I'm working on the first part of the following problem on distribution on torus:
I was able to prove that the Foruier coefficients are slowly increasing, but how to see that the series converges?
My attempt: The series converges in distribution if it gives pointwise convergence on $C^{\infty}$ functions, say $\sum g(k) \overline{F\phi}$ converges. To use the absolute value condition, I think we need to make this series absolut converges...but how? It seems to me the polynomial growth is too fast...
Any help is appreciated, thanks!
Let $$ \hat{g}_K := \sum_{\kappa \in [-K,K]^n} g(\kappa) E_\kappa .$$ This defines the linear functional $$(\hat{g}_K, \phi) = \sum_{[-K,K]^n} g(\kappa) \check{\phi}(\kappa) .$$ The space of test functions for $\mathcal{D}'(\mathbb{T}^n)$ is (I assume due to convention, but this isn't clear from your quoted text) the space $C^\infty(\mathbb{T}^n)$ of smooth functions (the manifold $\mathbb{T}^n$ is already compact). Every $\phi \in C^\infty(\mathbb{T}^n)$ has (inverse) Fourier coefficients that decay faster than any polynomial: let $N' > 0$ and $\alpha \in \mathbb{N}^n$ with $\alpha = (N', \ldots, N')$, then $$ (2\pi)^{2N'} |1 - \kappa^2|^{N'} |(E_\kappa, \phi)| = |((1 - D^2)^{\alpha} E_\kappa, \phi)| = |(E_\kappa, (1 - D^2)^{\alpha} \phi)| \leq m(\mathbb{T}^n) \sup|D^\alpha \phi| < \infty,$$ hence $|\check{\phi}(\kappa)| \leq C_{N'} (1 + |\kappa|)^{-N'}$.
Therefore, to see that $(\hat{g}_K,\phi)$ converges as $K \to \infty$, note that for $N' = N + M$ with $M > 1$, $$ |(\hat{g}_K,\phi)| \leq C \sum_{\kappa \in [-K,K]^n} (1 + |\kappa|)^N |\check{\phi}(\kappa)| \leq C C_{N'} \sum_{\kappa \in [-K,K]^n} (1 + |\kappa|)^{-M} .$$ Further, in order that in the limit we have a continuous linear functional, we just need that there exist $C' > 0$ and $N'' \geq 0$ such that $$\lim_K |(\hat{g}_K, \phi)| \leq C' \sup\{|D^\alpha \phi(x)| | x \in \mathbb{T}^n, |\alpha| \leq N'' \} .$$ For every $\alpha \in \mathbb{N}^n$, we have $(E_\kappa, D^\alpha \phi) = \kappa^\alpha \check{\phi}(\kappa)$ so with $|\alpha| \gg N$, we have for every $K > 0$, $$ |(\hat{g}_K, \phi)| \leq C \left|\sum_{\kappa \in [-K,K]^n} \check{\phi}(\kappa) (1 + |\kappa|)^N \right| \leq C \left| \sum_{\kappa \in [-K,K]^n} \check{\phi}(\kappa) (\kappa^2)^\alpha \right| \leq C \sup |D^{2\alpha} \phi| .$$
Therefore, the linear functional $$ F: C^\infty(\mathbb{T}^n) \ni \phi \to \lim_K (\hat{g}_K, \phi) \in \mathbb{C} $$ is well-defined and continuous, hence it defines a distribution $F \in D'(U)$ and by construction it satisfies $\hat{F} = g$.