APPROXIMATION OF MONOTONE-FUNCTIONS BY MONOTONE POLYNOMIALS

The following theorem is proved for the case k + r > 2. Theorem. If k, r is-an-element-of N, I:= [-1, 1], and the function f = f(x) is nondecreasing on I and has r continuous derivatives, then for each positive integer n greater-than-or-equal-to r + k - 1 there is an algebraic polynomial P(n) = P(n) (x) of degree less-than-or-equal-to n that is nondecreasing on I and such that for all x is-an-element-of I \f(x) - P(n)(x)\ less-than-or-equal-to c(1/n2 + square-root 1-x2/n)(r)omega(k)(f(r); 1/n2 + square-root 1-x2/n), c = c(r, k), where omega(k)(f(r);t) is the kth-order modulus of continuity of the function f(r) = f(pr)(x) Bibliography: 16 titles.