Manuscript Em01-B@3 · Em01-B-v3.tex · lemma · 69189992

Lemma: Retarded multiplier and pole identity

Internal reconstruction and all-region frequency estimates › Exact retarded multiplier · lines 841–886 · proof included or linked

Em01-B@3#lem:pole, Em01-B@3#eq:partial-fractions, Em01-B@3#eq:pole

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The Fourier transform of the fixed kernel is

M_{R} (ω, ξ) = - δ ↓ 0 lim [(ω - i δ)^{2} - ρ^{2}]^{- 1} .

For $ρ > 0$ ,

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

Near a nonzero characteristic sheet $λ_{s} = ω - s ρ = 0$ , with $∣ λ_{s} ∣ < c ρ$ , $0 < c < 1$ ,

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

The full multiplier is homogeneous of degree $- 2$ .

Proof. The spatial Fourier transform of $R$ is $Θ (t) sin (ρt) / ρ$ . Temporal damping gives

\int_{0}^{\infty} e^{- (δ + iω) t} \frac{sin ( ρt )}{ρ} d t = - [(ω - i δ)^{2} - ρ^{2}]^{- 1} .

The damped existing kernels converge in tempered distributions. Partial fractions give the two identities.

The physical kernel has homogeneity degree $- 2$ in dimension four, so its Fourier transform has degree $- 2$ . Alternatively positive dilation in the damped identity gives the same distributional conclusion.

Displayed formulas in this passage

Em01-B@3#eq:partial-fractions · line 851
$M_{R} = - \frac{1}{2 ρ} [(ω - ρ - i 0)^{- 1} - (ω + ρ - i 0)^{- 1}] .$
Em01-B@3#eq:pole · line 861
$M_{R} = - \frac{s}{2 ρ} (λ_{s} - i 0)^{- 1} + \frac{s}{2 ρ ( ω + s ρ )} .$
unlabeled · line 844
$M_{R} (ω, ξ) = - δ ↓ 0 lim [(ω - i δ)^{2} - ρ^{2}]^{- 1} .$
unlabeled · line 874
$\int_{0}^{\infty} e^{- (δ + iω) t} \frac{sin ( ρt )}{ρ} d t = - [(ω - i δ)^{2} - ρ^{2}]^{- 1} .$

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section: Exact retarded multiplier