Manuscript Em01-A@1 · Em01-A.tex · lemma · 880b67af

Lemma: Unique normal coefficients

A precise supported normal-jet interface › Support and normal order must be explicit · lines 1208–1259 · proof included or linked

Em01-A@1#lem:normal-coefficients, Em01-A@1#eq:normal-form

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Every distribution of supported normal order at most $m$ has a unique representation

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

Proof. Choose $ρ \in C_{c}^{\infty} (R^{k})$ equal to one near zero, and define

J_{γ} ψ (s, z) = (- 1)^{∣ γ ∣} \frac{z ^{γ}}{γ !} ρ (z) ψ (s) .

For $∣ α ∣, ∣ γ ∣ \leq m$ ,

\partial_{z}^{α} J_{γ} ψ (s, 0) = (- 1)^{∣ γ ∣} δ_{α γ} ψ (s) .

Consequently

⟨ E_{α} b, J_{γ} ψ ⟩ = δ_{α γ} ⟨ b, ψ ⟩ .

This proves uniqueness and supplies the coefficient extraction formula

⟨ b_{γ}, ψ ⟩ = T (J_{γ} ψ) .

These coefficients are distributions because $J_{γ}$ is continuous on test spaces.

For a general test $φ$ , subtract its finite normal Taylor polynomial with the cutoff $ρ$ :

φ (s, z) - ρ (z) ∣ γ ∣ \leq m \sum \frac{z ^{γ}}{γ !} \partial_{z}^{γ} φ (s, 0) .

The resulting test has all normal derivatives through order $m$ equal to zero along $S$ . Applying the defining annihilation property of $T$ proves (eq:normal-form).

Displayed formulas in this passage

Em01-A@1#eq:normal-form · line 1212
$T = ∣ γ ∣ \leq m \sum E_{γ} b_{γ}, b_{γ} \in D^{'} (S_{0}) .$
unlabeled · line 1222
$J_{γ} ψ (s, z) = (- 1)^{∣ γ ∣} \frac{z ^{γ}}{γ !} ρ (z) ψ (s) .$
unlabeled · line 1228
$\partial_{z}^{α} J_{γ} ψ (s, 0) = (- 1)^{∣ γ ∣} δ_{α γ} ψ (s) .$
unlabeled · line 1234
$⟨ E_{α} b, J_{γ} ψ ⟩ = δ_{α γ} ⟨ b, ψ ⟩ .$
unlabeled · line 1240
$⟨ b_{γ}, ψ ⟩ = T (J_{γ} ψ) .$
unlabeled · line 1248
$φ (s, z) - ρ (z) ∣ γ ∣ \leq m \sum \frac{z ^{γ}}{γ !} \partial_{z}^{γ} φ (s, 0) .$

Lemma: Unique normal coefficients

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