Manuscript Em02-D@4 · Em02-D-v4.tex · section · fcf2fac7

Fixed smoothing profile

Smoothing, vertex, and exact rows › Fixed smoothing profile · lines 447–489

Em02-D@4#eq:smoothing

Let

b_{0} (v) = {exp (- 1/ (1 - v)), 0, 0 \leq v < 1, v \geq 1, p (X) = N_{p} b_{0} (∣ X ∣_{E}^{2}), N_{p} = (π^{2} \int_{0}^{1} v b_{0} (v) d v)^{- 1} .

Then $p$ is smooth, even, radial, nonnegative, compactly supported, and normalized. Put

p_{δ} (X) = δ^{- 4} p (X / δ), ρ (P, Q) = p (P) p (Q) .

The crossed array is

C_{μν ∣ I} (U, V) = δ_{μ 0} δ_{ν 0} R (U) R (- V), I = 1, 2.

Choose $χ \in C_{c}^{\infty} (R^{8})$ equal to one near the relative image of $supp ζ$ , and smooth $χ C$ by $ρ_{δ}$ .

For sufficiently small scales, on a neighborhood of the retained support,

D_{δ} (S, x) = A_{δ} (x) A_{δ} (S - x), A_{δ} = p_{δ} * R .

This identity and all its derivatives hold there because translated profile supports stay in the plateau of $χ$ .

Displayed formulas in this passage

Em02-D@4#eq:smoothing · line 480
$D_{δ} (S, x) = A_{δ} (x) A_{δ} (S - x), A_{δ} = p_{δ} * R .$
unlabeled · line 450
$b_{0} (v) = {exp (- 1/ (1 - v)), 0, 0 \leq v < 1, v \geq 1, p (X) = N_{p} b_{0} (∣ X ∣_{E}^{2}), N_{p} = (π^{2} \int_{0}^{1} v b_{0} (v) d v)^{- 1} .$
unlabeled · line 463
$p_{δ} (X) = δ^{- 4} p (X / δ), ρ (P, Q) = p (P) p (Q) .$
unlabeled · line 468
$C_{μν ∣ I} (U, V) = δ_{μ 0} δ_{ν 0} R (U) R (- V), I = 1, 2.$