Manuscript XV-F-patch@1 · XV-F-patch.tex · section · 6d6ef16d

Generators, coefficients, and the reconstruction rule

Patch I: the local amplitude module and reconstruction › Generators, coefficients, and the reconstruction rule · lines 1211–1237

XV-F-patch@1#eq:pole

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{Defines in the source. XIV-I.tex; eq:generators, def:amodule, eq:amodule, eq:reconstruction; the registered definition is reported at lines 873–901; the generator display is adjacent to that definition.}

Use the source Fourier convention

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

The inverse transform has factor $(2 π)^{- 4}$ . Let $M_{R}$ denote the Fourier transform of the existing kernel $R$ . Its source pole representation is

M_{R} = - \frac{s}{2 ρ} (ω - s ρ - i 0)^{- 1} + \frac{s}{2 ρ ( ω + s ρ )}, ρ = ∣ ξ ∣ > 0, ∣ ω - s ρ ∣ < c ρ, 0 < c < 1.

No new choice of boundary value is made here.

Displayed formulas in this passage

XV-F-patch@1#eq:pole · line 1228
$M_{R} = - \frac{s}{2 ρ} (ω - s ρ - i 0)^{- 1} + \frac{s}{2 ρ ( ω + s ρ )}, ρ = ∣ ξ ∣ > 0, ∣ ω - s ρ ∣ < c ρ, 0 < c < 1.$
unlabeled · line 1219
$u (ω, ξ) = \int e^{- i (t ω + x \cdot ξ)} u (t, x) d t d^{3} x, κ = (ω, ξ) .$

Generators, coefficients, and the reconstruction rule

Displayed formulas in this passage

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