Pointwise estimation of comonotone approximation
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.
English version (Springer): Ukrainian Mathematical Journal 46 (1994), no. 11, pp 1620-1626.
Citation Example: Dzyubenko H. A. Pointwise estimation of comonotone approximation // Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1467–1472.