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

Theorem: Complete null criterion for the specified module

Finite normal jets and complete reconstruction-null tests › Smooth multiplication and the specified module · lines 892–936 · proof included or linked

XV-B@2#thm:jetnull, XV-B@2#eq:fcoeff, XV-B@2#eq:B, XV-B@2#eq:jetnormal, XV-B@2#eq:completekernel

Back to manuscript

For $a = (a_{α}) \in P_{r}^{jet} (T)$ , define the smooth functions

f_{γ} (a) (x) = α \geq γ ∣ α ∣ \leq r \sum (- 1)^{∣ α - γ ∣} (γ α) (\partial_{q}^{α - γ} a_{α}) (x, 0)

and the distributions

B_{γ} (a) = f_{γ} (a) T .

Then

Rec_{T} (a) = ∣ γ ∣ \leq r \sum E_{γ} B_{γ} (a),

and

ker Rec_{T} = {a : B_{γ} (a) = 0 in D^{'} (X) for every ∣ γ ∣ \leq r} .

Thus (eq:completekernel) is a complete distributional null-relation test.

Proof. Apply Lemma lem:leibniz to each summand $a_{α} E_{α} T$ and collect the coefficient of $E_{γ}$ . This gives (eq:fcoeff)–(eq:jetnormal). Lemma lem:extraction then gives (eq:completekernel).

Displayed formulas in this passage

XV-B@2#eq:fcoeff · line 896
$f_{γ} (a) (x) = α \geq γ ∣ α ∣ \leq r \sum (- 1)^{∣ α - γ ∣} (γ α) (\partial_{q}^{α - γ} a_{α}) (x, 0)$
XV-B@2#eq:B · line 906
$B_{γ} (a) = f_{γ} (a) T .$
XV-B@2#eq:jetnormal · line 911
$Rec_{T} (a) = ∣ γ ∣ \leq r \sum E_{γ} B_{γ} (a),$
XV-B@2#eq:completekernel · line 917
$ker Rec_{T} = {a : B_{γ} (a) = 0 in D^{'} (X) for every ∣ γ ∣ \leq r} .$

Theorem: Complete null criterion for the specified module

Displayed formulas in this passage

Nearby in this section