Manuscript XV-A@1 · XV-A.tex · proposition · f53f3515

Proposition: Finite normal-jet characterization

The supported-sector reference and its conditional interface › A support-explicit normal-jet theorem · lines 1289–1342 · proof included or linked

XV-A@1#prop:normal, XV-A@1#eq:normal

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A distribution has supported normal order at most $r$ if and only if it has a unique representation

t = ∣ γ ∣ \leq r \sum E_{γ} b_{γ}, b_{γ} \in D^{'} (S_{0}) .

Proof. Choose $χ \in D (V)$ equal to one near $0$ and set

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

For $∣ β ∣, ∣ γ ∣ \leq r$ ,

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

If $t$ has the stated normal order, define

⟨ b_{γ}, ψ ⟩ = ⟨ t, J_{γ} ψ ⟩ .

The lift is continuous on test-function spaces, so $b_{γ}$ is a distribution.

For $ϕ \in D (U)$ , put

a_{γ} (x) = (- 1)^{∣ γ ∣} \partial_{q}^{γ} ϕ (x, 0) .

The difference

ϕ - ∣ γ ∣ \leq r \sum J_{γ} a_{γ}

has all normal jets through order $r$ equal to zero. Applying $t$ therefore gives

⟨ t, ϕ ⟩ = ∣ γ ∣ \leq r \sum ⟨ b_{γ}, a_{γ} ⟩,

which is (eq:normal).

Conversely, each insertion in (eq:insert) is supported in $S$ and depends only on the indicated jet, proving the stated normal order. Testing (eq:normal) on $J_{γ} ψ$ extracts $b_{γ}$ and proves uniqueness.

Displayed formulas in this passage

XV-A@1#eq:normal · line 1293
$t = ∣ γ ∣ \leq r \sum E_{γ} b_{γ}, b_{γ} \in D^{'} (S_{0}) .$
unlabeled · line 1302
$J_{γ} ψ (x, q) = \frac{( - 1 ) ^{∣ γ ∣}}{γ !} q^{γ} χ (q) ψ (x) .$
unlabeled · line 1308
$(- 1)^{∣ β ∣} \partial_{q}^{β} J_{γ} ψ (x, 0) = δ_{β γ} ψ (x) .$
unlabeled · line 1314
$⟨ b_{γ}, ψ ⟩ = ⟨ t, J_{γ} ψ ⟩ .$
unlabeled · line 1322
$a_{γ} (x) = (- 1)^{∣ γ ∣} \partial_{q}^{γ} ϕ (x, 0) .$
unlabeled · line 1327
$ϕ - ∣ γ ∣ \leq r \sum J_{γ} a_{γ}$
unlabeled · line 1332
$⟨ t, ϕ ⟩ = ∣ γ ∣ \leq r \sum ⟨ b_{γ}, a_{γ} ⟩,$

Proposition: Finite normal-jet characterization

Displayed formulas in this passage

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