Finite switching and nonvacuum mode mismatch

A compactly switched temporal capture cannot exactly select the nonzero band-limited packet $p$. We construct an explicit approximation, including a low-frequency integrability repair.

Choose real $\chi \in {C_c}^{\infty }((-2,2))$, with $0\le \chi \le 1$ and $\chi =1$ on $[-1,1]$. Choose $\psi \in {C_c}^{\infty }((-2,2))$ with $\int \psi (t)\mathrm{d}t=1$. For $L>0$, set

For sufficiently large $L$, $v_L\ne 0$. Normalize $v_L$ and choose its constant phase so that

Write

When ${\theta }_L=0$, omit $g_L$ and all mismatch terms.