Copositive Polynomial and Spline Approximation

✍ Scribed by Y.K. Hu; D. Leviatan; X.M. Yu

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 439 KB
• Volume: 80
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1995.1015

✦ Synopsis

We 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 spliners with (n-1) equally spaced knots can be estimated by (C \omega_{2}(f, 1 / n)), where (C) is an absolute constant. We also show that the degree of copositive polynomial approximation to (f \in \mathbf{C}^{1}[0,1]) can be estimated by (C n^{-1} \omega,\left(f^{\prime}, 1 / n\right)), 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 \geqslant 1), (f \in C^{\prime}[0,1]). In addition, the estimates involved (\mathrm{Cn}^{-r}\left(\rho\left(f^{(r)}, 1 / n\right)\right.) 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 (\omega_{2}) estimate of the degree of copositive polynomia approximation of (f \in \mathbf{C}[0,1]), and another shows that the degree of copositive spline approximation cannot reach (\omega_{4}), just as in the case of polynomials. 1995 Academic Press. Inc.