Multivariable versions of a lemma of Kaluza's

Abstract

Let $d\in \mathbb{N}$ and $f(z)={\sum }_{\alpha \in {\mathbb{N}}_0^d}{c}_{\alpha }{z}^{\alpha }$ be a convergent multivariable power series in $z=(z_1,\dots ,z_d)$. In this paper we present two conditions on the positive coefficients $c_{\alpha }$ which imply that $f(z)=\frac{1}{1-{\sum }_{\alpha \in {\mathbb{N}}_0^d}{q}_{\alpha }{z}^{\alpha }}$ for non-negative coefficients $q_{\alpha }$. 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 $$f(z)={\int }_{[0,1{]}^d}\frac{1}{1-{\sum }_{j=1}^d{t}_j{z}_j}d\mu (t)$$ satisfy one of our conditions, whenever $d\mu (t)=d{\mu }_1(t_1)\times \cdots \times d{\mu }_d(t_d)$ is a product of probability measures ${\mu }_j$ on $[0,1]$. Our results have applications to the theory of Nevanlinna-Pick kernels.

References in corpus (3)

• Regenerative random permutations of integers
• Hilbert spaces of analytic functions with a contractive backward shift
• Radially weighted Besov spaces and the Pick property