Convex Polynomial Approximation in the Uniform Norm: Conclusion

Abstract

Abstract. Estimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some interesting open questions. In this paper we give final answers to all those open problems. We are able to say, for each r-th differentiable convex function, whether or not its degree of convex polynomial approximation in the uniform norm may be estimated by a Jackson-type estimate involving the weighted Ditzian-Totik kth modulus of smoothness, and how the constants in this estimate behave. It turns out that for some pairs (k, r) we have such estimate with constants depending only on these parameters. For other pairs the estimate is valid, but only with constants that depend on the function being approximated, while there are pairs for which the Jackson-type estimate is, in general, invalid.

Coconvex approximation in the uniform norm: the final frontier

“…To this end, for 0 < ε < 1/2, set g ε := g 6 (x + ε). Then g ε ∈ C 6 ϕ , g (6) ε ϕ 6 < 1, g (3) ε (x) > 0, and g (5) ε (x) > 0, x ∈ [−1, 1], and finally M ε := g ε 2M 6 . Now we take ε so small that g (3) ε (−1) > m 6 (A + 2M 6 ),…”

“…Let f ∈ C 2 ϕ , n 4, and let the polynomials p 1 and p n of degree 4 be such that p (3.24) f − p n In cn −2 ω ϕ 3,2 (f , 1/n). We end this section by recalling that for f ∈ C r ϕ , it was shown in [6] (see inequalities (3.4) and (3.5) there), that…”

“…Despite the simplicity of its statement, Theorem 1.1 remained unresolved for quite some time, and while its particular cases have been known from as early as 1986, in its final form it appeared only very recently in [6] where the case for α = 4 (which surprisingly turned out to be the most evasive case of all) has been proved (see [6] for more details).…”

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

“…(f ) = E 1 (f ) 1 N α n −α .For α ∈[3,5], Theorem 1.1 immediately follows from (4.1) and the following lemma (see[3, Theorem 2] and [5, lines 25 in the table on p. 5]). Acta Mathematica Hungarica 123, 2009 &&…”

“…On the other hand, it follows from the well known Dzyadyk type inequality (see, e.g., [5,Lemma 5.2]) that for any polynomial Q m ∈ P m ,…”