论文标题
几个变量中的最佳近似值和正交多项式
Optimal approximants and orthogonal polynomials in several variables
论文作者
论文摘要
我们讨论了多变量再现核希尔伯特空间中最佳多项式近似值的概念。特别是，我们分析了在一个变量中不存在的多变量情况下出现的困难，例如，加权空间中最佳近似值和正交多项式之间的更复杂的关系。弱内部函数（最佳近似值都是恒定的）提供了极端情况，即无法从最佳近似值中恢复非平凡的正交多项式。提出了具体的例子来说明一般理论，并用于反驳一些关于几个变量最佳近似值零的自然猜想。
We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants and orthogonal polynomials in weighted spaces. Weakly inner functions, whose optimal approximants are all constant, provide extreme cases where nontrivial orthogonal polynomials cannot be recovered from the optimal approximants. Concrete examples are presented to illustrate the general theory and are used to disprove certain natural conjectures regarding zeros of optimal approximants in several variables.
