论文标题
Kaluza的引理的多变量版本
Multivariable versions of a lemma of Kaluza's
论文作者
论文摘要
令$ d \ in \ mathbb {n} $和$ f（z）= \ sum_ {α\ in \ mathbb {n} _0^d}c_αz^α$是$ z =（z_1，\ dots，z_d）$中的收敛性多变量功率系列。在本文中，我们介绍了两个条件$c_α$，这意味着$ f（z）= \ frac {1} {1- \ sum_ {α\ in \ mathbb {n} n} _0^d}q_αz^α} $ forn-ne-ne-negative un-negative系数$q_α$ $q_α$。如果$ d = 1 $，那么我们的两个结果都会减少到卡鲁扎的引理。对于$ d> 1 $，我们提出示例，以表明我们的两个条件彼此独立。事实证明，类型$$ f（z）= \ int _ {[0,1]^d} \ frac {1} {1} {1- \ sum_ {j = 1}^d t_j z_j z_j} d t_j z_j}dμ（t）$ $每当我们的条件下，每当$ d $dμ（t）=dμ_1（t _1（t _1）$ d \ d \ d \ d d d d d y d d d d y d d d d y d d d d y d d d d d d d d d d d d ddμ_1（t _1）是$ [0,1] $上的概率测量$μ_j$的产物。 我们的结果适用于Nevanlinna-Pick核的理论。
Let $d\in \mathbb{N}$ and $f(z)= \sum_{α\in \mathbb{N}_0^d} c_αz^α$ be a convergent multivariable power series in $z=(z_1,\dots,z_d)$. In this paper we present two conditions on the positive coefficients $c_α$ which imply that $f(z)=\frac{1}{1-\sum_{α\in \mathbb{N}_0^d} q_αz^α}$ for non-negative coefficients $q_α$. If $d=1$, then both of our results reduce to a lemma of Kaluza's. For $d>1$ we present examples to show that our two conditions are independent of one another. It turns out that functions of the type $$f(z)= \int_{[0,1]^d} \frac{1}{1-\sum_{j=1}^d t_j z_j} dμ(t)$$ satisfy one of our conditions, whenever $dμ(t) = dμ_1(t_1) \times \dots \times dμ_d(t_d)$ is a product of probability measures $μ_j$ on $[0,1]$. Our results have applications to the theory of Nevanlinna-Pick kernels.
