Let $\tau>0$, and let $T\in \mathcal{D}'(\mathbb{R})$ be a $\tau$-periodic distribution (that is, $ \langle T, \varphi(\cdot+\tau)\rangle= \langle T,\varphi\rangle $ for all $\varphi \in \mathcal{D}(\mathbb{R})$). Then $$ T=\sum_{n\in \mathbb{Z}} c_n e^{i 2\pi t/\tau}, $$ for some $c_n\in \mathbb{C}$, and where the equality means that the symmetric partial sums of the series on the right hand side converge in $\mathcal{D}'(\mathbb{R})$ to $T$. What are the $c_n$s in terms of $T$? One would think that they are given by $c_n=\langle T, e^{-in2\pi /\tau}\rangle/\tau$, but $e^{-in2\pi/\tau}$ is not a test function in $\mathcal{D}(\mathbb{R})$.

Let $\tau>0$, and let $T\in \mathcal{D}'(\mathbb{R})$ be a $\tau$-periodic distribution (that is, $ \langle T, \varphi(\cdot+\tau)\rangle= \langle T,\varphi\rangle $ for all $\varphi \in \mathcal{D}(\mathbb{R})$). Then $$ T=\sum_{n\in \mathbb{Z}} c_n e^{i 2\pi t/\tau}, $$ for some $c_n\in \mathbb{C}$, and where the equality means that the symmetric partial sums of the series on the right hand side converge in $\mathcal{D}'(\mathbb{R})$ to $T$. What are the $c_n$s in terms of $T$? One would think that they are given by $c_n=\langle T|_{(0,2\pi)}, e^{-in2\pi /\tau}\rangle/\tau$, but $e^{-in2\pi/\tau}$ is not a test function in $\mathcal{D}((0,2\pi))$.