PhyslibAlpha.Mathematics.Order.PositiveDual.Interpolation

Let $E$ be an ordered real vector space. Suppose we have two finite families of positive linear functionals $\{\theta_i\}_{i=1}^n$ and $\{\theta'_j\}_{j=1}^m$ on $E$ such that their total sums are equal: $$\sum_{i=1}^n \theta_i = \sum_{j=1}^m \theta'_j.$$ Then there exists a "refinement table" of positive linear functionals $\{\tau_{ij}\}_{1 \le i \le n, 1 \le j \le m}$ such that the sum of each row $i$ equals $\theta_i$ and the sum of each column $j$ equals $\theta'_j$: $$\forall i, \sum_{j=1}^m \tau_{ij} = \theta_i \quad \text{and} \quad \forall j, \sum_{i=1}^n \tau_{ij} = \theta'_j.$$

Given an ordered real vector space $E$ with an order unit $u$, let $v = (a, c)$ be a pair of finite families of elements $a_i \in E$ (for $i \in \{1, \dots, n\}$) and $c_j \in E$ (for $j \in \{1, \dots, m\}$). The set of slack values is the set of all $\mu \in \mathbb{R}$ such that there exists an element $g \in E$ satisfying $a_i - g \le \mu u$ for all $i$ and $g + c_j \le \mu u$ for all $j$.

Given an ordered real vector space $E$ with an order unit $u$, let $v = (a, c)$ be a pair of finite families of elements $a = (a_1, \dots, a_n)$ and $c = (c_1, \dots, c_m)$ in $E$. The interpolation gauge of $v$ is the infimum of the set of all real numbers $\mu$ for which there exists an interpolant $g \in E$ satisfying the inequalities $a_i - g \le \mu u$ for all $i \in \{1, \dots, n\}$ and $g + c_j \le \mu u$ for all $j \in \{1, \dots, m\}$.

Let $E$ be an ordered real vector space with an order unit $u$. For any pair of finite families of elements $a_i \in E$ (for $i \in \{1, \dots, n\}$) and $c_j \in E$ (for $j \in \{1, \dots, m\}$), the set of slack values $\mu \in \mathbb{R}$ such that there exists an element $g \in E$ satisfying $a_i - g \le \mu u$ for all $i$ and $g + c_j \le \mu u$ for all $j$ is non-empty.

Let $E$ be an ordered real vector space with an order unit $u$. For any pair of finite families of elements $a = (a_1, \dots, a_n)$ and $c = (c_1, \dots, c_m)$ in $E$, let $S$ be the set of slack values $\mu \in \mathbb{R}$ such that there exists an element $g \in E$ satisfying $a_i - g \le \mu u$ for all $i \in \{1, \dots, n\}$ and $g + c_j \le \mu u$ for all $j \in \{1, \dots, m\}$. Then the set $S$ is bounded below.

Let $E$ be an ordered real vector space with an order unit $u$. Let $v = (a, c)$ be a pair of finite families of elements $a = (a_1, \dots, a_n)$ and $c = (c_1, \dots, c_m)$ in $E$. If $\mu \in \mathbb{R}$ is a slack value for $v$ (that is, $\mu \in \text{slacks}(u, v)$, meaning there exists an element $g \in E$ such that $a_i - g \le \mu u$ and $g + c_j \le \mu u$ for all $i, j$), then the interpolation gauge of $v$ is less than or equal to $\mu$, i.e., $\text{gauge}(u, v) \le \mu$.

Let $E$ be an ordered real vector space with an order unit $u$. Let $v, w \in (\text{Fin } n \to E) \times (\text{Fin } m \to E)$ be pairs of finite families of elements in $E$. If $\mu \in \mathbb{R}$ is a slack value for $v$ (i.e., $\mu \in \text{slacks}(u, v)$) and $\nu \in \mathbb{R}$ is a slack value for $w$ (i.e., $\nu \in \text{slacks}(u, w)$), then their sum $\mu + \nu$ is a slack value for the component-wise sum of the families $v + w$ (i.e., $\mu + \nu \in \text{slacks}(u, v + w)$).

Let $E$ be an ordered real vector space with an order unit $u$. For any two pairs of finite families of elements $v, w \in E^n \times E^m$, the interpolation gauge satisfies the subadditivity inequality: $$\text{gauge}_u(v + w) \le \text{gauge}_u(v) + \text{gauge}_u(w)$$ where the interpolation gauge $\text{gauge}_u(a, c)$ for families $a = (a_1, \dots, a_n)$ and $c = (c_1, \dots, c_m)$ is defined as the infimum of the set of all real numbers $\mu$ for which there exists an interpolant $g \in E$ satisfying $a_i - g \le \mu u$ and $g + c_j \le \mu u$ for all $i \in \{1, \dots, n\}$ and $j \in \{1, \dots, m\}$.

Let $E$ be an ordered real vector space with an order unit $u$. Let $v = (a, b)$ be a pair consisting of a family of elements $a_i \in E$ (for $i \in \{1, \dots, n\}$) and a family $b_j \in E$ (for $j \in \{1, \dots, m\}$). For any positive real number $c > 0$, the set of slack values $\text{slacks}(u, c \cdot v)$ for the scaled families is equal to the set of slack values $\text{slacks}(u, v)$ scaled by $c$. That is, $\text{slacks}(u, c \cdot v) = \{ c\mu \mid \mu \in \text{slacks}(u, v) \}$, where a slack value $\mu$ is a real number for which there exists an interpolant $g \in E$ satisfying $a_i - g \le \mu u$ and $g + b_j \le \mu u$ for all $i, j$.

Let $E$ be an ordered real vector space with an order unit $u$. Let $v = (a, b)$ be a pair of finite families of elements in $E$, where $a = (a_1, \dots, a_n)$ and $b = (b_1, \dots, b_m)$. For any positive real number $c > 0$, the interpolation gauge of the scaled pair $c \cdot v = (ca, cb)$ is $c$ times the interpolation gauge of $v$, i.e., $$\text{gauge}(u, c \cdot v) = c \cdot \text{gauge}(u, v)$$ where the interpolation gauge $\text{gauge}(u, v)$ is defined as the infimum of the set of all real numbers $\mu$ for which there exists an interpolant $g \in E$ satisfying $a_i - g \le \mu u$ for all $i$ and $g + b_j \le \mu u$ for all $j$.

Let $E$ be an ordered real vector space with an order unit $u$. For the pair of zero families $(a, c)$ where $a_i = 0$ for all $i \in \{1, \dots, n\}$ and $c_j = 0$ for all $j \in \{1, \dots, m\}$, the interpolation gauge is zero, i.e., $\text{gauge}(u, (0, 0)) = 0$.

Let $E$ be an ordered real vector space with an order unit $u$. Let $v = (a, c)$ be a pair of finite families of elements in $E$, where $a = (a_1, \dots, a_n)$ and $c = (c_1, \dots, c_m)$. If every element of both families is non-positive, i.e., $a_i \le 0$ for all $i \in \{1, \dots, n\}$ and $c_j \le 0$ for all $j \in \{1, \dots, m\}$, then the interpolation gauge of $v$ is non-positive: $$\text{gauge}_u(v) \le 0$$ where $\text{gauge}_u(v)$ is defined as the infimum of the set of all real numbers $\mu$ for which there exists an element $g \in E$ satisfying $a_i - g \le \mu u$ and $g + c_j \le \mu u$ for all $i, j$.

Let $E$ be an ordered real vector space with an order unit $u$. Let $t = (a, b)$ be a pair of finite families of elements in $E$, where $a = (a_1, \dots, a_n)$ and $b = (b_1, \dots, b_m)$. For any real number $c \in \mathbb{R}$, the interpolation gauge $\text{gauge}_u(t)$ satisfies the scaling inequality: $$c \cdot \text{gauge}_u(t) \le \text{gauge}_u(c \cdot t)$$ where $c \cdot t$ denotes the scalar multiplication of the families, i.e., $(c a_1, \dots, c a_n, c b_1, \dots, c b_m)$. This inequality holds for both positive and negative scaling factors.

Let $E$ be an ordered real vector space with an order unit $u$. For any pair of finite families of elements $t \in E^n \times E^m$, there exists a linear functional $\Lambda : E^n \times E^m \to \mathbb{R}$ such that for all $v \in E^n \times E^m$, $\Lambda(v) \le \text{gauge}_u(v)$, and $\Lambda(t) = \text{gauge}_u(t)$, where $\text{gauge}_u$ is the interpolation gauge associated with the order unit $u$.

Given a linear functional $\Lambda : E^n \times E^m \to \mathbb{R}$ defined on a product of finite families of elements in a real vector space $E$, and given an index $i \in \{1, \dots, n\}$, the $i$-th lower part is the linear functional on $E$ defined by applying $\Lambda$ to a pair of vectors where the $i$-th entry of the first vector is $x \in E$ and all other entries in both vectors are zero. This represents the contribution to the total value of $\Lambda$ coming from the $i$-th "lower observable."

Given a linear functional $\Lambda : E^n \times E^m \to \mathbb{R}$ on a product of ordered real vector spaces, and an index $j \in \{1, \dots, m\}$, the $j$-th upper part of $\Lambda$ is the linear functional on $E$ that maps an element $x \in E$ to the value of $\Lambda$ evaluated at $(0, \text{single}_j(x))$, where $\text{single}_j(x) = (0, \dots, x, \dots, 0)$ is the vector in $E^m$ with $x$ at the $j$-th position and zero elsewhere. This defines the component of $\Lambda$ that acts on the $j$-th "upper" coordinate in the product space.

Let $E$ be a real vector space and let $\Lambda : E^n \times E^m \to \mathbb{R}$ be a linear functional. For any element $v = (x, y) \in E^n \times E^m$, where $x = (x_1, \dots, x_n) \in E^n$ and $y = (y_1, \dots, y_m) \in E^m$, the value of $\Lambda$ at $v$ is given by the sum of its components: $$\Lambda(v) = \sum_{i=1}^n (\text{lowerPart}_i \, \Lambda)(x_i) + \sum_{j=1}^m (\text{upperPart}_j \, \Lambda)(y_j)$$ where $\text{lowerPart}_i \, \Lambda$ is the linear functional on $E$ corresponding to the $i$-th component of the first factor, and $\text{upperPart}_j \, \Lambda$ is the linear functional on $E$ corresponding to the $j$-th component of the second factor.

Let $E$ be an ordered real vector space and $\Lambda : E^n \times E^m \to \mathbb{R}$ be a linear functional. For any index $i \in \{1, \dots, n\}$ and any element $x \in E$, if $x \ge 0$, then the $i$-th lower part of $\Lambda$ evaluated at $x$ is non-negative, that is, $(\text{lowerPart}_i \, \Lambda)(x) \ge 0$.

Let $E$ be an ordered real vector space and $\Lambda : E^n \times E^m \to \mathbb{R}$ be a linear functional. For any index $j \in \{1, \dots, m\}$ and any element $x \in E$, if $x \ge 0$, then the $j$-th upper part of $\Lambda$ evaluated at $x$ is non-negative, that is, $(\text{upperPart}_j \, \Lambda)(x) \ge 0$.

Let $E$ be an ordered real vector space. Suppose $\Lambda$ is a linear functional defined on the product space $E^n \times E^m$ of finite families of elements (specifically, a functional bounded above by the interpolation gauge). For any element $h \in E$, the sum of the $i$-th lower parts of $\Lambda$ evaluated at $h$ is equal to the sum of the $j$-th upper parts of $\Lambda$ evaluated at $h$: $$\sum_{i=1}^n \text{lowerPart}_i(\Lambda)(h) = \sum_{j=1}^m \text{upperPart}_j(\Lambda)(h)$$ where the $i$-th lower part $\text{lowerPart}_i(\Lambda)$ is the component of $\Lambda$ acting on the $i$-th element of the first family, and the $j$-th upper part $\text{upperPart}_j(\Lambda)$ is the component acting on the $j$-th element of the second family.

Let $E$ be an ordered real vector space with an order unit $u$. Suppose that the positive functionals on $E$ form a lattice. Given finite families of elements $\{a_i\}_{i=1}^n$ and $\{b_j\}_{j=1}^m$ in $E$ such that $a_i \le b_j$ for all $i \in \{1, \dots, n\}$ and $j \in \{1, \dots, m\}$, then the interpolation gauge satisfies: $$\text{gauge}_u(a, -b) \le 0$$ where the interpolation gauge $\text{gauge}_u(a, -b)$ is the infimum of the set of all real numbers $\mu$ for which there exists an element $g \in E$ such that $a_i - g \le \mu u$ and $g - b_j \le \mu u$ for all indices $i$ and $j$.

Let $E$ be an ordered real vector space with an order unit $u$, such that the positive functionals on $E$ form a lattice. Given finite families of elements $\{a_i\}_i$ and $\{b_j\}_j$ in $E$ (where $a_i$ are lower observables and $b_j$ are upper observables such that $a_i \le b_j$ for all $i, j$), then for every $\epsilon > 0$, there exists an element $g \in E$ such that $a_i \leq g + \epsilon \cdot u$ for all $i$ and $g \leq b_j + \epsilon \cdot u$ for all $j$.