Abstract

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-1ωr-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].

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1137/0733020

Keywords

Computer algorithm, Degree of copositive approximation, Polynomial approximation, Spline approximation, Splines with equally spaced knots

Publication Date

1-1-1996

Journal Title

SIAM Journal on Numerical Analysis

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