Fourier transform basic

Every function below is belonged to the Schwartz space $S(\mathbb{R}).$

The Fourier transform on S(R)
Fourier Transform:
$\begin{align*}
\widehat{f}(x) = \int_{-\infty}^{\infty} f(x) e^{-2\pi ix\xi} dx.
\end{align*}$

Propositions 1:
related with frequency and translations
(1) $f(x+h) \rightarrow \widehat{f}(\xi) e^{2\pi ih \xi} \ \textit{whenever} \ h \in \mathbb{R}.$
(2) $f(x) e^{-2\pi ixh} \rightarrow \widehat{f}(\xi + h) \ \textit{whenever} \ h \in \mathbb{R}.$

related with scaling
(3) $f(\delta x) \rightarrow \delta^{-1} \widehat{f}(\delta^{-1} \xi) \ whenever \ \delta > 0.$

related with differentiation
(4) $f'(x) \rightarrow 2\pi i \xi \widehat{f}(\xi).$
(5) $\begin{align*}
-2\pi i x f(x) \rightarrow \frac{d}{d\xi} \widehat{f}(\xi).
\end{align*}$

Theorem 1:
$\textit{If} \ f \in S(\mathbb{R}), \ \textit{then} \ \widehat{f} \in S(\mathbb{R}).$

The Gaussians as good kernels
Theorem 2.1:
$\textit{If} \ f(x) = e^{-\pi x^2}, \ \textit{then} \ \widehat{f}(\xi) = f(\xi).$ (why? Proposition 1-4, 1-5.)

Corollary 2.1:
$\textit{If} \ \delta > 0 \ \textit{and} \ K_{\delta}(x) = \delta^{-1/2}e^{-\pi x^2 / \delta}, \ \textit{then} \ \widehat{K_{\delta}}(\xi) = e^{-\pi \delta \xi^2}.$

Theorem 2.2:
$\textit{The collection} \ \{ K_{\delta} \}_{\delta > 0} \ \textit{is a family of good kernels as} \ \delta \rightarrow 0.$

Corollary 2.2:
$(f * K_{\delta})(x) \rightarrow f(x) \ \textit{uniformly in x as} \ \delta \rightarrow 0.$

The Fourier inversion
Proposition 3:
$\begin{align*}
\int_{-\infty}^{\infty} f(x) \widehat{g}(x) \textit{dx} = \int_{-\infty}^{\infty} \widehat{f}(y) g(y) \textit{dy}.
\end{align*}$ (why? Fubini's Theorem.)

Fourier inversion:
$\begin{align*}
f(x) = \int_{-\infty}^{\infty} \widehat{f}(\xi) e^{2\pi i x \xi} \textit{d}\xi.
\end{align*}$ (why? Cool properties of Gaussians which are stated above.)

The Plancherel formula
Propositions 4:
(1) $f * g \in S(\mathbb{R}).$
(2) $f * g = g * f.$
(3) $(\widehat{f*g})(\xi) = \widehat{f}(\xi)\widehat{g}(\xi).$

Plancherel Theorem:
$\begin{align*}
\int_{-\infty}^{\infty} \left | \widehat{f}(\xi) \right |^2 \textit{d}\xi = \int_{-\infty}^{\infty} \left | f(x) \right |^2 \textit{dx}.
\end{align*}$