Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II - Kopotun, K. A.; Leviatan, D.; Shevchuk, I. A. (Ukrains’kyi Matematychnyi Zhurnal, 2010)
In Part I of the paper, we have proved that, for every α > 0 and a continuous function f, which is either convex (s = 0) or changes convexity at a finite collection Y_s = {y_i}^s_i = 1 of points y_i ∈ (-1, 1), {n^αE^(2)_n(f,Y_s): n ≥ N^*} ≤ c(α,s) {n^αE_n(f): n ≥ 1 }, where E_n (f) and E^(2)_n (f, Y_s) denote, respectively, the degrees of the best unconstrained and (co)convex approximations and c(α, s) is a constant depending only on α and s. Moreover, it has been shown that N^∗ may be chosen to be 1 for s = 0 or s = 1, α ≠ 4, and that it must depend on Y_s and α for s = 1, α = 4 or s ≥ 2. In Part II of the paper, we show that a more general inequality {n^αE^(2)_n(f,Y_s): n ≥ N^*} ≤ c(α, N, s) {n^αE_n(f): n ≥ N }, is valid, where, depending on the triple (α,N,s) the number N^∗ may depend on α,N,Y_s, and f or be independent of these parameters.