Manuscript Em01-C@3 · Em01-C-v3.tex · proposition · f0953702

Proposition: Exact complete-sum scale identities

Scale identities and the direct convergence target · lines 1267–1300 · proof included or linked

Em01-C@3#prop:scale, Em01-C@3#eq:Dorigin, Em01-C@3#eq:increment, Em01-C@3#eq:Dvertex, Em01-C@3#eq:increment-vertex

The family $t_{δ}$ is smooth in the strong distributional topology on compact subintervals of $(0, \infty)$ . Set

D_{δ} = δ \partial_{δ} t_{δ}, t_{n} = t_{2^{- n}}, Δ_{n} = t_{n + 1} - t_{n} .

Then

D_{δ} (Φ) Δ_{n} (Φ) = Ω_{Φ} (ϑ_{δ}), = Ω_{Φ} (ζ_{2^{- n}}),

and equivalently

D_{δ} Δ_{n} = i V (K_{ret}, ϑ_{δ} * (χ C)), = i V (K_{ret}, ζ_{2^{- n}} * (χ C)) .

Proof. On a compact positive scale interval the profiles and their scale derivatives have common compact support and are smooth in every test seminorm. The estimate (eq:Omega-order) makes their evaluations smooth uniformly on bounded joint test sets. Differentiate or subtract (eq:origin-evaluation).

Displayed formulas in this passage

Em01-C@3#eq:Dorigin · line 1277
$D_{δ} (Φ) Δ_{n} (Φ) = Ω_{Φ} (ϑ_{δ}), = Ω_{Φ} (ζ_{2^{- n}}),$
Em01-C@3#eq:increment · line 1277
$D_{δ} (Φ) Δ_{n} (Φ) = Ω_{Φ} (ϑ_{δ}), = Ω_{Φ} (ζ_{2^{- n}}),$
Em01-C@3#eq:Dvertex · line 1284
$D_{δ} Δ_{n} = i V (K_{ret}, ϑ_{δ} * (χ C)), = i V (K_{ret}, ζ_{2^{- n}} * (χ C)) .$
Em01-C@3#eq:increment-vertex · line 1284
$D_{δ} Δ_{n} = i V (K_{ret}, ϑ_{δ} * (χ C)), = i V (K_{ret}, ζ_{2^{- n}} * (χ C)) .$
unlabeled · line 1272
$D_{δ} = δ \partial_{δ} t_{δ}, t_{n} = t_{2^{- n}}, Δ_{n} = t_{n + 1} - t_{n} .$

Proposition: Exact complete-sum scale identities

Displayed formulas in this passage

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