The following definition comes from my professors notes. $\hat{f}$ refers to the Fourier transform of $f$.
Given $\gamma \in \mathbb{R}^d$, a bounded family of functions $(f_\alpha)_{\alpha > 0}$ is called a $\gamma$-net in $L^1(\mathbb{R}^d)$ if the following conditions hold:
- $\hat{f}_\alpha(\gamma)=1$ for all $\alpha > 0$.
- $\lim_{\alpha \rightarrow \infty} f_\alpha * f = 0$ for all $f \in L^1(\mathbb{R}^d)$ with $\hat{f}(\gamma)=0$.
As an example, he claims that the family of Poisson kernels in the upper half-plane $f_\alpha(x) = \displaystyle{\frac{1}{\pi}\cdot \frac{\frac{1}{2\pi \alpha}}{\left( \frac{1}{2\pi \alpha} \right)^2 + x^2}}$ is a $0$-net in $L^1(\mathbb{R})$. Indeed, a direct computation shows that $\hat{f}_\alpha(\xi) = \displaystyle{ \frac{1}{\sqrt{2\pi}} e^{-\frac{|\xi|}{2\pi \alpha}}}$. Not quite what we need, but we get condition one if we multiply by $\sqrt{2\pi}$ so its fine.
What I am unclear on is condition two. He does not mention in what sense we are to interpret convergence, but by the context I think it means convergence in $L^1(\mathbb{R}^d)$. I am able to show uniform convergence; if we replace $\alpha$ with $1/\alpha$ in the definition of $f_\alpha$ then by Holder's inequality
\begin{equation}
\| f_\alpha * f \|_\infty \leq \| f_\alpha \|_\infty \|f \|_1 = 2\| f\|_1/\alpha \rightarrow 0
\end{equation}
as $\alpha \rightarrow \infty$, for any $f \in L^1(\mathbb{R})$, but the same trick doesn't work for $L^1$ since $\|f_\alpha\|_1 = 1$. So, $\| f_\alpha * f\|_1 \leq \|f\|_1$.
Now, this isn't even the question! The exercise this is all in relation to is the following:
Given $\gamma \in \mathbb{R}^d$, provide an example of a $\gamma$-net in $L^1(\mathbb{R}^d)$ such that for all $\alpha > 0$ there is a neighborhood $V=V_\alpha$ of $\gamma$ with $\hat{f}_\alpha \mid_V \equiv 1$.
I figure that in $\mathbb{R}$ and for $\gamma=0$ I could construct such a $\gamma$-net by taking the exponential functions $\hat{f}_\alpha$ above, forcing them to equal one on an interval $[-1/\alpha, 1/\alpha]$, and then taking the inverse Fourier transform. For instance, $g_\alpha = \mathcal{F}^{-1}[(1-B_\alpha)\hat{f}_\alpha + B_\alpha]$, where $B_\alpha(x) = 1$ for $x \in [-1/\alpha, 1/\alpha]$ and $0$ otherwise. However, the details of whether these $g_\alpha$ are even well defined, let alone satisfy properties one and two, elude me.