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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Online Access: | https://arxiv.org/abs/2208.07349 |
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| _version_ | 1866916146943361024 |
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| author | Richter, Stefan Sautel, Jesse |
| author_facet | Richter, Stefan Sautel, Jesse |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_07349 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Multivariable versions of a lemma of Kaluza's Richter, Stefan Sautel, Jesse Functional Analysis 46E22 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. |
| title | Multivariable versions of a lemma of Kaluza's |
| topic | Functional Analysis 46E22 |
| url | https://arxiv.org/abs/2208.07349 |