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} e^{- i 2 \pi x \cdot \xi} f(x) \, \ell(d x)
\end{equation}
|
| (iii) |
\begin{equation}
\mathfrak{G}_f (\xi) : = \int_{\mathbb{R}^N} e^{- i x \cdot \xi} f(x) \, \ell(d x)
\end{equation}
|
Then
| (1) |
\begin{equation}
\mathfrak{F} \in \mathfrak{L}^1(M)
\quad \implies \quad
\mathfrak{F}^{-1} (\mathfrak{F}_f)
\overset{a.e.}{=} f
\end{equation}
|
| (2) |
\begin{equation}
\mathfrak{G} \in \mathfrak{L}^1(M)
\quad \implies \quad
\mathfrak{G}^{-1} (\mathfrak{G}_f)
\overset{a.e.}{=} \frac{1}{(2 \pi)^N} f
\end{equation}
|
