Lemma: Orthonormal finite windows with infrared control

For sufficiently large $L$:

1. $f_{1,L},f_{2,L}$ are orthonormal smooth tests supported in $(-2L,2L)$, and each has zero integral;

2. their Fourier transforms converge columnwise to $h_1,h_2$ in the Schwartz topology and in the weighted occupation norm;

3. the participating plane

   $${\mathfrak{v}}_L=\mathrm{ran}H+\mathrm{ran}{F}_L$$

   has dimension at most four and a finite occupation matrix.

Proof. Each $p_i$ is Schwartz, and $\int p_i=0$ because $h_i$ vanishes near zero. Hence

decays faster than every inverse power of $L$. The windowed error and the correcting bump tend to zero in every Schwartz seminorm. Thus $G_L\to I$; the inverse square root and the polar factors tend to $I$. Their finite matrix operations preserve supports and zero integrals.

For $\omega >0$,

A Fourier test $w$ with $w(0)=0$ satisfies

On $[1,\infty )$, $n$ is bounded. Schwartz convergence therefore implies weighted convergence. It also proves finite occupation for every vector in ${\mathfrak{v}}_L$.