Remark: Independence of the basis

For $u\in {\mathcal{H}}_k$ and $\rho =k-4$, the zero-moment construction reads $⟨{\mathfrak{R}}_{\mathrm{L}}(u),\phi ⟩=⟨u,W_{\rho }\phi ⟩$ on test functions, using the unique extension of $u$ to functions vanishing to order $\rho$ at $0$. Hence ${\mathfrak{R}}_{\mathrm{L}}{|}_{{\mathcal{H}}_k}$ does not depend on ${\mathcal{B}}_k$. Combined with Proposition prop:RL(3), the restriction identity is likewise basis-free.