Manuscript En01-B@2 · En01-B-v2.tex · lemma · f226d1eb

Lemma: Radial autocorrelation after Gaussian smearing

Gaussian regularization: sign, margin, and validated integration › A sign theorem for every finite Gaussian width · lines 1665–1698 · proof included or linked

En01-B@2#lem:radial

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For a homogeneous sphere and $r_{c} > 0$ , its smeared autocorrelation

w_{r_{c}} = w_{0} * g_{2 r_{c}}

is positive, smooth, radial, and strictly decreasing at every radius $r > 0$ .

Proof. The unsmeared $w_{0}$ is nonnegative, radial, and decreasing on $0 < r < 2 R$ , and vanishes outside that ball. Its layer-cake representation is a positive mixture of indicators of balls of nonzero radii.

For a ball of radius $a > 0$ , let

L_{a} (r) = \int_{∣ z ∣ \leq a} g_{2 r_{c}} (r e - z) d z .

By the divergence theorem,

L_{a}^{'} (r) = - \int_{∣ z ∣ = a} (e \cdot n_{z}) g_{2 r_{c}} (r e - z) d S (z) .

Pair a point with $e \cdot n_{z} > 0$ with its reflection across the plane orthogonal to $e$ . For $r > 0$ , the Gaussian value at the nearer point is strictly larger. The paired integral before the minus sign is positive. Thus $L_{a}^{'} (r) < 0$ . Integrate this strict inequality against the positive layer-cake mixture. Smoothness and positivity follow from Gaussian convolution.

Displayed formulas in this passage

unlabeled · line 1668
$w_{r_{c}} = w_{0} * g_{2 r_{c}}$
unlabeled · line 1682
$L_{a} (r) = \int_{∣ z ∣ \leq a} g_{2 r_{c}} (r e - z) d z .$
unlabeled · line 1686
$L_{a}^{'} (r) = - \int_{∣ z ∣ = a} (e \cdot n_{z}) g_{2 r_{c}} (r e - z) d S (z) .$

Lemma: Radial autocorrelation after Gaussian smearing

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