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

Definition: Finite retarded-pole amplitude module; transcribed

Patch I: the local amplitude module and reconstruction › Generators, coefficients, and the reconstruction rule · lines 1238–1270

XV-F-patch@1#def:amodule, XV-F-patch@1#eq:generators, XV-F-patch@1#eq:amodule, XV-F-patch@1#eq:reconstruction

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For an adapted chart $V$ and an integer $m \geq - 2$ , define

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

The module $A_{R}^{m} (V)$ consists of finite expressions

A (w; θ, κ) = j + ∣ β ∣ \leq m + 2 \sum c_{j β} (w) A_{j β} (θ, κ), c_{j β} \in C^{\infty} (V),

with the finite inherited tensor components understood. The coefficients carry their usual compact-set smooth seminorms. The frequency factors are distributions, not ordinary smooth symbols at the characteristic pole.

The module is filtered by the polynomial degree $j + ∣ β ∣$ . Its reconstruction is

Rec_{V, m} (A) = j + ∣ β ∣ \leq m + 2 \sum c_{j β} (w) δ^{(j)} (q) \partial_{r}^{β} R (r) .

Normal inverse Fourier transformation is performed on each tensor factor, and coefficient multiplication is performed afterward.

Displayed formulas in this passage

XV-F-patch@1#eq:generators · line 1241
$A_{j β} (θ, κ) = (i θ)^{j} (iκ)^{β} M_{R} (κ) .$
XV-F-patch@1#eq:amodule · line 1247
$A (w; θ, κ) = j + ∣ β ∣ \leq m + 2 \sum c_{j β} (w) A_{j β} (θ, κ), c_{j β} \in C^{\infty} (V),$
XV-F-patch@1#eq:reconstruction · line 1261
$Rec_{V, m} (A) = j + ∣ β ∣ \leq m + 2 \sum c_{j β} (w) δ^{(j)} (q) \partial_{r}^{β} R (r) .$

Definition: Finite retarded-pole amplitude module; transcribed

Displayed formulas in this passage

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