论文信息 - Variation diminishing properties of Bernstein polynomials on triangles

Variation diminishing properties of Bernstein polynomials on triangles

Abstract We show that if f is any function on a triangle T, then the variation of the gradient of the nth Bernstein polynomial Bn(f) on T cannot exceed the variation of the gradient of the nth Bezier net \ tf n on T. We deduce that if f is in C2(T), then the variation of the gradient of Bn(f) on T is bounded by a given constant (depending on T) times the variation of the gradient of f. It is further shown that the variation of Bn(f) on T is bounded by 2n (n + 1) times the variation of \ tf n on T.

Tim N. T. Goodman | T. Goodman

[1] Philip J. Davis,et al. The convexity of Bernstein polynomials over triangles , 1984 .

[2] George Polya,et al. Remarks on de la Vallée Poussin means and convex conformal maps of the circle. , 1958 .

[3] W. Schempp,et al. Multivariate Approximation Theory IV , 1989 .

[4] G. Farin,et al. Bézier polynomials over triangles and the construction of piecewise Cr polynomials , 1980 .

[5] C. Micchelli. On a numerically efficient method for computing multivariate B-splines , 1979 .