I have problems with this lemma, page 45 of the book "Introduction to the theory of distributions" by Friendlander and Joshi.
$Lemma$. Let $I=(0,1)^N$ be the unit cube in $\mathbb{R}^N$ with $N>1$. If $\varphi \in \mathcal{D}(I)$ then one can find functions $\psi_{jk} \in \mathcal{D}(0,1)$ with $j=1,2,...$ and $k=1,...,N$ such that the sequence $\varphi_m(z)=\sum_{j=1}^m \psi_{j1}(z_1) \cdot \cdot \cdot \psi_{jN}(z_N)$ converges to $\varphi$ in $\mathcal{D}(I)$.
$Proof$. Extend $\varphi$ to $\overline{I}$ by setting $\varphi=0$ on $\partial I$, and define a periodic function on $\mathbb{R}^N$ by setting $\overline{\varphi}(z)=\varphi(z')$ when $z\equiv z'(\mathrm{mod}) \mathbb{Z}^N$ where $\mathbb{Z}^N$ consisting of points with integer coordinates. One can expand $\overline{\varphi}$ as Fourier series \begin{align*} \displaystyle \overline{\varphi}=\sum_{g \in \mathbb{Z}^N} \widehat{\varphi}_g e^{2 \pi i g \cdot z} \end{align*} where $\widehat{\varphi_g}:=\int_{I} \varphi(z) e^{- 2 \pi i g \cdot z} dz$. It is well known that this series converges to $\overline{\varphi}$ in $\mathcal{E}(\mathbb{R}^N)$, and so to $\varphi$ in $\mathcal{E}(\overline{I})$. (*) We assume this without proof, and remark only that the proof follows from the fact, easily proved by partial integration, that $|g|^M \widehat{\varphi}_g \rightarrow 0$ as$|g| \rightarrow \infty$ for any $M \geq 0$. Now as $\varphi$ is supported in $I$, there is a $\delta > 0$ such that $\mathrm{supp}(\varphi) \subset [\delta, 1-\delta]^N$. Choose $\rho \in \mathcal{D}(0,1)$ such that $\rho=1$ su $(\delta/2 , 1- \delta/2)$ and set \begin{align*} \displaystyle \varphi_m = \sum_{|g_1| \leq m,...,|g_N| \leq m} \widehat{\varphi}_g \prod_{k=1}^N \rho(z_k) e^{2\pi i g_k z_k} \end{align*} (**)It is clear from Leibniz's theorem and the convergence of the Fourier series to $\varphi$ in $\mathcal{E}(\overline{I})$ that these functions, wich are of the requested form, and $\varphi \in \mathcal{D}(I)$.
Can you help me to prove the point (*)? and I did not understand what Leibitiz's theorem is in (**). Any reference?
Thanks for the answers
For the (*) part, the Riemann-Lebesgue lemma, I remember an argument saying that you must first prove it with step functions then use their density to approximate continuous functions. Here are two links from which you can get inspired : http://mathonline.wikidot.com/the-riemann-lebesgue-lemma and https://folk.ntnu.no/hanche/kurs/diffkomp/2006h/rieleb-a4.pdf