Joint compact-window construction

Ordinary temporal windowing alone would not establish integrability at zero using only ${\Gamma }_{\ell }\le 1$. The zero-integral correction avoids importing a low-frequency greybody asymptotic that has not been proved here.

Figure fig:windows is the temporal-mode interface used before capture. The zero-integral correction is what makes the Fourier test vanish at $\omega =0$. Ordinary compact windowing, without that correction, does not by itself give finite occupation from $n(\omega )\le 1/({\beta }_{\mathrm{H}}\omega )$.