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 alpha > 0 and a continuous function f, which is either convex (s = 0) or changes convexity at a finite collection Y(s) = {y(i)}(i=1)(s) of points y(i) is an element of (-1, 1), sup{n(alpha) E(n)((2)) (f, Y(s)): n >= N*} <= c(alpha, s) sup{n(alpha) E(n)((f)): n >= 1}, where En(f) and E(n)((2)) (f, Y(s)) denote, respectively, the degrees of the best unconstrained and (co)convex approximations and c.(alpha, s) is a constant depending only on alpha and s: Moreover, it has been shown that N* may be chosen to be 1 for s = 0 or s = 1; alpha not equal 4; and that it must depend on Y(s) and alpha for s = 1, alpha = 4 or s >= 2. In Part II of the paper, we show that a more general inequality sup{n(alpha) E(n)((2)) (f, Y(s)): n >= N*} <= c(alpha, N, s) sup{n(alpha) E(n)(f): n >= N}, is valid, where, depending on the triple (alpha, N, s) the number N* may depend on alpha, N, Y(s) and f or be independent of these parameters.