Twice Continuously Differentiable Extension of an Observation-Rate Family

Let $l$ and $\tilde{l}$ be natural numbers with $l\ge2$ and $\tilde{l}\ge1$, let $\tilde{\beta}$ be an observation-rate family on $l$ states with $\tilde{l}$ observation channels and rate bound $\tilde{B}$, let $\Delta^l\subset\mathbb{R}^l$ be the probability simplex, and let $\tilde{K}\ge0$ be a real number. Points of Euclidean space $\mathbb{R}^{l}$ are written $\Sigma$, with coordinates $\Sigma^1,\dots,\Sigma^l$. For a real-valued function $f$ on an open subset of $\mathbb{R}^{l}$ and indices $\gamma,\delta\in\{1,\dots,l\}$ we write $\partial_\gamma f$ for the partial derivative of $f$ with respect to the $\gamma$th variable, that is, with respect to the coordinate $\Sigma^\gamma$, which is unambiguous wherever it exists by Uniqueness of the Partial Derivative on a Euclidean Open Set, and $\partial_\delta\partial_\gamma f$ for the iterated partial derivative in the sense of clause 4 of that definition, namely $\partial_\delta$ applied to the function $\partial_\gamma f$.

A pair $(\tilde{U},\bar{\tilde{\beta}})$ is a twice continuously differentiable extension of the observation-rate family $\tilde{\beta}$ with derivative bound $\tilde{K}$ if it consists of an open set $\tilde{U}\subseteq\mathbb{R}^l$ with $\Delta^l\subset\tilde{U}$ and a family of functions $\bar{\tilde{\beta}}(\sigma,\upsilon,\cdot):\tilde{U}\to\mathbb{R}$, indexed by the pairs $(\sigma,\upsilon)$ with $\sigma\in\{1,\dots,l\}$ and $\upsilon\in\{1,\dots,\tilde{l}\}$, such that for every such pair:

1. (Extension.) $\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma)=\tilde{\beta}(\sigma,\upsilon,\Sigma)$ for all $\Sigma\in\Delta^l$.

2. (Regularity.) $\bar{\tilde{\beta}}(\sigma,\upsilon,\cdot)$ is of class $C^2$ on the open set $\tilde{U}$.

3. (Derivative bounds.) $|\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma)|\le\tilde{K}$ and $|\partial_\delta\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma)|\le\tilde{K}$ for all $\gamma,\delta\in\{1,\dots,l\}$ and all $\Sigma\in\tilde{U}$.

4. (Uniform continuity of second derivatives.) For every real $\varepsilon>0$ there is a real $\delta^\circ>0$ such that $|\partial_\delta\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma)-\partial_\delta\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma')|\le\varepsilon$ for all $\gamma,\delta\in\{1,\dots,l\}$, all pairs $(\sigma,\upsilon)$, and all $\Sigma,\Sigma'\in\tilde{U}$ whose Euclidean distance satisfies $d(\Sigma,\Sigma')\le\delta^\circ$.