On positive and copositive polynomial and spline approximation in L_p[-1, 1], 0 < p < ∞

For a function f ∈ L_p[-1, 1], 0 < p < ∞, with finitely many sign changes, we construct a sequence of polynomials P_n ∈ ∏_n which are copositive with f and such that ∥f-P_n∥_p ≤ Cω_φ(f, (n + 1)¹)_p, where ω_φ(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 ω_φ cannot be replaced by ω² if 1 < p < ∞. In fact, we show that even for positive approximation and all 0 < p < ∞ the same conclusion is true. Also, some results for (co)positive spline approximation, exact in the same sense, are obtained.