 (1993), p. 265-274 / Harvested from The Polish Digital Mathematics Library

Résumé

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 ≤ ∞.

Détail

Publié le : 1993-01-01

Zbl 0894.41009

EUDML-ID : urn:eudml:doc:210190

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