Square-integrable complex-valued function $y(t)$ is defined on a finite domain $t \in [0, 1]$ and satisfies the following constraint:

$$ \int_0^{1} |y(t)|^2 \, dt = 1 $$

I am seeking bounds on the Fourier transform:

$$ \tilde{y}(\omega) \equiv \frac{1}{\sqrt{2 \pi}}\int_0^{1} y(t) \, e^{-i \omega t} \, dt $$

for large $\omega$.

More specifically, I want to maximize $$B \equiv \int_{-\Omega/2}^{+\Omega/2} |\tilde{y}(\omega)|^2 d \omega$$

over all possible $y(t)$ for fixed $\Omega \gg 1$?

For constant $y(t) = 1 $ it is straightforward to compute:

$$B= \frac{2}{\pi} \, \rm{Si}(\Omega/2) -\frac{4}{\pi \Omega}\left [1-\cos (\Omega/2) \right ] \sim 1 - \frac{4}{ \pi \Omega}$$

I expect that by "flexing" $y(t)$ I can get $B$ closer to $1$ but lack a systematic approach to find $y(t)$ that gives the best asymptotics.

For this answer I gather that $y(t)$ can be constructed such that the approach of $B(\Omega)$ to 1 is faster than polynomial, but from this one I get that it can not be as fast as exponential. What would be the way to find the optimal shape (in the above sense) given my complete freedom to choose $y(t)$?


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