Manuscript XIV-J@1 · XIV-J.tex · proposition · 1c0e72ca

Proposition: Finite normal-jet characterization

J2d: a conditional extension interface, not an extension map › Normal order as an algebraic test condition · lines 1452–1495 · proof included or linked

XIV-J@1#prop:characterization, XIV-J@1#eq:generalnormal, XIV-J@1#eq:extraction

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A distribution $t \in D^{'} (U)$ annihilates $I_{m}$ if and only if it has a unique expression

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

The coefficients are recovered continuously by

⟨ b_{α}, ψ ⟩ = (- 1)^{∣ α ∣} ⟨ t, J_{α} ψ ⟩ .

In particular such a distribution is supported on $S_{resp}$ .

Proof. An expression (eq:generalnormal) plainly annihilates $I_{m}$ . Its coefficients are unique by Lemma lem:jetinsert.

Conversely, choose the same cutoff $χ$ as in (eq:Jjet) and subtract the finite normal Taylor jet of a test:

φ (s, n) - χ (n) ∣ α ∣ \leq m \sum \frac{n ^{α}}{α !} \partial_{n}^{α} φ (s, 0) \in I_{m} .

If $t$ annihilates this difference, its value is determined by the finite displayed jet. Defining the coefficients by (eq:extraction) therefore gives (eq:generalnormal). Continuity follows from continuity of the test-extension maps $J_{α}$ .

A test supported off $S_{resp}$ has every jet zero, so $t$ vanishes on that test.

Displayed formulas in this passage

XIV-J@1#eq:generalnormal · line 1456
$t = ∣ α ∣ \leq m \sum E_{α} b_{α}, b_{α} \in D^{'} (S_{0}) .$
XIV-J@1#eq:extraction · line 1462
$⟨ b_{α}, ψ ⟩ = (- 1)^{∣ α ∣} ⟨ t, J_{α} ψ ⟩ .$
unlabeled · line 1480
$φ (s, n) - χ (n) ∣ α ∣ \leq m \sum \frac{n ^{α}}{α !} \partial_{n}^{α} φ (s, 0) \in I_{m} .$

Proposition: Finite normal-jet characterization

Displayed formulas in this passage

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