One Counterexample of Comonotone Approximation of $2\pi$-periodic Function on Trigonometric Polynomials

TL;DR

This paper constructs a specific counterexample demonstrating limitations of comonotone approximation of periodic functions by trigonometric polynomials, showing that certain approximation bounds do not always hold.

Contribution

The paper provides the first explicit counterexample illustrating the failure of comonotone approximation bounds for periodic functions.

Findings

Counterexample functions exist with specific properties.

Shows limitations of comonotone approximation bounds.

Highlights the need for caution in approximation theory.

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 counterexample of comonotone approximation is proved. Example. For each $k\in\Bbb N$, $k>3$, and $n\in\Bbb N$ there a function $f(x):=f(x;s,Y,n,k)$ exists, such that $f\in\bigtriangleup^{(1)}(Y)\bigcap{\Bbb C}^{(1)}$ and $$ E_n^{(1)}(f;Y)>B_Yn^{\frac k3 -1}\frac 1n\omega_k\left(f';\frac 1n\right), $$ where $B_Y=$const, depending only on $Y$ and $k$; $\omega_k$ is the modulus of smoothness of order $k$, of $f$.