Jackson's type estimate of nearly coconvex approximation

TL;DR

This paper develops a Jackson-type estimate for nearly coconvex approximation of periodic functions, providing a method to construct trigonometric polynomials that closely approximate the function while preserving convexity properties except near specified points.

Contribution

The paper introduces a new Jackson-type estimate for nearly coconvex approximation, extending classical results to functions with changing convexity points and providing explicit bounds.

Findings

Constructs trigonometric polynomials matching convexity except near specified points.

Provides approximation bounds involving the modulus of continuity of order 4.

Establishes dependence of approximation quality on the spacing of convexity change points.

Abstract

Suppose that a continuous on the real axis $2\pi$-periodic function $f$ changes its convexity at $2s,\ s\in\Bbb N,$ points $y_i$ on each period: $-\pi\le y_{2s}<y_{2s-1}<...<y_1<\pi,$ and for the rest $i\in\Bbb Z,$ the points $y_i$ are defined periodically. In the paper, for each $n\ge N,$ a trigonometric polynomial $P_n$ of order $cn$ is found such that: $P_n$ has the same convexity as $f,$ everywhere except, perhaps, the small neighborhoods of the $y_i:$ $$ (y_i-\pi/n,y_i+\pi/n) $$ and $$ \|f-P_n\|\le c(s)\,\omega_4(f,\pi/n), $$ where $N$ is a constant depending only on $\min\limits_{i=1,...,2s}\{y_i-y_{i+1}\},\ c$ and $c(s)$ are constants depending only on $s,\ \omega_4(f,\cdot)$ is the modulus of continuity of the $4$-th order of the function $f,$ and $\|\cdot\|$ is the max-norm.