Manuscript IX-C@1 · IX-C.tex · proposition · 03ce8e6b

Proposition: Numerical observation descendants

Observation compatibility, continuity, and one parameter · lines 1141–1167 · proof included or linked

IX-C@1#prop:descendants

For the fixed scalar calibration (eq:Cal), and a fixed smooth-coefficient differential operator $D$ on $X$ , the corresponding numerical descendant is

a D (u_{μ}^{ρ} + cr) .

For example,

N_{c} (\partial_{t}^{n} f, g, f^{'}) = ⟨ (- \partial_{x^{0}})^{n} (u_{μ}^{ρ} + cr), f \otimes g \otimes f^{'} ⟩ .

Under Assumption ass:calibration, IX-B supplies the associated strong numerical finite-part estimate with test-order loss $64 \to 64 + ord D$ .

Proof. The displayed equality is the definition of distributional differentiation. Smooth multiplication and differential operators act linearly, so the parent shift $cr$ becomes $c a D r$ . The quantified transfer is inherited from IX-B IXB (thm:descendants, cor:conversion).

Displayed formulas in this passage

unlabeled · line 1146
$a D (u_{μ}^{ρ} + cr) .$
unlabeled · line 1150
$N_{c} (\partial_{t}^{n} f, g, f^{'}) = ⟨ (- \partial_{x^{0}})^{n} (u_{μ}^{ρ} + cr), f \otimes g \otimes f^{'} ⟩ .$

Proposition: Numerical observation descendants

Displayed formulas in this passage

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