Let $M = (\mathbb{R}^N, \mathcal{L}, \ell)$ be a
D1744: Lebesgue measure space such that
| (i) |
$f : \mathbb{R}^N \to \mathbb{C}$ is an D1921: Absolutely integrable function on $M$
|
| (ii) |
\begin{equation}
\mathfrak{F}_f(\xi)
: = \int_{\mathbb{R}^N} f(x) e^{- 2 \pi i x \cdot \xi} \, \ell(d x)
\end{equation}
|
| (iii) |
\begin{equation}
\mathfrak{G}_f(\xi)
: = \int_{\mathbb{R}^N} f(x) e^{- i x \cdot \xi} \, \ell(d x)
\end{equation}
|
Let $a \in \mathbb{R} \setminus \{ 0 \}$ be a
D993: Real number.
Then
| (1) |
\begin{equation}
\mathfrak{F}_{x \mapsto f(x / a)} (\xi)
= |a|^N \mathfrak{F}_f (a \xi)
\end{equation}
|
| (2) |
\begin{equation}
\mathfrak{F}_{x \mapsto f(a x)} (\xi)
= |a|^{- N} \mathfrak{F}_f \left( \frac{\xi}{a} \right)
\end{equation}
|
| (3) |
\begin{equation}
\mathfrak{G}_{x \mapsto f(x / a)} (\xi)
= |a|^N \mathfrak{G}_f (a \xi)
\end{equation}
|
| (4) |
\begin{equation}
\mathfrak{G}_{x \mapsto f(a x)} (\xi)
= |a|^{- N} \mathfrak{G}_f \left( \frac{\xi}{a} \right)
\end{equation}
|
