Nearly Comonotone Approximation of Periodic Functions

Abstract

Suppose that a continuous $2\pi$-periodic function $f$ on the real axis changes its monotonicity at points $y_i: -\pi\le y_{2s}< y_{2s-1}< \cdots< y_1<\pi,\ s\in\Bbb N$. In this paper, for each $n\ge N,$ a trigonometric polynomial $P_n$ of order $cn$ is found such that: $P_n$ has the same monotonicity as $f,$ everywhere except, perhaps, the small intervals$$(y_i-\pi/n,y_i+\pi/n)$$and$$\|f-P_n\|\le c(s)\omega_3(f,\pi/n),$$where $N$ is a constant depending only on $\min\limits_{i=1,\cdots,2s}\{y_i-y_{i+1}\},\ c,\ c(s)$ are constants depending only on $s,\ \omega_3(f,\cdot)$ is the modulus of smoothness of the $3$-rd order of the function $f,$ and $\|\cdot\|$ is the max-norm.