Physlib.Mathematics.Distribution.Basic

Let $\mathbb{k}$ be a field (typically $\mathbb{R}$ or $\mathbb{C}$) satisfying the `RCLike` property, $E$ be a normed vector space over $\mathbb{R}$, and $F$ be a normed vector space over $\mathbb{k}$. An $F$-valued distribution on $E$, denoted by $E \to d[\mathbb{k}] F$, is defined as a continuous linear map from the Schwartz space $\mathcal{S}(E, \mathbb{k})$ to $F$. The Schwartz space $\mathcal{S}(E, \mathbb{k})$ consists of smooth functions $E \to \mathbb{k}$ whose derivatives are rapidly decreasing. This definition treats distributions as tempered distributions, providing a generalization of functions from $E$ to $F$.

The notation $E \to_{d[\mathbb{k}]} F$ denotes the space of tempered distributions from a normed vector space $E$ over $\mathbb{R}$ to a normed vector space $F$ over the field $\mathbb{k}$ (where $\mathbb{k}$ is typically $\mathbb{R}$ or $\mathbb{C}$). A distribution in this space is defined as a continuous linear map from the Schwartz space $\mathcal{S}(E, \mathbb{k})$ to the vector space $F$.

Given a linear map $u: \mathcal{S}(E, \mathbb{k}) \to F$ from the Schwartz space to a normed vector space $F$ and a finite set $s$ of index pairs $(k, n) \in \mathbb{N} \times \mathbb{N}$, this definition constructs a distribution (a continuous linear map) in the space $E \to d[\mathbb{k}] F$. The construction is valid provided that $u$ satisfies a continuity condition: there exists a constant $C \ge 0$ such that for every test function $\eta \in \mathcal{S}(E, \mathbb{k})$, there exist indices $(k, n) \in s$ and a point $x \in E$ such that $$\|u(\eta)\| \le C \|x\|^k \|D^n \eta(x)\|,$$ where $D^n \eta(x)$ denotes the norm of the $n$-th iterated Fréchet derivative of the function $\eta$ at the point $x$. This condition ensures that the linear map $u$ is continuous with respect to the topology of the Schwartz space, allowing it to be treated as a tempered distribution.

Let $\mathbb{k}$ be an `RCLike` field (typically $\mathbb{R}$ or $\mathbb{C}$), $E$ be a real normed vector space, and $F$ be a normed vector space over $\mathbb{k}$. Let $u: \mathcal{S}(E, \mathbb{k}) \to F$ be a $\mathbb{k}$-linear map from the Schwartz space to $F$. Suppose $u$ satisfies a continuity condition defined by a finite set $s \subseteq \mathbb{N} \times \mathbb{N}$ and a constant $C \geq 0$, such that for every test function $\eta \in \mathcal{S}(E, \mathbb{k})$, there exist indices $(k, n) \in s$ and a point $x \in E$ satisfying $\|u(\eta)\| \leq C \|x\|^k \|D^n \eta(x)\|$, where $\|D^n \eta(x)\|$ denotes the norm of the $n$-th iterated Fréchet derivative of $\eta$ at $x$. Then, the distribution constructed from this linear map (denoted by `ofLinear`) applied to any test function $\eta$ is equal to $u(\eta)$.

Let $E$ be a finite-dimensional real normed vector space and $F$ be a normed vector space over a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$). The Fréchet derivative operator for distributions is a $\mathbb{k}$-linear map that transforms a distribution $u: E \to d[\mathbb{k}] F$ into a distribution valued in the space of continuous linear maps $\mathcal{L}(E, F)$. For a distribution $u$ and a test function $\eta \in \mathcal{S}(E, \mathbb{k})$, the derivative $Du$ is the distribution such that for any vector $v \in E$, the evaluation is given by $((Du) \eta)(v) = -u(\partial_v \eta)$, where $\partial_v \eta(x) = D\eta(x)v$ is the directional derivative of the test function $\eta$ in the direction $v$. In this framework, unlike traditional functions, every distribution is infinitely Fréchet differentiable.

Let $E$ be a finite-dimensional real normed vector space, $\mathbb{k}$ be a field (typically $\mathbb{R}$ or $\mathbb{C}$), and $F$ be a normed vector space over $\mathbb{k}$. For a distribution $u: E \to d[\mathbb{k}] F$, a Schwartz test function $\eta \in \mathcal{S}(E, \mathbb{k})$, and a vector $v \in E$, the evaluation of the Fréchet derivative of the distribution, denoted $Du$, is given by $$((Du) \eta)(v) = -u(\partial_v \eta)$$ where $\partial_v \eta$ is the Schwartz function representing the directional derivative of $\eta$ in the direction $v$, i.e., $(\partial_v \eta)(x) = D\eta(x)v$.

Let $E$ be a normed vector space over $\mathbb{R}$ and $F$ be a complex normed vector space. The Fourier transform $\mathcal{F}$ is a $\mathbb{C}$-linear map from the space of $F$-valued distributions on $E$ to itself, denoted as $(E \to d[\mathbb{C}] F) \to (E \to d[\mathbb{C}] F)$. Given a distribution $u$ and a test function $\eta \in \mathcal{S}(E, \mathbb{C})$, the action of the Fourier transform of $u$ (denoted $\hat{u}$ or $\mathcal{F}\{u\}$) is defined by $$\langle \mathcal{F}\{u\}, \eta \rangle = \langle u, \mathcal{F}\{\eta\} \rangle$$ where $\mathcal{F}\{\eta\}$ is the Fourier transform of the Schwartz function $\eta$.

Let $E$ be a real normed vector space and $F$ be a complex normed vector space. For a distribution $u \in E \to d[\mathbb{C}] F$ and a Schwartz test function $\eta \in \mathcal{S}(E, \mathbb{C})$, the action of the Fourier transform of $u$ (denoted $\mathcal{F}u$) on $\eta$ is equal to the action of $u$ on the Fourier transform of $\eta$ (denoted $\mathcal{F}\eta$): $$\langle \mathcal{F}u, \eta \rangle = \langle u, \mathcal{F}\eta \rangle.$$

Given a vector $c \in F$, the constant distribution is the element of the space of distributions $E \to d[\mathbb{k}] F$ that maps a Schwartz test function $\eta \in \mathcal{S}(E, \mathbb{k})$ to the integral $$\int_E \eta(x) \cdot c \, dx$$ where $dx$ denotes the volume measure on $E$. This construction requires that the volume measure on $E$ has temperate growth.

Let $\mathbb{k}$ be a field (such as $\mathbb{R}$ or $\mathbb{C}$), $E$ be a normed vector space over $\mathbb{R}$, and $F$ be a normed vector space over $\mathbb{k}$. Suppose the volume measure on $E$ has temperate growth. For any constant vector $c \in F$ and any Schwartz test function $\eta \in \mathcal{S}(E, \mathbb{k})$, the action of the constant distribution associated with $c$ on $\eta$ is given by the integral: $$\langle \text{const } c, \eta \rangle = \int_E \eta(x) \cdot c \, dx$$ where $dx$ denotes the volume measure on $E$.

Let $E$ be a finite-dimensional real normed vector space equipped with an additive Haar measure, and let $F$ be a normed space over $\mathbb{R}$. For any constant vector $c \in F$, the Fréchet derivative of the constant distribution associated with $c$ is the zero distribution.

Given a point $a$ in a normed vector space $E$ over $\mathbb{R}$, the Dirac delta distribution $\delta_a$ is a distribution in the space $E \to d[\mathbb{k}] \mathbb{k}$. It is defined as the continuous linear map that takes a Schwartz test function $\eta \in \mathcal{S}(E, \mathbb{k})$ to its value at $a$, denoted by $\langle \delta_a, \eta \rangle = \eta(a)$.

Let $E$ be a normed vector space over $\mathbb{R}$ and $\mathbb{k}$ be a field (such as $\mathbb{R}$ or $\mathbb{C}$) satisfying the `RCLike` property. For any point $a \in E$ and any Schwartz test function $\eta \in \mathcal{S}(E, \mathbb{k})$, the action of the Dirac delta distribution $\delta_a$ on $\eta$ is given by evaluation at $a$: $$\langle \delta_a, \eta \rangle = \eta(a)$$

Let $E$ be a normed vector space over $\mathbb{R}$ and $F$ be a normed vector space over a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$ satisfying the `RCLike` property). Given a point $a \in E$ and a vector $v \in F$, the vector-valued Dirac delta distribution is a distribution in the space $E \to d[\mathbb{k}] F$. It is defined as the continuous linear map that takes a Schwartz test function $\eta \in \mathcal{S}(E, \mathbb{k})$ and maps it to the scalar-vector product of the function's value at $a$ and the vector $v$: $$\langle \delta_a^v, \eta \rangle = \eta(a) \cdot v$$ Intuitively, this represents an infinitely intense vector field concentrated at a single point $a$ pointing in the direction $v$.

Let $E$ be a normed vector space over $\mathbb{R}$ and $F$ be a normed vector space over a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$). For any point $a \in E$, vector $v \in F$, and Schwartz test function $\eta \in \mathcal{S}(E, \mathbb{k})$, the action of the vector-valued Dirac delta distribution $\delta_a^v$ on $\eta$ is given by the scalar-vector product: $$\langle \delta_a^v, \eta \rangle = \eta(a) \cdot v$$

For a given $d \in \mathbb{N}$, the Heaviside step distribution is a tempered distribution on the Euclidean space $\mathbb{R}^{d+1}$. Its action on a Schwartz test function $\eta \in \mathcal{S}(\mathbb{R}^{d+1}, \mathbb{R})$ is defined by the integral of $\eta$ over the upper half-space where the last coordinate is positive: $$\langle H, \eta \rangle = \int_{\{x \in \mathbb{R}^{d+1} \mid x_{d+1} > 0\}} \eta(x) \, dx$$ where $x_{d+1}$ denotes the last component of the vector $x$.

For any natural number $d$ and any Schwartz test function $\eta \in \mathcal{S}(\mathbb{R}^{d+1}, \mathbb{R})$, the action of the Heaviside step distribution $H$ on $\eta$ is defined by the integral of $\eta$ over the upper half-space where the last coordinate is positive: $$\langle H, \eta \rangle = \int_{\{x \in \mathbb{R}^{d+1} \mid x_{d+1} > 0\}} \eta(x) \, dx$$ where $x_{d+1}$ denotes the last coordinate of the vector $x$.