AI 中文总结
该研究证明了圆周上的n次Bernstein算子可保持正定性，通过显式系数公式和两区间正性论证完成证明，并指出朴素仿射Bernstein算子在二维球面S²上无法保持正定性。
AI 中文摘要
我们证明，对每个n≥1，区间[0,π]上的n次Bernstein算子可保持圆周S¹上的正定性。等价地，若[0,π]上的连续函数定义了S¹上的正定性各向同性核，则其任意固定次数的Bernstein多项式逼近也满足该性质。该证明将问题转化为Bernstein像Q_{n,m}=Bₙ[cos(mx)]的余弦系数非负性，我们通过显式系数公式与两区间正性论证完成证明。我们还讨论了高维球面的类似情况，发现朴素仿射Bernstein算子在S²上已无法保持正定性锥。
英文摘要
We prove that, for every $n\ge1$, the degree-$n$ Bernstein operator on $[0,π]$ preserves positive-definiteness on the circle $S^1$. Equivalently, if a continuous function on $[0,π]$ defines a positive-definite isotropic kernel on $S^1$, then its Bernstein polynomial approximation of any fixed degree does as well. The proof reduces the problem to the nonnegativity of the cosine coefficients of the Bernstein images $Q_{n,m}=B_n[\cos(mx)]$, which we prove using an explicit coefficient formula and a two-regime positivity argument. We also discuss the higher-dimensional sphere analogue and show that the naive affine Bernstein operator fails to preserve the positive-definite cone already on $S^2$.
Comments15 pages
