Are the Degrees of Best (Co)Convex and Unconstrained Polynomial Approximation the Same?

Analysis Seminar
Wednesday, January 26, 2011 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 005
Prof. Dany Leviatan – Tel Aviv University
Doron Lubinsky
Let C[-1, 1] be the space of continuous functions on [-1, 1], and denote by \Delta^2 the set of convex functions f \in C[-1, 1]. Also, let E_n(f) and En^{(2)}_n(f) denote the degrees of best unconstrained and convex approximation of f \in \Delta^2 by algebraic polynomials of degree < n, respectively. Clearly, E_n(f) \le E^{(2)}_n (f), and Lorentz and Zeller proved that the opposite inequality E^{(2)}_n(f) \le CE_n(f) is invalid even with the constant C = C(f) which depends on the function f \in \Delta^2. We prove, for every \alpha > 0 and function f \in \Delta^2, that sup{n^\alpha E^{(2)}_n(f) : n \ge 1} \le c(\alpha)sup{n^\alpha E_n(f): n \ge 1}, where c(\alpha) is a constant depending only on \alpha. Validity of similar results for the class of piecewise convex functions having s convexity changes inside (-1,1) is also investigated. It turns out that there are substantial differences between the cases s \le 1 and s \ge 2.