The degree of copositive approximation and a computer algorithm

The main results are as follows. (1) Let f ∈ C[0, 1] change its sign a finite number of times; then the degree of copositive approximation of f by splines with n equally spaced knots is bounded by C ω₃(f, 1/n) for n large enough. This rate is the best in the sense that ω₃ cannot be replaced by ω₄. (2) An algorithm is developed based on the proof. (3) The first result above holds for a copositive polynomial approximation of f. (4) If f ∈ C¹[0, 1], then the degree of approximation by copositive splines of order r is bounded by Cn^-1ω_r-1(f′, 1/n). The results on f ∈ C[0, 1] fill a gap left by S. P. Zhou [Israel J. Math., 78 (1992), pp. 75-83], and Y. K. Hu, D. Leviatan, and X. M. Yu [J. Anal., 1 (1993), pp. 85-90; J. Approx. Theory, 80 (1995), pp. 204-218].