Manuscript XIV-J@1 · XIV-J.tex · input · 1c0e72ca

Input statement: XIV-I amplitude and reconstruction

J0: copied premises and their domains › The internal pole module · lines 334–384

XIV-J@1#input:Imodule, XIV-J@1#eq:pole, XIV-J@1#eq:Rec

In the Fourier convention

u (ω, ξ) = \int e^{- i (t ω + x \cdot ξ)} u (t, x) d t d^{3} x,

write

ρ = ∣ ξ ∣, p (ω, ξ) = ω^{2} - ρ^{2}, λ_{s} = ω - s ρ, s \in {+ 1, - 1} .

On a conic neighborhood

ρ > 0, ∣ λ_{s} ∣ < c ρ, 0 < c < 1,

the fixed internal multiplier is

M_{R} = - \frac{s}{2 ρ} (ω - s ρ - i 0)^{- 1} + \frac{s}{2 ρ ( ω + s ρ )} .

This is the Fourier transform of the existing retarded kernel, not a prescription for a new graph product.

The finite internal amplitude generators are

A_{j β} (θ, κ) = (i θ)^{j} (iκ)^{β} M_{R} (κ) .

For smooth coefficient arrays the reconstruction is

Rec j, β \sum c_{j β} (w) A_{j β} = j, β \sum c_{j β} (w) δ^{(j)} (q) \partial_{r}^{β} R (r) .

For both $K_{5}$ placements, $j + ∣ β ∣ \leq 4$ . The module has uniform finite-seminorm estimates on normalized frequency annuli meeting the pole, uniform after finitely many coefficient derivatives on compact parameter sets. Its finite lower-degree remainder is an internal frequency-distribution estimate only.

The reconstruction is a joint distributional map with $z$ retained; it is not a claim that singular kernels have pointwise slices at every residual value XIVI.

Displayed formulas in this passage

XIV-J@1#eq:pole · line 352
$M_{R} = - \frac{s}{2 ρ} (ω - s ρ - i 0)^{- 1} + \frac{s}{2 ρ ( ω + s ρ )} .$
XIV-J@1#eq:Rec · line 367
$Rec j, β \sum c_{j β} (w) A_{j β} = j, β \sum c_{j β} (w) δ^{(j)} (q) \partial_{r}^{β} R (r) .$
unlabeled · line 337
$u (ω, ξ) = \int e^{- i (t ω + x \cdot ξ)} u (t, x) d t d^{3} x,$
unlabeled · line 343
$ρ = ∣ ξ ∣, p (ω, ξ) = ω^{2} - ρ^{2}, λ_{s} = ω - s ρ, s \in {+ 1, - 1} .$
unlabeled · line 348
$ρ > 0, ∣ λ_{s} ∣ < c ρ, 0 < c < 1,$
unlabeled · line 362
$A_{j β} (θ, κ) = (i θ)^{j} (iκ)^{β} M_{R} (κ) .$