Corollary: Transmission and narrow-band errors

Let $h$ be as in Theorem thm:main.

If an approximation ${\tilde{\Gamma }}_{\ell }$ satisfies

then

Writing $q(\omega )={\Gamma }_{\ell }(\omega )/(e^{{\beta }_{\mathrm{H}}\omega }-1)$, for every ${\omega }_0\in B$,

Thus normalized packets in bands shrinking to a fixed ${\omega }_0>0$ have expectation tending to $q({\omega }_0)$.

Proof. For (eq:gammaerror), integrate the absolute error against $|h{|}^2$, use the stated uniform transmission error, and use

For (eq:banderror), subtract $q({\omega }_0)$ inside (eq:packetlaw) and use $\int |h{|}^2=1$. Continuity follows from Lemma lem:scattering.