Manuscript XV-B@2 · XV-B.tex · section · 4d3f12e8

Insertion and extraction

Finite normal jets and complete reconstruction-null tests › Insertion and extraction · lines 718–755

XV-B@2#eq:insertion, XV-B@2#eq:testlift, XV-B@2#eq:extractjets

Fix a product chart

U = X \times V_{q}, S = X \times {0},

where $X \subset R^{d}$ and $V_{q} \subset R^{m}$ are open and $0 \in V_{q}$ . The notation and representation are tied to these fixed normal coordinates.

For $γ \in N^{m}$ and $b \in D^{'} (X)$ , define

⟨ E_{γ} b, ϕ ⟩ = (- 1)^{∣ γ ∣} ⟨ b, \partial_{q}^{γ} ϕ (x, 0)⟩, ϕ \in D (U) .

Thus $E_{γ} b = b \otimes \partial_{q}^{γ} δ_{0}$ in the chosen product coordinates.

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

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

These are continuous maps between the corresponding test-function spaces, and

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

Displayed formulas in this passage

XV-B@2#eq:insertion · line 729
$⟨ E_{γ} b, ϕ ⟩ = (- 1)^{∣ γ ∣} ⟨ b, \partial_{q}^{γ} ϕ (x, 0)⟩, ϕ \in D (U) .$
XV-B@2#eq:testlift · line 740
$J_{γ} ψ (x, q) = \frac{( - 1 ) ^{∣ γ ∣}}{γ !} q^{γ} χ (q) ψ (x), ψ \in D (X) .$
XV-B@2#eq:extractjets · line 749
$(- 1)^{∣ β ∣} \partial_{q}^{β} J_{γ} ψ (x, 0) = δ_{β γ} ψ (x) .$
unlabeled · line 721
$U = X \times V_{q}, S = X \times {0},$

Insertion and extraction

Displayed formulas in this passage

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