The problem is this;
For any $2\neq p\in[1,\infty]$, show that there exists $f\in L^p(\mathbb{R}^d)$ such that $\widehat{f}\in L^1_{loc}$ and $|\widehat{f}|\notin \widehat{L^p(\mathbb{R}^d)}$.
A hint is given in two steps;
(i) Use complex Gaussians to show that for any $N\in\mathbb{N}$, there is $f_N\in L^p\cap L^1$ with $\widehat{f_N}\in L^1$ such that $\|f_N\|_{L^p}\leq 1$, $|\widehat{f_N}|\in\widehat{L^p}$, but $\||\widehat{f}|^{\vee}|\|_{L^p}>N$.
(ii) Sum an infinite series with nearly orthogonal Fourier series.
To provide some basic information, a complex Gaussian is given by $$g_z(x)=e^{-\pi z|x|^2},\ \ \ \ \ \ \mathrm{Re}(z)>0$$ and $$\|g_z\|_{L^1}=\frac{1}{|\mathrm{Re}(z)|^{d/2}},$$ $$\widehat{g_z}(\xi)=\frac{1}{z^{d/2}}e^{-\pi|\xi|^2/z}.$$
What I could do so far is only the step (i) with $p>2$, letting $\mathrm{Re}(z)=1,\ \mathrm{Im}(z)=N$, but I still don't get what the step (ii) means.
Any suggestions would be appreciated!