Let $\phi:\mathbb{R}\to\mathbb{R}$ be an even function with support on $[-1,1]$. Assume that it is in $L^1\cap L^2$ and that its Fourier transform is also in $L^1\cap L^2$. Assume as well that $|\phi|_\infty = \phi(0)=1$. Define $F(x) = \int_x^\infty \widehat{\phi}(t) dt$.

Given these constraints, what is the choice of $\phi$ that minimizes $$\int_0^\infty |F(x)| dx?$$

What is the value of that minimum?