Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II

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),

$$ \sup \left\{ {{n^\alpha }E_n^{(2)}\left( {f,{Y_s}} \right):n \geqslant \mathcal{N}*} \right\} \leqslant c\left( {\alpha, s} \right)\sup \left\{ {{n^\alpha }{E_n}(f):n \geqslant 1} \right\}, $$

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 $\mathcal{N}*$ 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

$$ \sup \left\{ {{n^\alpha }E_n^{(2)}\left( {f,{Y_s}} \right):n \geqslant \mathcal{N}*} \right\} \leqslant c\left( {\alpha, \mathcal{N},s} \right)\sup \left\{ {{n^\alpha }{E_n}(f):n \geqslant \mathcal{N}} \right\}, $$

is valid, where, depending on the triple $\left( {\alpha, \mathcal{N},s} \right)$ the number $\mathcal{N}*$ may depend on $\alpha, \mathcal{N},{Y_s}$, and f or be independent of these parameters.