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

Theorem: Finite normal-jet characterization

Finite normal jets and complete reconstruction-null tests › Insertion and extraction · lines 794–833 · proof included or linked

XV-B@2#thm:normal, XV-B@2#eq:normal

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

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

Proof. Suppose $t$ has the stated normal order. Define

⟨ b_{γ}, ψ ⟩ = ⟨ t, J_{γ} ψ ⟩ .

Continuity of $J_{γ}$ makes $b_{γ}$ a distribution.

For $ϕ \in D (U)$ , set

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

Each $a_{γ}$ is a test on $X$ , and

ϕ - ∣ γ ∣ \leq r \sum J_{γ} a_{γ}

has zero normal jets through order $r$ . Consequently,

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

which is (eq:normal).

Conversely, every insertion is supported on $S$ and depends only on its indicated jet. The finite sum therefore has the required normal order. Lemma lem:extraction proves uniqueness.

Displayed formulas in this passage

XV-B@2#eq:normal · line 798
$t = ∣ γ ∣ \leq r \sum E_{γ} b_{γ}, b_{γ} \in D^{'} (X) .$
unlabeled · line 807
$⟨ b_{γ}, ψ ⟩ = ⟨ t, J_{γ} ψ ⟩ .$
unlabeled · line 814
$a_{γ} (x) = (- 1)^{∣ γ ∣} \partial_{q}^{γ} ϕ (x, 0) .$
unlabeled · line 819
$ϕ - ∣ γ ∣ \leq r \sum J_{γ} a_{γ}$
unlabeled · line 824
$⟨ t, ϕ ⟩ = ∣ γ ∣ \leq r \sum ⟨ b_{γ}, a_{γ} ⟩,$

Theorem: Finite normal-jet characterization

Displayed formulas in this passage

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