Pointwise estimation of comonotone approximation - Dzyubenko, H. A. (Ukrains’kyi Matematychnyi Zhurnal, 1994)
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.