On Positive and Copositive Polynomial and Spline

For a function f 2 L p ?1; 1], 0 < p < 1, with nitely many sign changes, we construct a sequence of polynomials P n 2 n which are copositive with f and such that kf ? P n k p C! ' (f; (n + 1) ?1) p , where ! ' (f; t) p denotes the Ditzian-Totik modulus of continuity in L p metric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, then ! ' can not be replaced by ! 2 if 1 < p < 1. In fact, we show that even for positive approximation and all 0 < p < 1 the same conclusion is true. Also, some results for (co)positive spline approximation, exact in the same sense, are obtained.