Manuscript XVI-K@1 · XVI-K.tex · section · aa7553bb

Taylor-complement tests

A coherent extension map on the finite span › Taylor-complement tests · lines 1150–1194

XVI-K@1#eq:Wrho, XVI-K@1#eq:W0

For $∣ α ∣, ∣ β ∣ \leq ρ$ , set

M^{(ρ)}_{β α} = \partial^{β} (\frac{z ^{α}}{α !} ϖ (z)) (0) .

Order multi-indices by increasing total degree. The matrix is triangular with diagonal entries one, because its entry vanishes unless $α \leq β$ . Define

w^{(ρ)}_{α} (z) W_{ρ} ψ = ∣ γ ∣ \leq ρ \sum ((M^{(ρ)})^{- 1})_{γ α} \frac{z ^{γ}}{γ !} ϖ (z), = ψ - ∣ α ∣ \leq ρ \sum \partial^{α} ψ (0) w^{(ρ)}_{α} .

Then

\partial^{β} w^{(ρ)}_{α} (0) = δ_{β α}, W_{ρ} w^{(ρ)}_{α} = 0.

Thus $W_{ρ}$ projects onto

D_{ρ} = {ψ \in C_{c}^{\infty} (R^{4}) : \partial^{α} ψ (0) = 0 for ∣ α ∣ \leq ρ} .

In particular,

w^{(0)}_{0} = ϖ, W_{0} ψ = ψ - ψ (0) ϖ .

The sans-serif projection $W_{0}$ is distinct from the positive-frequency distribution $W_{0}$ .

Displayed formulas in this passage

XVI-K@1#eq:Wrho · line 1160
$w^{(ρ)}_{α} (z) W_{ρ} ψ = ∣ γ ∣ \leq ρ \sum ((M^{(ρ)})^{- 1})_{γ α} \frac{z ^{γ}}{γ !} ϖ (z), = ψ - ∣ α ∣ \leq ρ \sum \partial^{α} ψ (0) w^{(ρ)}_{α} .$
XVI-K@1#eq:W0 · line 1186
$w^{(0)}_{0} = ϖ, W_{0} ψ = ψ - ψ (0) ϖ .$
unlabeled · line 1153
$M^{(ρ)}_{β α} = \partial^{β} (\frac{z ^{α}}{α !} ϖ (z)) (0) .$
unlabeled · line 1174
$\partial^{β} w^{(ρ)}_{α} (0) = δ_{β α}, W_{ρ} w^{(ρ)}_{α} = 0.$
unlabeled · line 1180
$D_{ρ} = {ψ \in C_{c}^{\infty} (R^{4}) : \partial^{α} ψ (0) = 0 for ∣ α ∣ \leq ρ} .$