Question 1: Suppose $a,b\in C^\infty(\mathbb{R})\cap L^\infty(\mathbb{R})$. Is it possible that the pointwise product $ab$ equals the constant function $1$ and both $a$ and $b$ have compactly supported Fourier transforms?
I would like to understand a specific example (if it exists) but don't know enough of the theory to find one.
Question 2: A more specific problem that I have in mind is whether there exist functions $a$ and $b$ defined on $\mathbb{R}\backslash U$, where $U$ is a small open neighbourhood around $0$, such that:
- $a,b$ are odd functions,
- $a\rightarrow\pm 1$ at $\pm\infty$,
- $ab=1$ on $\mathbb{R}\backslash U$,
- both $a$ and $b$ have compactly supported Fourier transforms.
Remark: To provide some context for the question, my motivation for considering whether such $a$ and $b$ exist comes from trying to do functional calculus of a self-adjoint elliptic differential operator $D$, for example $D=-i\frac{d}{dx}$ on $\mathbb{R}$. The compactness of the supports of $\hat{a}$ and $\hat{b}$ have to do with $a(D)$ and $b(D)$ having "finite propagation".