Distribution (functional analysis)

Summary: usual functions can be reinterpreted like acting over an special set of functions: the test functions (functions with compact support). So we can think of functions like something similar to covectors. But there are more operators than the usual functions: they are the distributions. So distributions are a kind of covectors for the vector space of functions.
In other words: a function $f$, acting on another function $h$, is a kind of weighted sum of the values of $h$ (we say that $f$ is a distribution). Since there are more degenerate forms of calculating weighted sums of the values of $h$, we have more distributions than functions (for example Dirac delta, which is nothing but evaluating $h$). Since the functions $h$ are evaluated at points, and distributions generalize evaluation, we can thing of distributions as smeared out points. So, for me, distributions should be called "generalized points" instead of "generalized functions".

Motivation

Consider vectors. Given a basis, we can think of functions which extract components, which are called covectors ${\sigma }_1,{\sigma }_2,...$ But we can also consider linear combinations of this objects, and they are called covectors:

On the other hand, a function $h$ defined on ${\mathbb{R}}^n$ can be interpreted like a vector (it is, in fact, a vector) with a non-numerable basis indexed by $x\in {\mathbb{R}}^n$. Informally:

following physics convention.
And we can think of functionals $T_x$ given by

which recover the $x$-component.

Moreover, we can consider linear combinations of these functionals, for example

in analogy with equation $(1)$. Observe that $T$ is encoded by the function $f(x)=f_x$.

Now, observe that for a given function $h$ and a covector $T\equiv f$ we have a pairing

which is well-defined if, for example, $h$ has compact support. Observe that $⟨T,-⟩$ is a continuous linear map.

Definition

Let us define $D({\mathbb{R}}^n)$ as the set of ${\mathbb{C}}^{\mathrm{\infty }}$-functions with compact support. We call a distribution to every continuous linear transformation:

Usually, the action will be denoted, for $\phi \in D({\mathbb{R}}^n)$:

Any locally integrable function $f:\mathbb{R}\mapsto \mathbb{R}$ define a distribution $T_f$ in the following way

An important example of distribution, not really coming from a function, is the Dirac delta function $\delta$ (not really a function!)

About the nature of distributions as sheafs: see this.

Differentiation of distributions

We also can define the derivative of a distribution. It would be convenient that for usual functions $(T_f{)}^{\prime }=T_{f^{\prime }}$, that is

where we have integrated by parts.
So in general we define $T^{\prime }$ as

From this point of view it can be check that the Dirac delta is the derivative of the Heaviside function

Attention: value at 0 could be controversial.

At the same way that Dirac delta means evaluation at 0 of a test function, we can interpret that general distributions are "generalized points", or "matter distribution", or "bodies"; and the action over a function is "evaluate the function over that body". If the distribution have compact support I think we can choose any function, not only test functions.

Gradient of a distribution

In ${\mathbb{R}}^n$, the partial derivatives extend component by component. For $T\in {\mathcal{D}}^{\prime }({\mathbb{R}}^n)$, define ${\partial }_{i}T$ by:
$⟨{\partial }_{i}T,\phi ⟩=-⟨T,{\partial }_i\phi ⟩$

The gradient is then the $n$-tuple:
$\mathrm{\nabla }T=({\partial }_{1}T,\dots ,{\partial }_{n}T)\in {\mathcal{D}}^{\prime }({\mathbb{R}}^n{)}^n$

Key facts:

• Every distribution has all partial derivatives of all orders — no regularity of $T$ is required.
• For $T=T_f$ with $f\in C^1$, we recover ${\partial }_i{T}_f=T_{{\partial }_{i}f}$ (consistent with the classical gradient).
• Multiplication by $g\in C^{\mathrm{\infty }}$ is defined by $⟨gT,\phi ⟩=⟨T,g\phi ⟩$, since $g\phi \in \mathcal{D}$ whenever $\phi \in \mathcal{D}$.

Action of a vector field on a distribution

Let $F=(F_1,\dots ,F_n)$ with $F_i\in C^1({\mathbb{R}}^n)$. The directional derivative of $T$ along $F$ is:
$F\cdot \mathrm{\nabla }T=\sum _i{F}_i{\partial }_{i}T\in {\mathcal{D}}^{\prime }({\mathbb{R}}^n)$

where each $F_i{\partial }_{i}T$ uses multiplication by $F_i\in C^1$ followed by distributional differentiation. Explicitly, for $\phi \in C_c^1({\mathbb{R}}^n)$:
$⟨F\cdot \mathrm{\nabla }T,\phi ⟩=\sum _i⟨{\partial }_{i}T,F_i\phi ⟩=-\sum _i⟨T,{\partial }_i(F_i\phi )⟩=-⟨T,\mathrm{div}(\phi F)⟩$

So the equation $F\cdot \mathrm{\nabla }T=0$ in ${\mathcal{D}}^{\prime }$ is equivalent to:
$⟨T,\mathrm{div}(\phi F)⟩=0\mathrm{\forall }\phi \in C_c^1({\mathbb{R}}^n)$

When $T=T_{\sigma }$ for $\sigma \in L^{\mathrm{\infty }}\subset L_{\mathrm{loc}}^1$, this reads:
${\int }_{{\mathbb{R}}^n}\sigma (X)\mathrm{div}(\phi (X)F(X))dX=0\mathrm{\forall }\phi \in C_c^1$

This is the weak first integral condition: $F\cdot \mathrm{\nabla }\sigma =0$ in ${\mathcal{D}}^{\prime }$ says exactly that $\sigma$ is constant along orbits of $F$, in the distributional sense.

Related: there is a notion of invariant distribution through a flow of a vector field: see invariant distributions via vector field flow.