Copositive Polynomial and Spline Approximation

We prove that if a function f ∈ C [0, 1] changes sign finitely many times, then for any n large enough the degree of copositive approximation to f by quadratic spliners with n-1 equally spaced knots can be estimated by Cω₂(f, 1/n), where C is an absolute constant. We also show that the degree of copositive polynomial approximation to f ∈ C¹[0, 1] can be estimated by Cn^-1ω_r(f′, 1/n), where the constant C depends only on the number and position of the points of sign change. This improves the results of Leviatan (1983, Proc. Amer. Math. Soc.88, 101-105) and Yu (1989, Chinese Ann. Math.10, 409-415), who assumed that for some r ≥ 1, f ∈ C^r[0, 1]. In addition, the estimates involved Cn^-rω(f^r, 1/n) and the constant C dependended on the behavior of f in the neighborhood of those points. One application of the results is a new proof to our previous ω₂ estimate of the degree of copositive polynomia approximation of f ∈ C[0, 1], and another shows that the degree of copositive spline approximation cannot reach ω₄, just as in the case of polynomials.