Manuscript X-A-ST@1 · X-A-ST.tex · lemma · d9ccbde1

Lemma: Smooth smeared kernel

Support, continuity, and the smooth-kernel statement › The observation-smeared linear coordinate kernel · lines 1660–1689 · proof included or linked

X-A-ST@1#lem:kernel, X-A-ST@1#eq:smearedkernel

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With $b = E_{A} a$ ,

J_{ε} (a) [h] = \int h_{ρ σ} q_{a, ε}^{ρ σ} d^{4} x, q_{a, ε} = σ C^{*} (E_{A} a) .

This tensor is smooth, compactly supported on the retained slab, and extends smoothly by zero across $t = 0$ . Its support is contained in $J_{η}^{-} (supp a) \cap {t \geq 0}$ .

Proof. Transposition gives

\int a_{μ} w^{μ} = - \int b_{μ} \partial_{α} (σ a_{1}^{α μ} [h]) = \int σ (\partial_{α} b_{μ}) a_{1}^{α μ} [h] .

There is no initial boundary contribution because $σ$ is flat and zero at $t = 0$ ; the terminal one vanishes because $b$ is terminally zero. Insertion of $a_{1}$ gives (eq:smearedkernel). Advanced smoothness and the flat ramp give the asserted regularity and support.

Displayed formulas in this passage

X-A-ST@1#eq:smearedkernel · line 1663
$J_{ε} (a) [h] = \int h_{ρ σ} q_{a, ε}^{ρ σ} d^{4} x, q_{a, ε} = σ C^{*} (E_{A} a) .$
unlabeled · line 1678
$\int a_{μ} w^{μ} = - \int b_{μ} \partial_{α} (σ a_{1}^{α μ} [h]) = \int σ (\partial_{α} b_{μ}) a_{1}^{α μ} [h] .$

Lemma: Smooth smeared kernel

Displayed formulas in this passage

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