Filtered compact realization of a band coefficient

We reproduce the needed bulk approximation rather than assuming that a packet coefficient is already a compact source.

For a compact-band coefficient $u$, choose a time center $t_j$. Let

Set real Cauchy data by

They have coefficient ${\omega }^{-2}u$ at $t_j$ and belong to every integer Sobolev space.

Choose a real $\chi \in {C_c}^{\infty }((-2,2))$, equal to one on $[-1,1]$. For $L\ge 1$, let

Let $s_L$ be their homogeneous Cauchy solution at $t_j$. With ${\eta }_T$ zero before $t_j-T$ and one after $t_j+T$, define

Its support is contained in