Bernstein Expansion with Non-Negative Coefficients and Corner Zeros
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.
Bernstein expansionpolynomials with non-negative coefficientsd-multiple form
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