Construct function with Fourier transform compactly supported in interval $[-k,k]$.

Question

I want to construct a function $f$ such that $\hat{f}$ is supported in $[-k,k]$.

My first attempt was to consider the function $$g(x) = \begin{cases} \exp\left( \frac{k}{k-x^2} \right), & -k \leq x \leq k, \\ 0, & \left| x \right| > k. \end{cases}$$

The integral of $\mathcal{F}^{-1}(g)$ does not converge however.

Answer 1

Take \begin{equation} g(x) = \begin{cases} \exp\left( \frac{1}{x^2-k} \right), & -k \leq x \leq k, \\ 0, & \left| x \right| > k. \end{cases} \end{equation} This function is in $C^\infty_c(\mathbb{R})$ and so its inverse Fourier decays rapidly.