Piecewise monotone pointwise approximation

Let r, k, s be three integers such that r > 1, For All k, or r = 1, k less than or equal to 3. We prove the following: Proposition. Let Y := {y(i)}(i=t)(s) be a fixed collection of distinct points y(i) is an element of (-1, 1) and Pi(x) := (x - y(t)). (.)...(.) (x - y(s)). Let I := [-1, 1]. If f is an element of C-(r )(I) and f'(x)Pi(x) greater than or equal to 0, x is an element of I, then for each integer n greater than or equal to k + r - 1 there is an algebraic polynomial P-n = P-n(x) of degree less than or equal to n such that P-n'(x)Pi(x) greater than or equal to 0 and (1) \f(x) - P-n(x)\ less than or equal to B(1/n(2) + 1/n root 1-x(2))(r) omega(k) (f((r)); 1/n(2) + 1/n root 1-x(2)) for all x is an element of I, where omega(k) (f((r)); t) is the modulus of smoothness of the kth order of the function f((r)) and B is a constant depending only on r, k, and Y. If s = 1. the constant B does nor depend on Y except in the case (r = 1, k = 3). In addition it is shown that (l) does not hold for r = 1, k > 3.