Copositive polynomial and spline approximation (Q1890564)

Here the authors prove that if a function \(f \in \mathbb{C} [0.1]\) changes sign finitely many times then for any \(n\) large enough the degree of copositive approximation to \(f\) by quadratic splines with \(n-1\) equally spaced knots can be estimated by a constant multiple of \(\omega_2 (f,1/n)\). Somewhat a similar theorem on copositive polynomial approximation is also given for \(f \in C^1 [0.1]\) which improves upon the results of two of these authors: (1) \textit{D. Leviatan} [Proc. Am. Math. Soc. 88, 101-105 (1983; Zbl 0532.41007)] and (2) \textit{X. Yu} [Chin. Ann. Math., Ser. B 10, No. 3, 409-415 (1989; Zbl 0683.41013)]. Couple of applications of these theorems are also given, including an alternative proof of one of their earlier theorems [J. Anal 1, 85-90 (1993; Zbl 0770.41008)].