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Multivariable versions of a lemma of Kaluza's

Stefan Richter, Jesse Sautel

Abstract

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.

Multivariable versions of a lemma of Kaluza's

Abstract

Let and be a convergent multivariable power series in . In this paper we present two conditions on the positive coefficients which imply that for non-negative coefficients . If , then both of our results reduce to a lemma of Kaluza's. For we present examples to show that our two conditions are independent of one another. It turns out that functions of the type satisfy one of our conditions, whenever is a product of probability measures on . Our results have applications to the theory of Nevanlinna-Pick kernels.
Paper Structure (7 sections, 14 theorems, 79 equations)

This paper contains 7 sections, 14 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on graded monoids
  3. Kaluza's lemma for strongly graded Monoids
  4. The proof of Theorem \ref{['1']}
  5. The proof of Theorem \ref{['2']}
  6. Examples for Theorem \ref{['2']}
  7. Applications to reproducing kernels

Key Result

Theorem 1.1

(Kaluza's Lemma) Let $M>0$ and let $\{c_n\}_{n \geq 0}$ be a sequence of positive real numbers with $c_0=1$. Define a sequence of real numbers $\{r_n\}_{n \ge 1}$ by $r_n : = \dfrac{c_n}{c_{n-1}}$ for each $n\in \mathbb N$. If $\{r_n\}_{n \geq 1}$ is a non-decreasing sequence that is bounded above b are all non-negative.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • proof
  • Lemma 2.5
  • Lemma 2.6
  • ...and 22 more