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

Xiang Wu¹ ; Song Zhou¹

¹

Colloquium Mathematicum, Tome 64 (1993) no. 2, pp. 265-274.

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 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 ≤ ∞.

Détail

DOI : 10.4064/cm-64-2-265-274