$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\supp}{\operatorname{supp}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$
In this Chapter we extend notion of function. These new "functions" (actually most of them are not functions at all) are called distributions and are very useful for PDE (and not only). We define them as linear forms on the test functions which are some nice functions. For usual function $f$ such form is \begin{equation*} f(\varphi)=\int f(x)\varphi(x)\,dx. \end{equation*} We also extend the notion of solution.
We introduce three main spaces of test functions:
Definition 1. Let
Loran Schwartz who provided the first systematic theory of distributions used these notations and they became traditional. However we need to explain what does it mean convergence of test function:
Definition 2.
Theorem 1. \begin{equation} \mathcal{D} \subset \mathcal{S} \subset \mathcal{E} \label{eq-11.1.3} \end{equation} where $\mathcal{K}_1\subset\mathcal{K}_2$ means not only that all elements of $\mathcal{K}_1$ are also elements of $\mathcal{K}_2$ but also that $\varphi_n\overset{\mathcal{K}_1}{\to}\varphi$ implies that $\varphi_n\overset{\mathcal{K}_2}{\to}\varphi$. Also in (\ref{eq-11.1.3}) each smaller space $\mathcal{K}_1$ is dense in the larger one $\mathcal{K}_2$: for each $\varphi\in \mathcal{K}_2$ there exists a sequence $\varphi_n\in \mathcal{K}_1$ converging to $\varphi$ in $\mathcal{K}_2$.
Remark 1. Those who studies Real Analysis heard about Topological Vector Spaces but we are not going to introduce topology (which is ridiculously complicated on $\mathcal{D}$), just convergence is sufficient for all needs. The same approach is also used in the very advanced cources.
Definition 3.
Distribution $f$ (over $\mathcal{K}$) is a continuous linear form on $\mathcal{K}$: $f:\mathcal{K}\to \mathbb{C}$ such that \begin{gather*} f(\alpha_1 \varphi_1 + \alpha_2 \varphi_2)= \alpha_1 f(\varphi_1)+\alpha_2 f(\varphi_2)\qquad \forall \varphi_1,\varphi_2\in \mathcal{K}\; \forall \alpha_1,\alpha_2\in \mathbb{C};\\ \varphi_n\overset{\mathcal{K}}{\to}\varphi \implies f(\varphi_n)\to f(\varphi). \end{gather*}
The space of such linear forms is denoted by $\mathcal{K}'$.
Theorem 1 immediately implies
Theorem 2. \begin{equation} \mathcal{D}' \supset \mathcal{S}' \supset \mathcal{E}' \label{eq-11.1.4} \end{equation} where $\mathcal{K}'_1\supset\mathcal{K}'_2$ means not only that all elements of $\mathcal{K}_2$ are also elements of $\mathcal{K}_1$ but also that $f_n\overset{\mathcal{K}'_2}{\to} f$ implies that $f_n\overset{\mathcal{K}'_2}{\to}f$. Also in (\ref{eq-11.1.4}) each smaller space $\mathcal{K}'_2$ is dense in the larger one $\mathcal{K}'_1$: for each $f\in \mathcal{K}'_1$ there exists a sequence $f_n\in \mathcal{K}'_2$ converging to $f$ in $\mathcal{K}'_1$.
So far we have not introduced the convergence of distributions, so we do it right now:
Definition 4. $f_n\overset{\mathcal{K}'}{\to}f$ iff $f_n(\varphi)\to f(\varphi)$ for all $\varphi\in \mathcal{K}$.
Remark 2.
Example 1. $\delta:=\delta (x)$ is an element of $\mathcal{E}'$ defined as $\delta(\varphi)=\varphi(0)$. It is traditionally called $\delta$-function or Dirac $\delta$-function despite not being a function but a distribution.
We introduce operations on distributions as an extension of operations on ordinary functions as long as they make sense.
Definition 5. Linear operations: \begin{equation} (\alpha_1 f_1 +\alpha_2 f_2)(\varphi)= \alpha_1 f_1(\varphi) +\alpha_2 f_2(\varphi) \label{eq-11.1.6} \end{equation} as $\alpha_1,\alpha_2\in \mathbb{C}$.
Exercise 1. Check that for ordinary functions $f_1,f_2$ we get a standard definition of $\alpha_1 f_1 +\alpha_2 f_2$ (in the framework of (\ref{eq-11.1.5})).
Definition 6. Shift. Let $T_a$ denote a shift of $\varphi$: $(T_a\varphi)(x) =\varphi (x-a)$. Then \begin{equation} (T_a f)(\varphi)= f(T_{-a}\varphi). \label{eq-11.1.7} \end{equation} We will write $T_af$ as $f(x-a)$.
Exercise 2.
Definition 7. Linear change of variables. Let $R_A$ denote a linear change of variables: $(R_A\varphi )(x)= \varphi(Ax)$ where $A$ is a non-degenerate linear transformation. Then \begin{equation} (R_A f)(\varphi)= |\det A|^{-1} f(R_{A^{-1}}\varphi) \label{eq-11.1.8} \end{equation} We will write $R_Af$ as $f(Ax)$.
Exercise 3.
Definition 8. Derivative. Then \begin{equation} (\partial f)(\varphi)= -f (\partial \varphi) \label{eq-11.1.9} \end{equation} where $\partial $ is a first order derivative.
Exercise 4.
Let $g\in C^\infty$. Observe that for $g\varphi \in \mathcal{D}$ and $g\varphi \in \mathcal{E}$ for $\varphi \in \mathcal{D}$ and $\varphi \in \mathcal{E}$ respectively. Therefore the following definition makes sense:
Definition 9. Multiplication by a function. Let either $f\in \mathcal{D}'$ or $f\in \mathcal{E}'$. Then $gf\in \mathcal{D}'$ or $gf\in \mathcal{E}'$ respectively is defined as \begin{equation} (g f)(\varphi)= f (g \varphi). \label{eq-11.1.10} \end{equation}
Exercise 5.
We cannot define in general the product of two distributions. However in some cases it is possible, f.e. when distributions are of different arguments.
Definition 10. Direct product. Let $f,g$ be distributions. Then $f(x)g(y)$ (also denoted as $f\otimes g$) is defined as \begin{equation} (f g) (\varphi) = f( g(\varphi)) \label{eq-11.1.11} \end{equation} where $\varphi=\varphi (x,y)$, then applying $g$ to it we get $\psi (x):=g(\varphi)$ a test function, and then applying $f$ we get a number. Similarly we get the same $fg$ if we apply first $f$ and then $g$.
Exercise 6.