The Degree of Copositive Approximation and a Computer Algorithm

Abstract

The main results are as follows. (1) Let $f \in {\bf C}[0,1]$ 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 $C\omega _3 (f,{1 / n})$ for n large enough. This rate is the best in the sense that $\omega _3 $ cannot be replaced by $\omega _4 $. (2) An algorithm is developed basedon the proof. (3) The first result above holds for a copositive polynomial approximation of f. (4) If $f \in C^1 [0,1]$, then the degree of approximation by copositive splines of order r is bounded by $Cn^{ - 1} \omega _{r - 1} (f',{1 / n})$. The results on $f \in {\bf C}[0,1]$ 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].