Physlib.SpaceAndTime.Space.Translations

For a translation vector $a \in \mathbb{R}^d$, this definition represents the continuous linear map $\tau_a: \mathcal{S}(\text{Space } d, X) \to \mathcal{S}(\text{Space } d, X)$ that acts on a Schwartz function $\eta$ by translating its argument. Specifically, for any $x \in \text{Space } d$, the value of the translated function is given by $(\tau_a \eta)(x) = \eta(x - v)$, where $v \in \text{Space } d$ is the vector corresponding to $a$ under the standard orthonormal basis.

For any translation vector $a \in \mathbb{R}^d$, any Schwartz function $\eta \in \mathcal{S}(\text{Space } d, X)$, and any point $x \in \text{Space } d$, the translation operator $\tau_a$ acting on $\eta$ satisfies the pointwise evaluation formula: $$(\tau_a \eta)(x) = \eta(x - v)$$ where $v \in \text{Space } d$ is the vector obtained from the coordinates $a$ using the standard orthonormal basis.

For any dimension $d \in \mathbb{N}$, given a translation vector $a \in \mathbb{R}^d$ and a Schwartz function $\eta \in \mathcal{S}(\text{Space } d, X)$, the translated function $\tau_a \eta$ is equal to the function $x \mapsto \eta(x - v_a)$, where $v_a \in \text{Space } d$ is the vector corresponding to the coordinates $a$ under the standard orthonormal basis of $\text{Space } d$.

For a translation vector $a \in \mathbb{R}^d$, this definition defines a linear map $\tau_a$ on the space of $X$-valued distributions on $\text{Space } d$. For any distribution $T \in \text{Space } d \to d[\mathbb{R}] X$, the translated distribution $\tau_a T$ is defined by its action on a test function $\eta$ in the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{R})$ as $(\tau_a T)(\eta) = T(\tau_{-a} \eta)$, where $\tau_{-a}$ is the translation operator on the Schwartz space.

For any dimension $d \in \mathbb{N}$, given a translation vector $a \in \mathbb{R}^d$, a distribution $T$ on $\text{Space } d$ with values in $X$, and a real-valued Schwartz test function $\eta \in \mathcal{S}(\text{Space } d, \mathbb{R})$, the action of the translated distribution $\tau_a T$ on $\eta$ is defined as the action of the distribution $T$ on the test function translated by $-a$. That is, $(\tau_a T)(\eta) = T(\tau_{-a} \eta)$.

Let $V = \text{Space } d$ be a $d$-dimensional real inner product space. For any translation vector $a \in \mathbb{R}^d$ and any scalar-valued distribution $T \in \mathcal{D}'(V, \mathbb{R})$, the distributional gradient of the translated distribution $\tau_a T$ is equal to the translation of the distributional gradient of $T$. That is, $$\nabla(\tau_a T) = \tau_a(\nabla T)$$ where $\nabla$ is the distributional gradient operator and $\tau_a$ is the translation operator on distributions.

For a natural number $d$, let $f: \text{Space}(d+1) \to X$ be a distribution-bounded function, and let $T_f$ be its associated distribution. For any translation vector $a \in \mathbb{R}^{d+1}$, the translation of the distribution $T_f$ by $a$ (denoted $\tau_a T_f$) is equal to the distribution associated with the function $x \mapsto f(x - v_a)$, where $v_a$ is the vector in $\text{Space}(d+1)$ corresponding to $a$ under the standard basis. That is, $$\tau_a T_f = T_{x \mapsto f(x - v_a)}.$$

For any dimension $d \in \mathbb{N}$, given a translation vector $a \in \mathbb{R}^d$ and an $\mathbb{R}^d$-valued distribution $T$ on $\text{Space } d$, the distributional divergence of the translated distribution $\tau_a T$ is equal to the translation of the distributional divergence of $T$. That is, $$\text{distDiv}(\tau_a T) = \tau_a(\text{distDiv } T)$$ where $\tau_a$ denotes the translation operator on distributions and $\text{distDiv}$ denotes the distributional divergence operator.