Consider a function $f\in L^2(\mathbb{R}^d)$ with $\|f\|_{2}=1$ and such that $\hat{f}$ is supported on the ball $B(0,1)$. I am wondering which is the best decay that $f$ can have.

I read on "G. Björck: Linear partial differential operators and generalized distributions" that you can construct an $f$ for which $$|f(x)|\leqslant Ce^{-|x|^\alpha}, x\in\mathbb{R}^d$$ where $0<\alpha<1$. Are there better estimates? And in particular for this estimate, which is the minimal value we can take of $C$ (provided that $\|f\|_{2}=1$)?