Stechkin-type estimate for nearly copositive approximation of periodic functions

Abstract

Under the conditions that a continuous $2\pi$-periodic function $f$ on the real axis changes its sign at $2s$ points $y_i\colon {-\pi}\le y_{2s}<y_{2s-1}<\ldots <y_1<\pi,$ $s\in\Bbb N,$ the other points $y_i,$ $i\in\Bbb Z,$ are defined by periodicity, and natural $n>N(k,y_i)$, where $N(k,y_i)$ is a constant that depends only on $k\in \Bbb N$ and $\min _{i=1,\ldots ,2s}\{y_i-y_{i+1}\}$, we find a trigonometric polynomial $P_n$ of order $\le n$ such that the signs of $P_n$ and $f$ are the same everywhere with the possible exception for small neighborhoods of the points $y_i\colon  (y_i-\pi/n,y_i+\pi/n),$ $ P_n(y_i)=0,$ $i\in\Bbb Z,$ and $\|f-P_n\|\le c(k,s)\,\omega_k(f,\pi/n),$ where $c(k,s)$ is a constant that depends only on $k$ and $s$; $\omega_k(f,\cdot)$ is the $k$th modulus of smoothness of $f,$ and $\|\cdot\|$ is the max-norm.