Fourier Transform of a Series is the Series of Fourier Transforms?

Question

If $\phi_k$ ($k\in\mathbb{Z})$ are in $L^2(\mathbb{R})$ and $\displaystyle\sum_{k\in\mathbb{Z}}\phi_k$ converges in $L^2(\mathbb{R})$, then is it true that $$ \mathcal{F}\left(\sum_{k\in\mathbb{Z}}\phi_k\right)=\sum_{k\in\mathbb{Z}}\mathcal{F}(\phi_k)\,? $$ I'm led to think this is true and maybe Plancherel can be used to give a positive answer?

Answer 1

Let us treat the case of one-sided sums. Since the Fourier transform is linear, it suffices to prove that if $u_n\to u$ in $\mathbb L^2\left(\mathbb R\right)$, then $\mathcal F(u_n)\to\mathcal F(u)$ in $\mathbb L^2\left(\mathbb R\right)$, as we can use the result with $u_n$ the $n$th partial sum. Since $$\left\lVert\mathcal F(u_n)-\mathcal F(u)\right\rVert_2=\left\lVert\mathcal F(u_n-u)\right\rVert_2,$$ we can use Plancherel to conclude.