带零点的非负系数Bernstein展开
Bernstein Expansion with Non-Negative Coefficients and Corner Zeros
徐嘉 1姚勇 2秦小林2
作者信息
- 1. 西南民族大学数学学院,成都 610041
- 2. 中国科学院成都计算机应用研究所,成都 610213
- 折叠
摘要
一个熟知的结论是说,如果多项式f ∈ R[x]在单位方体In=[0,1]n上的值是严格正的,则f可以用带正系数的Bernstein基表示.但是,当f在单位方体In上存在零点时,上述结论不再成立.文章研究了 f带有角零点(单位方体的顶点)的情况,找到了在仅有角零点的假设下f存在非负系数的Bernstein展开需要满足的充分必要条件.文章通过引入d-多重型,将问题转化为d-多重型的系数问题.
Abstract
It is well known that if a polynomial f ∈ R[x]is strictly positive on the unit box In=[0,1]n,then f can be written as a Bernstein expansion with strictly positive coefficients.However,the above conclusion no longer holds if f has zero points on In.In this paper,we consider the case of f with corner zero points(vertices of In).As a result,we provide a necessary and sufficient condition for the Bernstein expansion of f with non-negative coefficients when the zeros are only at corner of In.Our method relies on constructing the d-multiple form whose terms are homogeneous,the problem is transformed into the verification of coefficients of a given d-multiple form.
关键词
Bernstein展开/非负系数多项式/d-多重型Key words
Bernstein expansion/polynomials with non-negative coefficients/d-multiple form引用本文复制引用
基金项目
中央高校一般项目(2020NYB40)
四川省科技计划(2019ZDZX0006)
四川省科技计划(2020YFQ0056)
出版年
2024
NETL
NSTL
万方数据