Manuscript Em01-D@2 · Em01-D-v2.tex · corollary · d3b49151

Corollary: Exact scale derivative

An intrinsic complete-sum translation representation · lines 1237–1266 · proof included or linked

Em01-D@2#cor:scale, Em01-D@2#eq:scaled

Put

θ (h) = - 8 ρ (h) - h \cdot \nabla ρ (h) = - div (h ρ (h)) .

Then

\int θ = 0, δ \partial_{δ} ρ_{δ} = θ_{δ},

and

δ \partial_{δ} t_{δ} (Φ) = Ω_{Φ} (θ_{δ}) .

The family is strongly smooth on compact positive-scale intervals.

Proof. Differentiate the compact profile in its test-function topology. On a compact positive-scale interval, all profiles and scale derivatives have common support and uniformly bounded seminorms. Estimate (eq:Omegaorder) justifies differentiation uniformly on bounded joint test sets.

Displayed formulas in this passage

Em01-D@2#eq:scaled · line 1250
$δ \partial_{δ} t_{δ} (Φ) = Ω_{Φ} (θ_{δ}) .$
unlabeled · line 1240
$θ (h) = - 8 ρ (h) - h \cdot \nabla ρ (h) = - div (h ρ (h)) .$
unlabeled · line 1245
$\int θ = 0, δ \partial_{δ} ρ_{δ} = θ_{δ},$

Corollary: Exact scale derivative

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