Pointwise Estimation of the Almost Copositive Approximation of Continuous Functions by Algebraic Polynomials

In the case where a function f continuous on a segment changes its sign at s points y (i) : - 1 < y (s) < y (s-1) < ... < y (1) < 1, for any n a a"center dot greater than a certain constant N(k, y (i) ) that depends only on k a a"center dot and we determine an algebraic polynomial P (n) of degree ae<currency> n such that: P (n) has the same sign as f everywhere except, possibly, small neighborhoods of the points and |f(x) - P (n) (x)| ae<currency> c(k, s)omega (k) (f, rho (n) (x)),aEuro integral x a [-1, 1], where c(k, s) is a constant that depends only on k and s and omega (k) (f, center dot) is the modulus of continuity of the function f of order k.