How to choose, $\phi \in C^{\infty}_{c}(\mathbb R)$ such that its Fourier transform $\hat{\phi}$ is 1 in some neighbourhood of the given point?

Question

Put $C_{c}^{\infty}(\mathbb R)=$ The space of $C^{\infty}$ functions on $\mathbb R$ whose support is compact.

Fix $x_{0}\in \mathbb R.$

My Question is : Can we expect to choose, $\phi \in C^{\infty}_{c}(\mathbb R)$ such that its Fourier transform $\hat{\phi}$ is 1 (constant) in some neighbourhood of the given point $x_{0}$ ; if answer is yes, how ?

Thanks,

Answer 1

No, this cannot be done. If $\phi\in C_c^\infty(R)$, then its Fourier transform extends to be an entire analytic function (cf. Paley--Wiener theorem). Should it be constant for (real) $x$ in some neighborhood of $x_0$, it would be identically constant by the uniqueness theorem for analytic functions, and hence zero, because the Fourier transform of a smooth compactly supported function decays at infinity.