Let $g_1,g_2\cdots,g_n\cdots$ be a sequence of functions in $L^2(\mathbb{R})$ such that $||g_n||_2=\frac{1}{2}$ and it's fourier transform $\hat{g_n}$ vanishes outside $[-n.n+1]$. Show that $\sum_ng_n$ converges to a function $g\in L^2(\mathbb{R})$ and compute $||g||_2$.
My attempt : As $g_n$'s are in $L^2(\mathbb{R})$ so we can apply Plancherel Theorem but how to use that I can't figure out. Any help/hint in this regards would be highly appreciated. Thanks in advance!
This does not seem to be true. Because of the disjoint fourier transforms, we have $$ \begin{align} \left\|\,g\,\right\|_2^2 &=\left\|\,\hat{g}\,\right\|_2^2\\ &=\left\|\,\sum_{n\in\mathbb{Z}}\hat{g}_n\,\right\|_2^2\\ &=\sum_{n\in\mathbb{Z}}\left\|\,\hat{g}_n\,\right\|_2^2\\ &=\sum_{n\in\mathbb{Z}}\left\|\,g_n\,\right\|_2^2\\ &=\sum_{n\in\mathbb{Z}}\frac14 \end{align} $$ which diverges.