Degree of nearly comonotone approximation of periodic functions

TL;DR

This paper investigates how well periodic functions with finitely many monotonicity changes can be approximated by polynomials that nearly preserve these monotonicity properties, revealing limitations on the approximation rate.

Contribution

It introduces the concept of nearly comonotone approximation and establishes bounds on the approximation rate, showing the impossibility of reaching certain rates even with minimal relaxation.

Findings

Nearly comonotone approximation achieves a rate of omega_3.

Exact comonotone approximation is restricted by omega_2.

Relaxing comonotonicity on small measure sets does not allow reaching omega_4.

Abstract

Let a $2\pi$-periodic function $f\in\Bbb C$ changes its monotonicity at a finitely even number of points $y_i$ of the period. The degree of approximation of this $f$ by trigonometric polynomials which are comonotone with it, i.e. that change their monotonicity exactly at the points $y_i$ where $f$ does, is restricted by $\omega_2(f,\pi/n)$ (with a constant depending on the location of these $y_i$). Recently, we proved that relaxing the comonotonicity requirement in intervals of length proportional to $\pi/n$ about the points $y_i$ (so called nearly comonotone approximation) allows the polynomials to achieve the approximation rate of $\omega_3$. By constructing a counterexample, we show here that even with the relaxation of the requirement of comonotonicity for the polynomials on sets with measures approaching $0$ (no matter how slowly or how fast) $\omega_4$ is not reachable.