Pointwise estimation of comonotone approximation

Abstract

We prove that, for a continuous function f(x) defined on the interval [−1,1] and having finitely many intervals where it is either nonincreasing or nondecreasing, one can always find a sequence of polynomials P _n (x) with the same local properties of monotonicity as the function f(x) and such that ¦f(x)−P _n (x) ¦≤Cω₂(f;n⁻²+n ⁻¹√1−x ²), whereC is a constant that depends on the length of the smallest interval.