Fourier Transformation: Properties

Linearity:

\[f(x),g(x), F(p),G(p),\quad a,b\in\mathbb{R}\]

\begin{eqnarray*}\F\left\{af(x) + bg(x)\right\}&=&a\F\left\{f(x)\right\} + b \F\left\{g(x)\right\}\\ &=&aF(p) + b G(p)\end{eqnarray*}
Differential Properties: $f(x)$ and derivative: $f'(x)=\frac{df}{dx}$ \begin{eqnarray*} \F\left\{f'(x)\right\}&=&ip\F\left\{f(x)\right\}=ip F(p)\\f^n(x)&=&\frac{d^nf}{dx^n}\;\;\mbox{$n$-th derivative: }\F\left\{f^n(x)\right\}=(ip)^n F(p)\\ &&\Rightarrow\mbox{ derivative leads to a factor $ip$} \end{eqnarray*}
Convolution theorem: \begin{eqnarray*} \F^{-1}\left\{\sqrt{2\pi}F(p)\cdot G(p)\right\}&=&\int\limits_{-\infty}^{+\infty}f(x-t)g(t) dt=f\star g(x)\\ \mbox{and } \F\left\{\sqrt{2\pi}f(x)g(x)\right\}&=&F\star G(p)=\int\limits_{-\infty}^{+\infty}F(p-p')G(p')dp' \end{eqnarray*}

Definition 34 Convolution $f \star g$ \begin{eqnarray*}f\star g(x)&:=&\int\limits_{-\infty}^{+\infty}f(x-t)g(t)dt\\ &&\mbox{t has negative and positive values ($\pm$direction)}\\ f\star g(x)&=&g\star f(x)\end{eqnarray*}