论文标题
基督徒功能的瓦解
A disintegration of the Christoffel function
论文作者
论文摘要
我们表明，基督徒函数（CF）将（或可以瓦解）分解为两个基督教函数的乘积，一个与边际函数相关，另一个与条件分布相关，本着“崩解的CF的精神，是CFS的分解”。在证明中，人们使用了一个明显被忽视的属性（但本身有趣），该属性表明，任何平方的多项式总和是某种线性形式的基督徒函数（在单变量的情况下具有代表度量）。对于基本的半代数集合的多项式凸锥也是如此。对CF的这种解释建立了多项式优化和正交多项式之间的另一个桥梁。
We show that the Christoffel function (CF) factorizes (or can be disintegrated) as the product of two Christoffel functions, one associated with the marginal and the another related to the conditional distribution, in the spirit of "the CF of the disintegration is the disintegration of the CFs". In the proof one uses an apparently overlooked property (but interesting in its own) which states that any sum-of-squares polynomial is the Christoffel function of some linear form (with a representing measure in the univariate case). The same is true for the convex cone of polynomials that are positive on a basic semi-algebraic set. This interpretation of the CF establishes another bridge between polynomials optimization and orthogonal polynomials.
