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

• DOI: 10.1007/s11253-010-0362-2
• Publication Date: 2010-10-07T03:50:21Z
• Authors: K. Kopotun; D. Leviatan; I. A. Shevchuk
• Keywords: Convexity

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.