The bivariate Bernstein-Schoenberg operatorV T of degreem, introduced in , is a spline approximation operator that generalizes the Bernstein polynomial operatorB m . It is shown here that for a convex functionf,f≤V T (f)≤B m (f). This result is then used to show that for a twice differentiable functiong, the asymptotic error limm(V T (g)-g) depends only on the asymptotic error for quadratic polynomials. The latter is evaluated explicitly in the special circumstances thatV T is, in a sense, asymptotically close toB m .
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C. de Boor (1976):Splines as linear combinations of B-splines: A Survey. In: Approximation theory II (G. G. Lorentz, C. K. Chui, L. L. Schumaker, eds.). New York: Academic Press, pp. 1–47.
E. Cohen, T. Lyche, R. Riesenfeld (1984):Discrete box splines and refinement algorithms. Computer Aided Geometric Design,1:131–148.
W. Dahmen, C. A. Micchelli (1983):Recent progress in multivariate splines. In: Approximation Theory IV (C. D. Chui, L. L. Schumaker, J. D. Ward, eds.). New York: Academic Press, pp. 17–121.
W. Dahmen, C. A. Micchelli (1984):Subdivision algorithms for the generation of box spline surfaces. Computer Aided Geometric Design,1:115–129.
T. N. T. Goodman, S. L. Lee (1981):Spline approximation operators of Bernstein-Schoenberg type in one and two variables. J. Approx. Theory,33:248–263.
G. G. Lorentz (1953): Bernstein Polynomials. Toronto: Univ. of Toronto Press.
L. L. Schumaker (1973):Constructive aspects of discrete polynomial spline functions. In: Approximation Theory (G. G. Lorentz, ed.). New York: Academic Press, pp. 469–476.
Communicated by Klaus Höllig.
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Goodman, T.N.T. Some properties of bivariate Bernstein-Schoenberg operators. Constr. Approx 3, 123–130 (1987). https://doi.org/10.1007/BF01890558
Key words and phrases
- Bernstein-Schoenberg operator
- Bernstein polynomial
- Asymptotic error