Lemma: Graded ideal

${\mathfrak{J}}_0$ is graded by field degree, and ${\mathfrak{J}}_0\cap {{\mathcal{A}}_{\mathrm{off}}}^{(1)}=\{F_{P_{m}v}:v\in \mathcal{D}\}$. Consequently, for $d\in \mathcal{D}$, $\hat{\varphi }(d)=0$ in ${\mathfrak{A}}_0$ if and only if $d\in P_m\mathcal{D}$.

Proof. A general element of the pointwise ideal is a finite sum ${\sum }_i{G}_i{F}_{P_m{u}_i}$ with $G_i\in {\mathcal{A}}_{\mathrm{off}}$. Decomposing each $G_i$ into homogeneous parts and using that the pointwise product of homogeneous functionals of degrees $p,q$ is homogeneous of degree $p+q$, every homogeneous component of the sum is again in ${\mathfrak{J}}_0$; so ${\mathfrak{J}}_0$ is graded. Its degree-one part consists of ${\sum }_i{\lambda }_i{F}_{P_m{u}_i}=F_{P_m({\sum }_i{\lambda }_i{u}_i)}$, ${\lambda }_i\in \mathbb{C}$. Finally $F_d\in {\mathfrak{J}}_0$ iff $F_d\in {\mathfrak{J}}_0\cap {{\mathcal{A}}_{\mathrm{off}}}^{(1)}$ iff $F_d=F_{P_{m}v}$ for some $v$, iff $d=P_{m}v$, since $F_d$ determines $d$ (test on $\phi \in \mathcal{D}$).