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

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 $f\in {C}_{[-1,1]}^{k}$, with ${f}^{\left(k\right)}\left(x\right)\ge 0$ for x ∈ [0,1] and ${f}^{\left(k\right)}\left(x\right)\le 0$ for x ∈ [-1,0], such that lim supn→∞ (en(k)(f)p) / (ωk+2+[1/p](f,n-1)p) = + ∞ where ${e}_{n}^{\left(k\right)}{\left(f\right)}_{p}$ 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 ${P}^{\left(k\right)}\left(x\right){f}^{\left(k\right)}\left(x\right)\ge 0$ for all x ∈ [-1,1]. This theorem, which is a particular case of a more general one, gives a complete solution...