$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$
In the previous Section 5.1 we introduced Fourier transform and Inverse Fourier transform \begin{align} & \hat{f}( k )= \frac{\kappa}{2\pi}\int_{-\infty}^\infty f(x)e^{-i k x}\,dx \tag{FT}\\ & \check{F}(x)= \frac{1}{\kappa} \int_{-\infty}^\infty F( k ) e^{i k x}\,d k \tag{IFT} \end{align} with $\kappa=1$ (but here we will be a bit more flexible):
Theorem 1. $F= \hat{f} \iff f=\check{F}$. (Already "proved")
Theorem 2.
Proof. Easy. Preservation of inner product follows from preservation of norm.
Remark 1.
Theorem 3.
$g(x)=f(x-a)\implies \hat{g}( k )=e^{-i k a}\hat{f}( k )$;
$g(x)=f(x)e^{ibx}\implies \hat{g}( k )=\hat{f}( k -b)$;
$g(x)=f'(x) \implies \hat{g}( k )=i k \hat{f}( k )$;
$g(x)=xf (x) \implies \hat{g}( k )=i \hat{f}{}'( k )$;
$g(x)=f(\lambda x)\implies \hat{g}( k )=|\lambda|^{-1} \hat{f}(\lambda^{-1} k )$;
Proof. Here for brevity we do not write that all integrals are over $\mathbb{R}$ and set $\kappa=2\pi$.
$\hat{g}=\int e^{-i k x}g(x)\,dx = \int e^{-i k x}f(x-a)\,dx= \int e^{-i k (x+a)}f(x)\,dx= e^{-i k a}\hat{f}( k )$. We replaced $x$ by $(x+a)$ in the integral.
$\hat{g}=\int e^{-i k x}g(x)\,dx = \int e^{-i k x}e^{ibx}f(x)\,dx= \int e^{-i ( k -b)x}f(x)\,dx= \hat{f}( k -b)$.
$\hat{g}=\int e^{-i k x}g(x)\,dx = \int e^{-i k x}f'(x)\,dx \overset{\text{by parts}}= -\int \bigl(e^{-i k x}\bigr)'f(x)\,dx= i k \hat{f}( k )$.
$\hat{g}=\int e^{-i k x}g(x)\,dx = \int e^{-i k x}xf(x)\,dx= \int i\partial_ k \bigl(e^{-i k x}\bigr) \, f(x)\,dx= i\hat{f}{}'( k )$.
$\hat{g}=\int e^{-i k x}g(x)\,dx = \int e^{-i k x}f(\lambda x )\,dx = \int e^{-i k |\lambda|^{-1}x}f(x)\,\lambda^{-1}dx= \lambda^{-1}\hat{f}(\lambda^{-1} k )$. Here we replaced $x$ by $\lambda^{-1}x$ in the integral and $|\lambda|^{-1}$ is an absolute value of Jacobian.
Corollary 1. $f$ is even (odd) iff $\hat{f}$ is even (odd).
Definition 1. Convolution of functions $f$ and $g$ is a function $f*g$: \begin{equation} (f*g)(x):=\int f(x-y)g(y)\,dy. \label{eq-5.2.3} \end{equation}
Theorem 4.
$h=f*g\implies \hat{h}( k )=\frac{2\pi}{\kappa}\hat{f}( k )\hat{g}( k )$;
$h(x)=f(x)g(x)$ $\implies \hat{h}=\kappa \hat{f}*\hat{g}$;
Proof.
\begin{equation*} \hat{h}(x)=\frac{\kappa}{2\pi} \int e^{-ix k }h(x)\,dx = \frac{\kappa}{2\pi} \iint e^{-ix k }f(x-y)g(y)\,dxdy; \end{equation*} replacing in the integral $x:=y+z$ we arrive to \begin{equation*} \frac{\kappa}{2\pi} \iint e^{-i(y+z) k }f(z)g(y)\,dzdy= \frac{\kappa}{2\pi} \int e^{-iz k }f(z)\,dz \times \int e^{-iy k }g(y)\,dz \end{equation*} which is equal to $\frac{2\pi}{\kappa}\hat{f}( k )\hat{g}( k )$.
Similarly $\hat{f}*\hat{g}$ is a Fourier transform of $\frac{\kappa_1}{2\pi}fg$ where $\kappa_1=\frac{2\pi}{\kappa}$.
Example 1. Let $f(x)=e^{-\alpha x}$ as $x>0$ and $f(x)=0$ as $x<0$. Here $\Re \alpha>0$. \begin{equation*} \hat{f}( k )= \int_0^\infty e^{-(\alpha +i k )x}\,dx = -(\alpha +i k )^{-1}e^{-(\alpha +i k )x}\bigr|_{x=0}^{x=\infty}= (\alpha +i k )^{-1} \end{equation*} provided $\kappa=2\pi$.
In the general case $\hat{f}( k )= \frac{\kappa}{2\pi}(\alpha+i k )^{-1}$.
Example 2. Let $f(x)=e^{-\frac{\alpha}{2}x^2}$ with $\Re\alpha\ge 0$. Here even for $\Re \alpha=0$ F.t. exists as integrals are converging albeit not absolutely.
Note that $f'=\alpha x f$. Applying Fourier transform and Theorem 3 (c),(d) to the left, right we get $i k \hat{f}= -i\alpha \hat{f}'$; solving it we arrive to $\hat{f}=Ce^{-\frac{1}{2\alpha} k ^2}$.
To find $C$ note that $C=\hat{f}(0)= \frac{\kappa}{2\pi}\int e^{-\frac{\alpha}{2}x^2}\,dx$ and for real $\alpha>0$ we make a change of variables $x=\alpha^{-\frac{1}{2}}z$ and arrive to $C=\frac{\kappa}{\sqrt{2\pi \alpha}}$ because $\int e^{-z^2/2}\,dz=\sqrt{2\pi}$. Therefore \begin{equation*} \hat{f}( k )= \frac{\kappa}{\sqrt{2\pi\alpha}}e^{-\frac{1}{2\alpha} k ^2}. \end{equation*} Knowing complex variables one can justify it for complex $\alpha $ with $\Re\alpha\ge 0$; we take a correct branch of $\sqrt{\alpha}$ (condition $\Re\alpha\ge 0$ prevents going around origin). In particular, $(\pm i )^{\frac{1}{2}}=e^{\pm \frac{i\pi}{4}}$ and therefore for $\alpha=\pm i\beta $ with for $\beta>0$ we get $f=e^{\mp\frac{i}{2\beta }x^2}$ and \begin{equation*} \hat{f}( k )=\frac{\kappa}{2\sqrt{\pi\beta}} (1\mp i)e^{\pm\frac{i}{2\beta} k ^2x}. \end{equation*}
Theorem 5. Let $f(x)$ be a continuous function on the line $(\infty,\infty)$ which vanishes for large $|x|$. Then for any $a>0$ \begin{equation} \sum_{n=-\infty}^\infty f(an) = \sum_{n=-\infty}^\infty \frac{2\pi }{a}\hat{f}(\frac{2\pi }{a}n) . \label{eq-5.2.4} \end{equation}
Proof. Observe that function \begin{equation*} g(x)= \sum_{n=-\infty}^\infty f(x+an) \end{equation*} is periodic with period $a$. Note that the Fourier coefficients of $g(x)$ on the interval $(-\frac{a}{2}, \frac{a}{2})$ are $b_m=\frac{2\pi}{a}\hat{f}(\frac{2\pi}{a})$, where $\hat{f}(k)$ is the Fourier transform of $f(x)$.
Finally, in the Fourier series of $g(x)$ on $(-\frac{a}{2}, \frac{a}{2})$ plug $x = 0$ to obtain $g(0)=\sum_m b_m$ which coincides with (\ref{eq-5.2.4}).
$\Leftarrow$ $\Uparrow$ $\Downarrow$ $\downarrow$ $\Rightarrow$