Let $u\in \mathscr{E}^{\prime}(-\pi,\pi)$ and define $a_{k}=\dfrac{1}{2\pi}\left\langle u,e^{-ikt}\right\rangle.$ Discuss series convergence $\displaystyle\sum a_{k}e^{ikt}$ in $\mathcal{D}^{\prime}(-\pi,\pi).$
I did an exercise that I think I can use here.
Let $(a_{k})_{k\in\mathbb{Z}}$ a sequence that satisfies $|a_{k}|\leq M |k|^{N}, k\neq 0, k \in \mathbb{Z},$ where $M,N$ be a constants. Then the series $\displaystyle \sum_{k=-\infty}^{\infty}a_{k}e^{ikt}$ converges in $\mathcal{D}^{\prime}(-\pi,\pi).$
Could someone help me please?
We have $$|a_k| = \dfrac{1}{2\pi} |\left\langle u, e^{-ikt} \right\rangle |.$$
Since $u \in \mathcal{E}'(-\pi, \pi)$ then exits $L$ compact in $(-\pi, \pi), C>0 $ and $n \in \mathbb{Z}_{+}$ such that $$ |\left\langle u, \varphi \right\rangle | \leq C \sum_{|\alpha| \leq n} \sup_{L} |\partial^{\alpha} \varphi |,$$ where $\varphi = e^{-ikt},$ i.e, $$|a_k| = \dfrac{1}{2\pi} |\left\langle u, \varphi \right\rangle | \leq \dfrac{1}{2\pi} C \sum_{|\alpha| \leq n} \sup_{L} |k^{|\alpha|} \varphi | \leq \tilde{C} |k|^N,$$ for some $N \in \mathbb{N}.$ Using the exercise you mentioned, it follows that $\sum a_k e^{ik t }$ is convergence in $\mathcal{D}'(-\pi,\pi).$