带零点的非负系数Bernstein展开

Bernstein Expansion with Non-Negative Coefficients and Corner Zeros

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.