Degree of nearly comonotone approximation of periodic functions

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.