Positive and copositive spline approximation in Lp[0, 1]

The order of positive and copositive spline approximation in the Lp-norm, 1 ≤ p < ∞, is studied; the main results are 1. (1) the error of positive approximation by splines is bounded by Cω2(f, 1/n)p if f has a nonnegative extension; 2. (2) the order deteriorates to ω1 if f does not have such an extension; 3. (3) the error of copositive spline approximation is bounded by Cω(f, 1/n)p; 4. (4) if f is also continuous, the error in (3) can be estimated in terms of the third τ-modulus τ3(f, 1/n)p. All constants in the error bounds are absolute.