Comonotone approximation of periodic functions

Abstract

Let $\tilde{C}$ be the space of continuous $2\pi$-periodic functions $f$, endowed with the uniform norm $\Vert f\Vert ≔{max}_{x\in R}\left|f\left(x\right)\right|$, and denote by ${\omega }_m(f,t)$, the $m$th modulus of smoothness of $f$. Denote by ${\tilde{C}}^r$, the subspace of $r$ times continuously differentiable functions $f\in \tilde{C}$, and let $T_n$, be the set of trigonometric polynomials $T_n$ of degree $<n$. If $f\in {\tilde{C}}^r$, has $2s$, $s\ge 1$, extremal points in $(-\pi ,\pi ]$, denote by $E_n^{\left(1\right)}\left(f\right)≔\underset{T_n\in T_n:f^{\prime }{T}_n^{\prime }\ge 0}{inf}\Vert f-T_n\Vert ,$ the error of its best comonotone approximation. We prove, that if $f\in {\tilde{C}}^r$, then for either $m=1$, or $m=2$ and $r=2s$, or $m\in N$ and $r>2s$, $E_n^{\left(1\right)}\left(f\right)\le \frac{c(m,r,s)}{n^r}{\omega }_m(f^{\left(r\right)},1/n),n\ge 1,$where the constant $c(m,r,s)$ depends only on $m$, $r$ and $s$.