带零点的非负系数Bernstein展开  

Bernstein Expansion with Non-Negative Coefficients and Corner Zeros

作　　者：徐嘉 姚勇[2] 秦小林[2] Xu Jia;Yao Yong;QIN Xiaolin(School of Mathematics,Southwest Minzu University,Chengdu 610041;Chengdu Institute of Computer Applications,Chinese Academy of Sciences,Chengdu 610213)

机构地区：[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

出　　处：《系统科学与数学》2024年第5期1282-1291,共10页Journal of Systems Science and Mathematical Sciences

基　　金：中央高校一般项目(2020NYB40);四川省科技计划(2019ZDZX0006,2020YFQ0056)资助课题。

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I^(n)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.It is well known that if a polynomial■is strictly positive on the unit box■then can be written as a Bernstein expansion with strictly positive coefficients.However,the above conclusion no longer holds if f has zero points on I^(n)In this paper,we consider the case of f with corner zero points(vertices ofI^(n)).As a result,we provide a necessary and sufficient condition for the Bernstein expansion of with non-negative coefficients when the zeros are only at corner of I^(n).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 

分 类 号：O174.14[理学—数学]