Comonotone Second Jackson's Inequality

TL;DR

This paper proves a comonotone version of Jackson's inequality, establishing that the best uniform approximation of certain periodic functions by comonotone polynomials improves at a rate proportional to 1/n^r.

Contribution

It introduces a comonotone analogue of second Jackson's inequality for periodic functions with specific monotonicity constraints.

Findings

Establishes an upper bound for the approximation error as c/n^r.

Shows the approximation rate depends only on r and the set Y.

Extends classical Jackson's inequality to comonotone functions.

Abstract

Let $2s$ points $y_i=-\pi\le y_{2s}<\ldots<y_1<\pi$ be given. Using these points, we define the points $y_i$ for all integer indices $i$ by the equality $y_i=y_{i+2s}+2\pi$. We shall write $f\in\bigtriangleup^{(1)}(Y)$ if $f$ is a $2\pi$-periodic function and $f$ does not decrease on $[y_i, y_{i-1}]$ if $i$ is odd; and $f$ does not increase on $[y_i, y_{i-1}]$ if $i$ is even. We denote $E_n^{(1)}(f;Y)$ the value of the best uniform comonotone approximation. In this article the following Theorem -- the comonotone analogue of second Jackson's Inequality -- is proved. Theorem. If $f\in\bigtriangleup^{(1)}(Y)\bigcap {\Bbb W}^r$, $r>2$, then $$E_n^{(1)}(f;Y)\le \frac c{n^r}, $$ where $c=c(r,Y)=const$ depending only on $r$ and $Y$, ${\Bbb W}^r$ Sobolev space.