In a proof of the Riemann-Lebesgue lemma I encountered the Fourier transform of the characteristic function $f$ of a cuboid $\prod_{k = 1}^{n} [a_k, b_k]$.
My lecture notes claim that
$$
\mathcal{F}(f(\xi))
=\frac{1}{(2 \pi)^{\frac{n}{2}}} \prod_{i = 1}^{n} \frac{e^{-i \langle b_i, \xi \rangle} – e^{-i \langle a_i, \xi \rangle}}{i \xi_i}.
$$
Besides the horrible misuse of $i$ as index (I presume the $i$ in the denominator is the imaginary unit) I don't think this is correct, as for one, $a_i, b_i \in \mathbb{R}$, so taking their scalar product with $\xi \in \mathbb{R}^n$ doesn't make so much sense.
Also, this doesn't work for $\xi_k = 0$. If $\xi_k = 0$ for some $k \in \{1, \ldots, n\}$, the corresponding factor is just $1$, right?
My approach to calculate $\mathcal{F}(f(\xi))$ would be as follows
\begin{align}
(2 \pi)^{-\frac{n}{2}}\int_{a_1}^{b_1} \ldots \int_{a_n}^{b_n} e^{-i \langle x, \xi \rangle} d x_1 \ldots d x_n
& = (2 \pi)^{-\frac{n}{2}} \int_{a_1}^{b_1} \ldots \int_{a_n}^{b_n} \exp\left(-i \sum_{j = 1}^{n} x_j \xi_j\right) d x_1 \ldots d x_n \\
& = (2 \pi)^{-\frac{n}{2}} \prod_{k = 1}^{n} \int_{a_k}^{b_k} e^{-i x_k \xi_k} d x_k \\
& = (2 \pi)^{-\frac{n}{2}}\prod_{k = 1}^{n} \frac{i(e^{-i b_k \xi_k} – e^{-i a_k \xi_k})}{\xi_k}.
\end{align}
Questions
- Is the result from the lecture notes correct? If not, is it just a typo that can be easily fixed?
- Is my calculation correct?
Best Answer
@2 Yes, your calculations are correct.
@1 And hence: You're right, the lecture notes are incorrect. I think the error is part uncareful notation, part typo and part thinking-A-but-typing-B. Yes, the index should be something other than $i$. Yes, $\langle a_k,\xi\rangle$ does not make sense. The scalar product is probably supposed to be $\langle a,\xi \rangle$ and $\langle b,\xi\rangle$ respectively, where $a:=(a_1,...,a_n), b:=(b_1,...,b_n)$. This scalar product is very useful for multidimensional Fourier transforms and you have used it correctly in your calculation. But since $\prod_k (e^{-i a_k \xi_k} - e^{-i b_k \xi_k}) \neq (\prod_k e^{-i a_k \xi_k} - \prod_k e^{-i a_k x_k})$ it cannot be used in any obvious way to simplify the result of the integral as the author of your lecture notes maybe had hoped.