Physlib.Mathematics.VariationalCalculus.IsLocalizedfunctionTransform

A function transformation $F : (X \to U) \to (Y \to V)$ is said to be **localized** if for every compact set $K \subseteq Y$, there exists a compact set $L \subseteq X$ such that for any two functions $\phi, \phi' : X \to U$, if $\phi$ and $\phi'$ agree on $L$ (i.e., $\phi(x) = \phi'(x)$ for all $x \in L$), then their transforms $F\phi$ and $F\phi'$ agree on $K$ (i.e., $(F\phi)(y) = (F\phi')(y)$ for all $y \in K$).

Let $F: (Y \to V) \to (Z \to W)$ and $G: (X \to U) \to (Y \to V)$ be function transformations. If both $F$ and $G$ are localized function transformations, then their composition $F \circ G$ is also a localized function transformation. A function transformation is localized if for every compact set $K$ in its target space, there exists a compact set $L$ in its source space such that if two functions $\phi, \phi'$ agree on $L$, then their transforms agree on $K$.

Let $F: (Y \to V) \to (Z \to W)$ and $G: (X \to U) \to (Y \to V)$ be function transformations. If both $F$ and $G$ are localized function transformations, then their composition, defined by $\phi \mapsto F(G(\phi))$, is also a localized function transformation. A function transformation is localized if for every compact set $K$ in the domain of the output functions, there exists a compact set $L$ in the domain of the input functions such that if two functions agree on $L$, their transforms agree on $K$.

The identity transformation $\text{id} : (Y \to V) \to (Y \to V)$ is a localized function transform. That is, for every compact set $K \subseteq Y$, there exists a compact set $L \subseteq Y$ such that for any two functions $\phi, \phi' : Y \to V$, if $\phi$ and $\phi'$ agree on $L$ (i.e., $\phi(y) = \phi'(y)$ for all $y \in L$), then $\text{id}(\phi)$ and $\text{id}(\phi')$ agree on $K$.

Let $F: (X \to U) \to (Y \to V)$ be a function transformation, where $V$ is a normed additive commutative group. If $F$ is a localized function transform, then the transformation defined by $\phi \mapsto -F\phi$ is also a localized function transform. A function transformation $F$ is said to be localized if for every compact set $K \subseteq Y$, there exists a compact set $L \subseteq X$ such that for any two functions $\phi, \phi' : X \to U$, if $\phi$ and $\phi'$ agree on $L$, then their transforms $F\phi$ and $F\phi'$ agree on $K$.

Let $V$ be a normed additive commutative group. Let $F, G: (X \to U) \to (Y \to V)$ be two function transformations. If $F$ and $G$ are both localized function transforms, then their pointwise sum, defined by $\phi \mapsto F\phi + G\phi$ (where $(F\phi + G\phi)(y) = (F\phi)(y) + (G\phi)(y)$), is also a localized function transform. A function transformation $F$ is said to be **localized** if for every compact set $K \subseteq Y$, there exists a compact set $L \subseteq X$ such that for any two functions $\phi, \phi' : X \to U$, if $\phi$ and $\phi'$ agree on $L$ (i.e., $\phi(x) = \phi'(x)$ for all $x \in L$), then their transforms $F\phi$ and $F\phi'$ agree on $K$ (i.e., $(F\phi)(y) = (F\phi')(y)$ for all $y \in K$).

Let $F : (X \to \mathbb{R}) \to (X \to \mathbb{R})$ be a localized function transform, meaning that for every compact set $K \subseteq X$, there exists a compact set $L \subseteq X$ such that if any two functions $\phi, \phi' : X \to \mathbb{R}$ satisfy $\phi|_L = \phi'|_L$, then $(F\phi)|_K = (F\phi')|_K$. For any function $\psi : X \to \mathbb{R}$, the transform defined by $\phi \mapsto \psi \cdot F\phi$ (pointwise, $(\psi \cdot F\phi)(x) = \psi(x) \cdot (F\phi)(x)$) is also a localized function transform.

Let $X$ be a topological space. If $F : (X \to \mathbb{R}) \to (X \to \mathbb{R})$ is a localized function transform, then for any fixed function $\psi : X \to \mathbb{R}$, the transformation mapping each function $\phi : X \to \mathbb{R}$ to the pointwise product $(F\phi) \cdot \psi$ (defined by $x \mapsto (F\phi)(x) \cdot \psi(x)$) is also a localized function transform.

Let $V$ be a real normed space. If $F : (X \to U) \to (X \to V)$ is a localized function transform, then for any function $\psi : X \to \mathbb{R}$, the transformation mapping each function $\phi : X \to U$ to the pointwise scalar product $x \mapsto \psi(x) \cdot (F\phi)(x)$ is also a localized function transform. A transformation is localized if for every compact set $K \subseteq X$, there exists a compact set $L \subseteq X$ such that for any two functions $\phi, \phi'$, if $\phi|_L = \phi'|_L$, then $(F\phi)|_K = (F\phi')|_K$.

The divergence operator, which maps a vector field $\phi : \text{Space } d \to \mathbb{R}^d$ to the scalar field $x \mapsto \text{div } \phi(x)$, is a localized function transform. Specifically, for any compact set $K \subseteq \text{Space } d$, there exists a compact set $L \subseteq \text{Space } d$ such that for any two vector fields $\phi, \phi' : \text{Space } d \to \mathbb{R}^d$, if $\phi(x) = \phi'(x)$ for all $x \in L$, then $(\text{div } \phi)(y) = (\text{div } \phi')(y)$ for all $y \in K$.

The operator that maps a vector field $\phi: \text{Space } d \to \text{Space } d$ to the divergence of its basis representation, given by $x \mapsto \text{div}(\text{basis.repr} \circ \phi)(x)$, is a localized function transform. Specifically, for every compact set $K \subseteq \text{Space } d$, there exists a compact set $L \subseteq \text{Space } d$ such that for any two vector fields $\phi, \phi' : \text{Space } d \to \text{Space } d$, if $\phi$ and $\phi'$ agree on $L$ (i.e., $\phi(x) = \phi'(x)$ for all $x \in L$), then the divergence of their basis representations agree on $K$ (i.e., $\text{div}(\text{basis.repr} \circ \phi)(y) = \text{div}(\text{basis.repr} \circ \phi')(y)$ for all $y \in K$).

The gradient operator, which maps a scalar function $\psi : \text{Space } d \to \mathbb{R}$ to its gradient field $x \mapsto \nabla \psi(x)$, is a localized function transform. This means that for any compact set $K \subseteq \text{Space } d$, there exists a compact set $L \subseteq \text{Space } d$ such that for any two scalar functions $\psi, \psi'$, if $\psi$ and $\psi'$ agree on $L$, then their gradients $\nabla \psi$ and $\nabla \psi'$ agree on $K$.

The gradient operator, which maps a function $\psi: \text{Space } d \to \mathbb{R}$ to its gradient field $x \mapsto \nabla \psi(x)$, is a localized function transform. That is, for every compact set $K \subseteq \text{Space } d$, there exists a compact set $L \subseteq \text{Space } d$ such that for any two functions $\psi, \psi' : \text{Space } d \to \mathbb{R}$, if $\psi(x) = \psi'(x)$ for all $x \in L$, then their gradients agree on $K$, i.e., $\nabla \psi(y) = \nabla \psi'(y)$ for all $y \in K$.

Let $X$ be a space and $U, V$ be normed spaces over $\mathbb{R}$. Let $\mathcal{L}(U, V)$ denote the space of continuous linear maps from $U$ to $V$. For any function $f : X \to \mathcal{L}(U, V)$, the function transformation $F : (X \to U) \to (X \to V)$ defined by $(F\phi)(x) = f(x)(\phi(x))$ is a localized function transform. That is, for every compact set $K \subseteq X$, there exists a compact set $L \subseteq X$ such that for any two functions $\phi, \phi' : X \to U$, if $\phi(x) = \phi'(x)$ for all $x \in L$, then $(F\phi)(x) = (F\phi')(x)$ for all $x \in K$.

Let $U$ be a normed additive commutative group and a normed space over $\mathbb{R}$. The function transformation that maps a function $\phi : \mathbb{R} \to U$ to its derivative $\phi'$ is a localized function transform.

Let $X$ be a proper normed space over $\mathbb{R}$ and $U$ be a normed space over $\mathbb{R}$. For any fixed vector $dx \in X$, the function transformation $F : (X \to U) \to (X \to U)$ defined by $(F\phi)(x) = \text{D}\phi(x)(dx)$, where $\text{D}\phi(x)$ denotes the Fréchet derivative of $\phi$ at $x$, is a localized function transform. Specifically, for every compact set $K \subseteq X$, there exists a compact set $L \subseteq X$ such that for any two functions $\phi, \phi' : X \to U$, if $\phi|_L = \phi'|_L$, then $\text{D}\phi(x)(dx) = \text{D}\phi'(x)(dx)$ for all $x \in K$.

Let $F : (X \to U) \to (X \to W \times V)$ be a localized function transformation. Then the transformation mapping a function $\phi : X \to U$ to the function $x \mapsto (F \phi(x))_1$ (the first component of the result of $F\phi$ at $x$) is also a localized function transformation.

Let $F : (X \to U) \to (X \to W \times V)$ be a localized function transformation. Then the transformation $G : (X \to U) \to (X \to V)$, defined such that for any function $\phi : X \to U$ and point $x \in X$, $(G\phi)(x) = \text{proj}_2((F\phi)(x))$ (the second component of the output of $F$), is also a localized function transformation.

Let $F : (X \to U) \to (X \to W)$ and $G : (X \to U) \to (X \to V)$ be two localized function transformations. Then the product transformation $H : (X \to U) \to (X \to W \times V)$, defined such that for any function $\phi : X \to U$ and point $x \in X$, $(H\phi)(x) = (F\phi(x), G\phi(x))$, is also a localized function transformation.

Let $X$ and $Y$ be real inner product spaces, and let $X$ be a proper space (i.e., a space where every closed and bounded set is compact). For a fixed vector $dy \in Y$, the function transformation $F : (X \to Y) \to (X \to X)$ defined by $F(\phi)(x) = (D\phi(x))^\dagger(dy)$, where $(D\phi(x))^\dagger$ is the adjoint of the Fréchet derivative of $\phi$ at $x$, is a localized function transform. Specifically, for every compact set $K \subseteq X$, there exists a compact set $L \subseteq X$ such that for any two functions $\phi, \phi' : X \to Y$, if $\phi$ and $\phi'$ agree on $L$, then $F\phi$ and $F\phi'$ agree on $K$.