Let $\phi:\mathbb{R}\to \lbrack 0,\infty)$ be piecewise continuous, symmetric ($\phi(x)=\phi(-x)$) and with support on $(-1,1)$. Let $\Phi(x)=\int_{-\infty}^u \phi(x)$; assume $\Phi(1)=1$.

What is $\phi$ such that $$\frac{|\Phi|_2^2}{1-\max_{|t|\geq 1} |\widehat{\phi}(t)|}$$ is minimal?

Note that $\Phi(1)=1$ implies $|\widehat{\phi}(t)|\leq 1$ for all $t$.

Let $\phi:\mathbb{R}\to \lbrack 0,\infty)$ be piecewise continuous, symmetric ($\phi(x)=\phi(-x)$) and with support on $(-1,1)$. Let $\Phi(x)=\int_{-\infty}^x \phi(u) du$; assume $\Phi(1)=1$.

What is $\phi$ such that $$\frac{|\Phi|_2^2}{1-\max_{|t|\geq 1} |\widehat{\phi}(t)|}$$ is minimal?

Note that $\Phi(1)=1$ implies $|\widehat{\phi}(t)|\leq 1$ for all $t$.