Manuscript IX-B@2 · IX-B-v2.tex · section · 8ce915d6

Residue and its product-test normalization

Fixed data and inherited inputs › Residue and its product-test normalization · lines 363–416

IX-B@2#eq:r, IX-B@2#eq:C

Following the VII-D residue definition, let

Z = R^{3} \times S^{2} \times (0, \infty)^{3},

Ψ (z, n, t, s, t^{'}) = ((t, z + t n), (s, z - s n), (t^{'}, z + t^{'} n)),

and

d ν = d^{3} z d Ω (n) d t d s d t^{'}, \int_{S^{2}} d Ω = 4 π .

The inherited positive order-zero distribution is

r = c_{0} Ψ_{*} ν, c_{0} = \frac{1}{64 ( 2 π ) ^{3}}, Rel = Ψ (Z) = supp r .

These are VII-D's prescribed residue and support VIID (def:residue, eq:rdef).

For the light-ray transform

A_{f} (y, n) = \int_{R} f (t, y + (t + s) n) d t,

the inherited product-test identity reads

⟨ r, f \otimes g \otimes f^{'} ⟩ = c_{0} \int g (y) d^{4} y \int_{S^{2}} A_{f} (y, n) A_{f^{'}} (y, n) d Ω (n) = C (f, g, f^{'}) .

Thus this pairing is exactly the VII-C light-ray coefficient, including every numerical factor VIIC (eq:residue) VIID (eq:rjoint, prop:ledger).

. The symbol $μ > 0$ denotes the subtraction scale and $ϑ_{ρ}$ the mollifier profile, as in VII-D; VII-C used $μ$ for the profile. Here $r$ always denotes the residue distribution, and $λ$ denotes a radial variable when needed. The reflected relation $Rel$ is distinct from the observation equality diagonals $Δ_{ij}$ .

Displayed formulas in this passage

IX-B@2#eq:r · line 382
$r = c_{0} Ψ_{*} ν, c_{0} = \frac{1}{64 ( 2 π ) ^{3}}, Rel = Ψ (Z) = supp r .$
IX-B@2#eq:C · line 397
$⟨ r, f \otimes g \otimes f^{'} ⟩ = c_{0} \int g (y) d^{4} y \int_{S^{2}} A_{f} (y, n) A_{f^{'}} (y, n) d Ω (n) = C (f, g, f^{'}) .$
unlabeled · line 366
$Z = R^{3} \times S^{2} \times (0, \infty)^{3},$
unlabeled · line 369
$Ψ (z, n, t, s, t^{'}) = ((t, z + t n), (s, z - s n), (t^{'}, z + t^{'} n)),$
unlabeled · line 376
$d ν = d^{3} z d Ω (n) d t d s d t^{'}, \int_{S^{2}} d Ω = 4 π .$
unlabeled · line 392
$A_{f} (y, n) = \int_{R} f (t, y + (t + s) n) d t,$