This question is the anti-theorem of the following exercise and the solution of original exercise is here.
Exercise 2.3.8 from Classical Fourier Analysis by Loukas Grafakos
Suppose $(c_{k})_{k\in\mathbb{Z}^{d}}$ is a sequence that satisfies$\left|c_{k}\right|\leq A(1+\left|k\right|)^{M}$ for all $k$, where$A,M>0$ are fixed. Then $$\sum_{\left|k\right|\leq N}c_{k}\delta_{k}$$converges to a tempered distribution$u\in\mathcal{S}'(\mathbb{R}^{d})$ as $N\rightarrow\infty$.
I want to prove it by constructing some Schwartz function. If we only consider the case of $d=1$ to simplify the statement, my thinking is:
Take $\varphi\in C_C^\infty(\mathbb R)$ satisfying $\operatorname{supp}\varphi=(-1,1)$ and $\varphi(0)=1$. We know that if $|c_n|$ cannot be dominated by a polynomial then we can find a positive (or negative) subsequence of $\{c_n\}$ satisfying $\{c_{a_n}\}$ is also cannot be dominated by a polynomial and $a_{n+1}-a_{n}\ge 3\cdot 2^n$ for all $n$. Notice that $\operatorname{supp}\varphi(\frac {x-a_i}{2^i})\cap \operatorname{supp}\varphi(\frac {x-a_j}{2^j})=\varnothing$ for $i\neq j$. And we have $\psi(x)\to0$ when $|x|\to \infty$, so $\psi\in \mathcal S$. Then $\langle u,\psi\rangle\ge\sum\limits_{i=0}^N c_{a_i}$ for any $N\in \mathbb N_+$. I want to show contradiction by $\sum\limits_{i=0}^N |c_{a_i}|=\left|\sum\limits_{i=0}^N c_{a_i}\right|\le \sum\limits_{|\alpha|,|\beta|\le p}|x^\alpha \partial^\beta \psi(x)|$.
The only thing I cannot finish is to show $\sum\limits_{|\alpha|,|\beta|\le p}|x^\alpha \partial^\beta \psi(x)|$ can be dominated by a polynomial. If that holds, it will be contradicted to $|c_n|$ cannot be dominated by a polynomial.
Does this idea work? If so, then how to deal with the case of higher dimension?