Solution of the equation $(\cos x)T = 0$ in $D'(\mathbb R)$

Question

I was thinking about the solution of the equation $(\cos x) T = 0$ in space $D'(\mathbb R)$. What is clear to me is that $T_k = c \delta_{a_k}$ with $a_k = \frac{(2k+1)\pi}2$ and $k \in \mathbb Z$ is a solution. Can you help me figure out if there are other solutions, or how to prove that these are the only solutions? Thanks!

Answer 1

What you want here is the possibility to "glue" two distributions defined on different domains.

You can show that, indeed, if $\Omega_{i}$, $i=1,2$ are two open domains $T_i\in D'(\Omega_i)$ such that $T_1|_{\Omega_1\cap\Omega_2}=T_2|_{\Omega_1\cap\Omega_2}$ in the sense of distributions, then there exists a unique distribution $T_0\in D'({\Omega_1\cup\Omega_2})$ such that $$T_0|_{\Omega_i}= T_i.$$

In you case you can easily show the uniqueness of solution if you restrain the equation on the domain $(k\pi , k\pi+\pi)$, and then "glue" these solutions together to obtain the final result.