Manuscript XV-C@1 · XV-C.tex · lemma · 5afd124d

Lemma: Coefficient extraction

Complete null tests in a finite normal-jet module › A specified local module · lines 538–570 · proof included or linked

XV-C@1#lem:extraction

For distributions $b_{γ} \in D^{'} (X)$ ,

∣ γ ∣ \leq r \sum E_{γ} b_{γ} = 0 ⟺ b_{γ} = 0 for every ∣ γ ∣ \leq r .

Proof. Choose $χ \in D (V_{q})$ equal to one near zero and define

J_{γ} ψ (x, q) = \frac{( - 1 ) ^{∣ γ ∣}}{γ !} q^{γ} χ (q) ψ (x), ψ \in D (X) .

For all multi-indices $β$ ,

(- 1)^{∣ β ∣} \partial_{q}^{β} J_{γ} ψ (x, 0) = δ_{β γ} ψ (x) .

The fact that $χ$ is constant near zero makes all its positive order derivatives at zero vanish.

Testing the finite insertion sum against $J_{γ} ψ$ therefore gives $⟨ b_{γ}, ψ ⟩$ . This proves the forward implication; the reverse is immediate.

Displayed formulas in this passage

unlabeled · line 541
$∣ γ ∣ \leq r \sum E_{γ} b_{γ} = 0 ⟺ b_{γ} = 0 for every ∣ γ ∣ \leq r .$
unlabeled · line 550
$J_{γ} ψ (x, q) = \frac{( - 1 ) ^{∣ γ ∣}}{γ !} q^{γ} χ (q) ψ (x), ψ \in D (X) .$
unlabeled · line 558
$(- 1)^{∣ β ∣} \partial_{q}^{β} J_{γ} ψ (x, 0) = δ_{β γ} ψ (x) .$