TY - JOUR
T1 - Nearly Comonotone Approximation of Periodic Functions
AU - G. A. Dzyubenko
JO - Analysis in Theory and Applications
VL - 1
SP - 74
EP - 92
PY - 2017
DA - 2017/01
SN - 33
DO - http://doi.org/10.4208/ata.2017.v33.n1.7
UR - https://global-sci.org/intro/article_detail/ata/4617.html
KW - Periodic functions, comonotone approximation, trigonometric polynomials, Jackson-type estimates.
AB - <p style="text-align: justify;">Suppose that a continuous $2\pi$-periodic function $f$ on the real axis changes its monotonicity at points $y_i: -\pi\le y_{2s}&lt; y_{2s-1}&lt; \cdots&lt; y_1&lt;\pi,\ s\in\Bbb N$. 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.</p>