This paper is concerned with Bernstein polynomials on k-simploids by which we mean a crossproduct of k lower dimensional simplices. Specifically, we show that if the Bernstein polynomials ofa given function f on a k-s...This paper is concerned with Bernstein polynomials on k-simploids by which we mean a crossproduct of k lower dimensional simplices. Specifically, we show that if the Bernstein polynomials ofa given function f on a k-simploid form a decreasing sequence then f+l, where l is any correspondingtensor product of affine functions. achieves its maximum on the boundary of the k-simploid. Thisextends recent results in [1] for bivariate Bernstein polynomials on triangles. Unlike the approachused in [1] our approach is based on semigroup techniques and the maximum principle for secondorder elliptic operators. Furthermore, we derive analogous results for cube spline surfaces.展开更多
基金This work was partially supported by NATO Grant No.DJ RG 639/84
文摘This paper is concerned with Bernstein polynomials on k-simploids by which we mean a crossproduct of k lower dimensional simplices. Specifically, we show that if the Bernstein polynomials ofa given function f on a k-simploid form a decreasing sequence then f+l, where l is any correspondingtensor product of affine functions. achieves its maximum on the boundary of the k-simploid. Thisextends recent results in [1] for bivariate Bernstein polynomials on triangles. Unlike the approachused in [1] our approach is based on semigroup techniques and the maximum principle for secondorder elliptic operators. Furthermore, we derive analogous results for cube spline surfaces.
