Definition 1.
Let us consider ordinary function $f$. Observe that if $f=0$ on open sets $\Omega_\iota$ (where $\iota $ runs any set of indices––finite, infinite or even non-enumerable) then $f=0$ on $\bigcup_\iota \Omega_\iota$. Therefore there exists a largest open set $\Omega$ such that $f=0$ on $\Omega$. Complement to this set is called support of $f$ and denoted as $\supp(f)$.
Definition 2.
Let us consider distribution $f$. We say that $f=0$ on open set $Omega$ if $f(\varphi)=0$ for any test function $\varphi$ such that $\supp \varphi \subset \Omega$.
Then the same observation as in (a) holds and therefore there exists a largest open set $\Omega$ such that $f=0$ on $\Omega$. Complement to this set is called support of $f$ and denoted as $\supp(f)$.
Definition 3.
Observe that $\supp(f)$ is always a closed set. If it is also bounded we say that $f$ has a compact support.
Exercise 1.
Prove that for two functions $f,g$ and for $f\in \mathcal{D}'$, $g\in \mathcal{E}$
\begin{gather}
\supp (gf)\subset \supp(f) \cap \supp(g),\label{eq-11.2.1}\\
\supp (\partial f)\subset \supp(f)\label{eq-11.2.2}
\end{gather}
where $\partial$ is a differentiation;
Prove that $\supp(f)=\emptyset$ iff $f=0$ identiacally;
Prove that $\supp(\delta_a)=\{a\}$. Prove that the same is true for any of its derivatives.
Remark 1.
In fact, the converse to Exercise 1(c) is also true: if $\supp(f)={a}$ then $f$ is a linear combination of $\delta (x-a)$ and its derivatives (up to some order).
Remark 2.
In the previous section we introduced spaces of test functions $\mathcal{D}$ and $\mathcal{E}$ and the corresponding spaces of distributions $\mathcal{D}'$ and $\mathcal{E}'$. However for domain $\Omega\subset \mathbb{R}^d$ one can introduce $\mathcal{D}(\Omega):= \{ \varphi \in \mathcal{D}:\, \supp \varphi \subset \Omega\}$ and $\mathcal{E}=C^\infty (\Omega)$. Therefore one can introduce corresponding spaces of distributions $\mathcal{D}'(\Omega)$ and $\mathcal{E}'(\Omega)=\{ f\in \mathcal{E}:\, \supp f\subset \Omega\}$. As $\Omega=\mathbb{R}^d$ we get our "old spaces".
Non-linear change of variables
Definition 4.
Let $f$ be a distribution with $\supp f\subset \Omega_1$ and let
$\Phi:\Omega_1\to \Omega_2 $ be one-to-one correspondence, infinitely smooth and with non-vanishing Jacobian $\det \Phi'$. Then $\Phi_* f$ is a distribution:
\begin{equation}
(\Phi_* f)(\varphi) = f( |\det \Phi'| \cdot \Phi^*\varphi )
\label{eq-11.2.3}
\end{equation}
where $(\Phi^*\varphi)(x)=\varphi(\Phi(x))$.
Mathematicians call $\Phi^*\varphi$ pullback of $\varphi$ and $\Phi_*f$ pushforward of $f$.
Exercise 2.
Check that for ordinary function $f$ we get $(\Phi_*f)(x)=f (\Phi^{-1}(x))$.
Fourier transform
Definition 5.
Let $f\in \mathcal{S}'$. Then Fourier transform $\hat{f}\in \mathcal{S}'$ is defined as
\begin{equation}
\hat{f}(\varphi) = f(\hat{\varphi})
\label{eq-11.2.4}
\end{equation}
for $\varphi \in \mathcal{S}$. Similarly, inverse Fourier transform $\check{f}\in \mathcal{S}'$ is defined as
\begin{equation}
\check{f}(\varphi) = f(\check{\varphi})
\label{eq-11.2.5}
\end{equation}
Exercise 3.
Check that for ordinary function $f$ we get a standard definition of $\hat{f}$ and $\check{f}$.
To justify Definition 5 one need to prove that
$f\in \mathcal{S}\iff \hat{f}\in \mathcal{S}$. Do it!
Prove that for $f\in \mathcal{E}'$ both $\hat{f}$ and $\check{f}$ are ordinary smooth functions
\begin{gather}
\hat{f}(k) = (2\pi)^{-d} f(e^{-ix\cdot k}), \label{eq-11.2.6}\\
\check{f}(k) = f(e^{ix\cdot k}).\label{eq-11.2.7}
\end{gather}
Check that all properties of Fourier transform (excluding with norms and inner products which may not exist are preserved.
Exercise 4.
Prove that Fourier transforms of $\delta (x-a)$ is
$(2\pi)^{-d}e^{-ix\cdot a}$.
Prove that Fourier transforms of $e^{ix\cdot a}$ is $\delta (x-a)$.
Exercise 5.
In dimension $1$
Prove that Fourier transforms of $\theta(x-a)$ and $\theta(-x+a)$ are respectively $(2\pi i)^{-1} (k-a-i0)^{-1}$ and $-(2\pi i)^{-1} (k-a+i0)^{-1}$ which are understood as limits in the sense of distributions of
$(2\pi i)^{-1}(k-a\mp i\varepsilon)^{-1}$ as $\varepsilon\to+0$. Recall that $\theta(x)$ is a Heaviside function.
As a corollary conclude that Fourier transform of $\operatorname{sgn}(x):=\theta(x)-\theta(-x)=x/|x|$ is
$(2\pi i)^{-1} \bigl((k-a-i0)^{-1}+ (k-a+i0)\bigr)^{-1}= \pi^{-1}(k-a)^{-1}$ with the latter understood in as principal value (see Exercise 11.1.4(f)).
As another corollary conclude that Fourier transform of $\theta(x)+\theta(-x)=1$ is
$(2\pi i)^{-1} \bigl((k-a-i0)^{-1}- (k-a+i0)\bigr)^{-1}$ and therefore
\begin{equation}
(2\pi i)^{-1} \bigl((k-a-i0)^{-1}- (k-a+i0)\bigr)^{-1}=\delta(x-a).
\label{eq-11.2.8}
\end{equation}
Convolution
Recall convolution (see
Definition 5.2.1) and its connection to Fourier transform.
Definition 6.
Let $f,g\in \mathcal{D}'$ (or other way around), $\varphi\in \mathcal{D}$ Then we can introduce $h(y) \in \mathcal{E}$ as
\begin{equation*}
h(y) = g ( T_y\varphi ),\qquad T_y \varphi(x):= \varphi (x-y).
\end{equation*}
Observe that $h \in \mathcal{D}$ provided $g\in \mathcal{E}'$. In this case we can introduce $h \in \mathcal{E}$ for $\varphi \in \mathcal{E}$.
Therefore if either $f\in \mathcal{E}'$ or $g\in \mathcal{E}'$ we introduce $f*g$ as
\begin{equation*}
(f*g)(\varphi) = f ( h ).
\end{equation*}
Exercise 6.
Check that for ordinary function $f$ we get a standard definition of the convolution;
Prove that convolution convolution has the same properties as multiplication;
Prove that $\partial (f*g)=(\partial f)*g = f*(\partial g)$;
Prove that $T_a (f*g)=(T_a f)*g = f*(T_a g)$
for operator of shift $T_a$;
Prove that $\supp (f*g) \subset \supp(f)+\supp(g)$ where arithmetic sum of two sets is defined as $A+B:=\{x+y:\, x\in A,\, y\in B\}$.
Remark 4.
One can prove that if a linear map $L:\mathcal{E}'\to \mathcal{D}'$ commutes with all shifts: $T_a (L f )=L(T_a f)$ for all $f\in \mathcal{E}'$ then there exists $g\in \mathcal{D}'$ such that $L$ is an operator of convolution: $Lf= g*f$;
One can extend convolution if none of $f,g$ has a compact support but some other assumption is fulfilled. For example, in one–dimensional case we can assume that either $\supp(f)\subset [a,\infty)$, $\supp(g)\subset [a,\infty)$ or that $\supp(f)\subset (-\infty,a]$, $\supp(g)\subset (-\infty,a]$.
Similarly in multidimensional case we can assume that $\supp(f)\subset C$, $\supp(g)\subset C$ where $C$ is a cone with angle $<\pi $ at its vertex $a$.
Fourier series
Definition 7.
We call one-dimensional distribution $f$ periodic with period $L$ if $f(x-L)=f(x)$.
More generally, let $\Gamma$ be a lattice of periods (see
Definition 4.B.1). We call distribution $f$ $\Gamma$-periodic if $f(x-n)=f(x)$ for all $n\in \Gamma$.
Periodic distributions could be decomposed into Fourier series: in one-dimensional case we have
\begin{equation}
f= \sum_{-\infty< m<\infty} c_n e^{\frac{2\pi imx}{L} }
\label{eq-11.2.9}
\end{equation}
and in multidimensional case
\begin{equation}
f= \sum_{ m\in \Gamma^*} c_m e^{im\cdot }
\label{eq-11.2.10}
\end{equation}
where $\Gamma^*$ is a dual lattice (see
Definition 4.B.3).
To define coefficients $c_m$ we cannot use ordinary formulae since integral over period (or elementary cell, again see the same definition) is not defined properly. Instead we claim that there exists $\varphi\in \mathcal{D}$ such that
\begin{equation}
\sum_{n\in \Gamma} \varphi(x-n) =1.
\label{eq-11.2.11}
\end{equation}
Indeed, let $\psi \in \mathcal{D}$ be non-negative and equal $1$ in some elementary cell. Then
$\varphi (x)= \psi (x)/ \bigl(\sum_{n\in \Gamma} \psi(x-n)\bigr)$ is an appropriate function.
Then
\begin{equation}
c_m= |\Omega|^{-1} (\varphi f)( e^{-im\cdot x})
\label{eq-11.2.12}
\end{equation}
where $|\Omega|$ is a volume of the elementary cell.
Exercise 7.
Find decomposition in Fourier series of one-dimensional distribution $f=\sum_{-\infty< n<\infty} \delta (x-n L)$;
Find Fourier transform of $f$ defined in (a);
Find the connection to Poisson summation formula (see Theorem 5.2.5).
Find decomposition in Fourier series of $d$-dimensional distribution $f=\sum_{n\in \Gamma } \delta (x-n)$;
Find Fourier transform of $f$ defined in (d);
Find the connection to multidimensional Poisson summation formula (see Remark 5.2A.3).