Comonotone Jackson's inequality

Let 2s points y(i) = -pi less than or equal to y(2s) < ... < y(1) < pi be given. Using these points, we define the points y(i) for all integer indices i by the equality y(i) = y(i+2s) + 2 pi. We shall write f is an element of Delta((1))(Y) if f is a 2 pi-periodic continuous function and f does not decrease on [y(i), y(i-1)], if i is odd; and f does not increase on [y(i), y(i-1)], if i is even. In this article the following Theorem 1-the comonotone analogue of Jackson's inequality-is proved. THEOREM 1. If f is an element of Delta((1))(Y), then for each nonnegative integer n there is a trigonometric polynomial tau(n)(x) of order less than or equal to n such that tau(n) is an element of Delta((1))(Y), mid \f(x) - pi(n)(x)\ less than or equal to c(s) omega(f; 1/(n + 1)), x is an element of R, where omega(f; t) is the modulus of continuity of f, c(s) = const. Depending only on s. (C) 1999 Academic Press.