COPOSITIVE POLYNOMIAL AND SPLINE APPROXIMATION

We prove that if a function f is-an-element-of 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 Comega2(f, 1/n), where C is an absolute constant. We also show that the degree of copositive polynomial approximation to f is-an-element-of C1[0, 1] can be estimated by Cn-1omega(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 greater-than-or-equal-to 1, f is-an-elment-of C(r)[0, 1]. In addition, the estimates involved Cn(-r)omega(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 omega2 estimate of the degree of copositive polynomia approximation of f is-an-element-of C[0, 1], and another shows that the degree of copositive spline approximation cannot reach omega4, just as in the case of polynomials. (C) 1995 Academic Press, Inc.