On Positive and Copositive Polynomial and Spline Approximation inLp[−1, 1], 0<p<∞

AbstractFor a functionf∈Lp[−1, 1], 0<p<∞, with finitely many sign changes, we construct a sequence of polynomialsPn∈Πnwhich are copositive withfand such that ‖f−Pn‖p⩽Cωϕ(f, (n+1)−1)p, whereωϕ(f, t)pdenotes the Ditzian–Totik modulus of continuity inLpmetric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, thenωϕcannot be replaced byω2if 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