Degree of nearly comonotone approximation of periodic functions

Abstract

Let a [Formula: see text]-periodic function [Formula: see text] change its monotonicity at a finitely even number of points [Formula: see text] of the period. The degree of approximation of this [Formula: see text] by trigonometric polynomials which are comonotone with it, i.e. that change their monotonicity exactly at the points [Formula: see text] where [Formula: see text] does, is restricted by [Formula: see text] (with a constant depending on the location of these [Formula: see text]). Recently, we proved that relaxing the comonotonicity requirement in intervals of length proportional to [Formula: see text] about the points [Formula: see text] (so-called nearly comonotone approximation) allows the polynomials to achieve the approximation rate of [Formula: see text]. By constructing a counterexample, we show that even with the relaxation of the comonotonicity requirement for the polynomials on sets with measures approaching [Formula: see text], [Formula: see text] is not reachable.