For readers convenience, I post the exercise2.3.8 here:
Let $(c_{k})_{k\in\mathbb{Z}^{n}}$ be a sequence that satisfies $|c_{k}|\leq A(1+|k|)^{M}$ for all $k$ and some fixed $M$ and $A>0$. Let $\delta_{k}$ denote the Dirac mss at the integer $k$. Show that the sequence of distributions $$\sum_{|k|\leq N}c_{k}\delta_{k}$$ converges to some tempered distribution $u$ in $\mathcal{S}'(\mathbb{R}^{n})$ as $N\to\infty$. Also show that $\hat{u}$ is the $\mathcal{S}'$ limit of the sequence of functions $$h_{N}(\xi)=\sum_{|k|\leq N}c_{k}e^{-2\pi i\xi\cdot k}$$.
Frankly speaking, I am confused by the $n$ dimension case where $n\geq 2$. For instance, when $n=2$, $$\sum_{|k|\leq N}\langle c_{k}\delta_{k},\varphi\rangle=\sum_{|k|\leq N}c_{k}\varphi(k)\leq \sum_{ |k|\leq N}A(1+|k|)^{M}\varphi(k)\leq \sum_{|k|\leq N}AC(1+|k|)^{-L+M}$$ because $\varphi$ is Schwartz function and so $\varphi(k)\leq C(1+|k|)^{-L}$ for any $L>0$.But does $\lim_{N\to\infty}\sum_{|k|\leq N}AC(1+|k|)^{-L+M}=\lim_{N\to\infty}\sum_{j=1}^{N}\sum_{|k|=j}AC(1+|k|)^{-L+M}$ convergence when $L$ large? When $n=1$, $\sum_{|k|=j}AC(1+|k|)^{-L+M}=2AC(1+|k|)^{-L+M}$ because $k=1$ and $-1$, and so in this case we can take $L=M+2$, but as $n\geq2$ , $\sum_{|k|=j}AC(1+|k|)^{-L+M}=?$ where $k=(k_{1},k_{2},\dots,k_{n})$ and $|k|=|k_{1}|+|k_{2}|+\dots+|k_{n}|$. It seems like a combinatorial problem.
In the note of "Differential analysis" by Dyatlov, there is a similar excercise Assume that the sequence $(a_{k})_{k\in\mathbb{Z}}$ satisfies $|a_{k}|\leq C(1+|k|)^{N}$ for some constant $C,N$. Show that fourier series $$\sum_{|k|\leq N}a_{k}e^{ikx}$$ convergence in $\mathcal{D}'(\mathbb{R})$, does $$\sum_{|k|\leq N}a_{k}e^{ikx}$$ belong to $\mathcal{D}'(\mathbb{R})$?
Thanks, any suggestions are wellcome!!!
Putting a unit cube around each lattice point you'll find that $$\sum_{k\in \Bbb{Z}^n,|k|=j} 1 \le Volume(\{ x\in \Bbb{R}^n , |x|\in [j,j+1]\}) = C_n(j+1)^n-C_n j^n = O(j^{n-1})$$ Whence $$\sum_{k\in \Bbb{Z}^n-0} |k|^{-r}$$ converges whenever $r > n$.