$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\supp}{\operatorname{supp}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$

In this Chapter we extend notion of function. These new

functions(actually most of them are not functions at all) are calleddistributionsand are very useful for PDE (and not only). We define them as linear forms on thetest functionswhich 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

- $\mathcal{D}=C_0^\infty$ is a space of infinitely smooth functions with compact support. It means that for each function $\varphi$ exists $a$ such that $\varphi(x)=0$ as $|x|\ge a$;
- $\mathcal{E}=C^\infty$ is a space of infinitely smooth functions;
- $\mathcal{S}$ is a space of infinitely smooth functions which decay at infinity (with all their derivatives faster than any power: \begin{equation} |\partial ^m \varphi| (1+|x|)^k \le M_{mk} \qquad \forall x\, \forall m,k. \label{eq-11.1.1} \end{equation}

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.**

- $\varphi_n \to \varphi $ in $\mathcal{D}$ iff $\max |\partial ^m(\varphi_n -\varphi)|\to 0$ as $n\to \infty$ for all $m$ and also there exist $a$ such that $\varphi_n(x)=0$ as $|x|\ge a$ for all $n$;
- $\varphi_n \to \varphi $ in $\mathcal{E}$ iff $\max_{|x|\le a} |\partial ^m(\varphi_n -\varphi)|\to 0$ as $n\to \infty$ for all $m$, $a$;
- $\varphi_n \to \varphi $ in $\mathcal{S}$ iff \begin{equation} |\partial ^m (\varphi_n-\varphi)| (1+|x|)^k \to 0 \qquad \forall x\, \forall m,k. \label{eq-11.1.2} \end{equation}

**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.**

- $\mathcal{E}'$ consists of distributions with compact support: $f\in \mathcal{D}'$ belongs to $\mathcal{E}'$ iff there exists $a$ such that $f(\varphi)=0$ for all $\varphi$ such that $\varphi (x)=0$ as $|x|\le a$.
- $\mathcal{S}'$ consists off
*temperate distributions*. - For $f\in L^1_{\mathrm{loc}}$ we can define action $f(\varpi)$ on $\mathcal{D}$
\begin{equation}
f(\varphi)=\int f(x)\varphi(x)\,dx
\label{eq-11.1.5}
\end{equation}
where integral is always understood as integral over the whole line $\mathbb{R}$ (or a whole space $\mathbb{R}^d$) and $L^1_{\mathrm{loc}}$ consists of
*locally integrable functions*(notion from the Real Analysis which means that $\int_{|x|\le a} |f(x)|\,dx <\infty$ for all $a$ but integral is a*Lebesgue integral*which is a natural extension of Riemann integral). One can prove that this form is continuous and thus $f\in \mathcal{D}'$. Due to this we sometimes nonâ€“rigorously will write (\ref{eq-11.1.5}) even for distributions which are not ordinary functions.

**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}
for $\alpha_1,\alpha_2\in \mathbb{C}$.

**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.**

- Check that for ordinary function $f$ we get a standard definition of $f(x-a)$ (in the framework of (\ref{eq-11.1.5})).
- Check that for $\delta$ we $\delta_a(x):=\delta(x-a)$ is defined as $\delta_a (\varphi)= \varphi (a)$.

**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.**

- Check that for ordinary function $f$ we get a standard definition of $R_A f$ (in the framework of (\ref{eq-11.1.5})).
- Check that for $\delta$ we get $ \delta (Ax)= |\det A|^{-1} \delta(x)$.
In particular as $|\det A|=1$ we have $\delta(Ax)=\delta(x)$ and as
$Ax=\lambda x$ (uniform dilatation) $\delta (\lambda x)=\lambda^{-d}\delta(x)$ where $d$ is a dimension. Therefore $\delta$ is
*spherically symmetric*and*positively homogeneous of degree $-d$*.

**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.**

- Check that for differentiable function $f$ we get a standard definition of $\partial f$ (in the framework of (\ref{eq-11.1.5})). Use integration by parts. Generalize to piecewise differentiable and continuous function.
- Check that for $\delta$ we get $\delta '$: $\delta'_a (\varphi)=-\varphi'(a)$ (in one dimension and similarly in higher dimensions). Calculate $\delta_a^{(n)}(\varphi)$ for $n=1,2,\ldots$
- Check that if $\theta(x)$ is a
*Heaviside function*: $\theta(x)=1$ as $x>0$ and $\theta(x)=0$ as $x\le 0$ then $\theta' (x)=\delta(x)$. - Check that if $f(x)$ is a smooth function as $x<\ a$ and as $x > a$ but has a jump at $a$ then \begin{equation*} f'=\overset{\circ}{f}{}'+ \bigl(f(a+0)-f(a-0)\bigr)\delta (x-a), \end{equation*} where $f'$ is understood in the sense of distributions and $\overset{\circ}{f}{}'(x)$ is an ordinary function equal to derivative of $f$ as $x\ne a$. Generalize to piecewise differentiable functions.
- Prove that if $f=\ln |x|$ then \begin{equation*} f'(\varphi)= pv \int x^{-1}\varphi (x)\,dx , \end{equation*} where integral is understood as a principal value integral.

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.**

- Check that for ordinary function $f$ we get a standard definition of $g f$ (in the framework of (\ref{eq-11.1.5})).
- Prove that $g \delta_a= g(a)\delta_a$ (use definitions);
- Calculate $g\delta '_a$, $g\delta ''_a$ (use definitions!).

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 \otimes 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 result if we apply first $f$ and then $g$.

**Exercise 6.**

- Let $(x\pm i0)^{\nu}= \lim _{\varepsilon\to +0} (x\pm \varepsilon i)^{\nu}$. Obviously, it exists for $\Re\nu >-1$.
- Using equality

\begin{equation} [(x\pm \varepsilon i)^{\nu}]'= \nu (x\pm \varepsilon i)^{\nu-1}, \label{eq-11.1.12} \end{equation} prove that these limits exist (in the sense of distributions) for $ \nu \ne -1,-2,\ldots$. - Using \begin{equation} [\ln (x\pm \varepsilon i)^{\nu}]'= (x\pm \varepsilon i)^{-1}, \label{eq-11.1.13} \end{equation} prove that these limits exist (in the sense of distributions) for $\nu =-1$. Then, using (\ref{eq-11.1.12}), prove that these limits exist (in the sense of distributions) for $\nu=-2,-3,\ldots$.
- Prove that \begin{gather} (x-i0)^{-1} + (x+i0)^{-1}=2 x^{-1}, \label{eq-11.1.14}\\ (x-i0)^{-1}- (x+i0)^{-1}=2\pi i \delta(x) \label{eq-11.1.15} \end{gather} with integral understood in the sense of principal value (See Exercise 4, use (\ref{eq-11.1.13}) and $\ln (x+i0)+\ln (x-i0)= 2\ln |x|$, $\ln (x+i0)-\ln (x-i0)= -2\pi i\theta(-x)$.

**Exercise 7.**
Consider functions
\begin{equation}
f^+_\nu(x):=\frac{1}{\Gamma(\nu+1)}x_+^\nu \qquad\text{with }
x_+^\nu=x^\nu \theta(x),
\label{eq-11.1.16}
\end{equation}
where $\Gamma(nu)$ is Euler Gamma-function.

- Prove that $f^+_\nu$ belongs to $L^1_{loc}$ for $\Re \nu >-1$ and that \begin{equation} f^+_{\nu+1}{}'=f^+_\nu. \label{eq-11.1.17} \end{equation} provided $\Re\nu>-1$.
- This equality (\ref{eq-11.1.17}) allows us to define $f^+_\nu $ as a distribution for $\Re >-2$ so that it be fulfilled, then for $\Re\nu>-3$ and recurrently all $\nu\in\mathbb{C}$.

Alternatively we can use analytic continuation with respect to $\nu$. - Observing that $f^+_0(x)=\theta(x)$ and using (\ref{eq-11.1.17}) prove that \begin{equation} f_{-n} =\delta^{(n-1)}(x) \qquad n=1,2,\ldots \label{eq-11.1.18} \end{equation}
- In the similar way define $gf^-_\nu(x)=f^+_\nu (-x)$ and establish their properties (similar to (\ref{eq-11.1.16})â€“(\ref{eq-11.1.18})).

**Exercise 8.**

Prove that $ \delta_{a_1}(x_1)\cdots \delta_{a_n}(x_n)= \delta_a(x)$ with $a=(a_1,\ldots,a_n)$, $x=(x_1,\ldots,x_n)$ and we have on the left product of $1$-dimensional $\delta$-functions and on the right $n$-dimensional.