I am interested in a smooth weight function $w_P(x_1, x_2) : \mathbb{R}^2 \rightarrow \mathbb{R}_{\geq 0}$ such that
we have $w_P(x_1, x_2) = 1$ if $ \|(x_1, x_2)\| \leq P$ and goes to $0$ as $ \|(x_1, x_2)\|$ gets larger, and it satisfies
$$
\sum_{ \substack{ \|(x_1, x_2)\| > P \\ x_1, x_2 \in \mathbb{Z}} } w_P(x_1, x_2) \leq C
$$
for some constant $C$ independent of $P$.
Could someone possibly give me an example of such weight function? (if such a function actually exists)
Thank you very much!
PS Here $\| \|$ is a sup norm.
Let $D = \{(x_1, x_2) \in \Bbb R^2 \mid \|(x_1, x_2)\| \leq P\}$. Then $D$ is compact, so for any $\varepsilon > 0$, there exists a smooth bump function which is equal to $1$ on $D$ and equal to $0$ at all points which are at least distance $\varepsilon$ away from $D$ (in the given norm). By taking $\varepsilon$ small enough, you can force the function to be zero at any integer point not in $D$.