Manuscript XV-C@1 · XV-C.tex · section · 5afd124d

A specified local module

Complete null tests in a finite normal-jet module › A specified local module · lines 498–537

XV-C@1#eq:insert, XV-C@1#eq:jetrec

Let

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

where $X \subset R^{d}$ and $V_{q} \subset R^{m}$ are open, and $0 \in V_{q}$ .

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

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

Fix $T \in D^{'} (X)$ and an integer $r \geq 0$ . Define the specific presentation space

P_{r}^{jet} (T) = ∣ α ∣ \leq r ⨁ C^{\infty} (U)

and its reconstruction

Rec_{T} (a) = ∣ α ∣ \leq r \sum a_{α} E_{α} T .

Multiplication here is by smooth functions and is defined.

This is a local test module. It is not declared to equal the inherited $P_{int}$ .

Displayed formulas in this passage

XV-C@1#eq:insert · line 510
$⟨ E_{γ} b, ϕ ⟩ = (- 1)^{∣ γ ∣} ⟨ b, \partial_{q}^{γ} ϕ (x, 0)⟩, ϕ \in D (U) .$
XV-C@1#eq:jetrec · line 527
$Rec_{T} (a) = ∣ α ∣ \leq r \sum a_{α} E_{α} T .$
unlabeled · line 501
$U = X \times V_{q}, S = X \times {0},$
unlabeled · line 521
$P_{r}^{jet} (T) = ∣ α ∣ \leq r ⨁ C^{\infty} (U)$