Manuscript Em01-C@3 · Em01-C-v3.tex · section · f0953702

The scalar family

Explicit scalar cone asymptotics › The scalar family · lines 1405–1443

Em01-C@3#eq:scalar-P, Em01-C@3#eq:scalar-moments

Fix $α, h > 0$ , with $β = α + h$ . In the response-normal coordinate $q = q_{R} = 4 β u$ ,

q_{1} = 4 α v - ∣ ξ ∣^{2}, q_{2} = \frac{h}{β} q - 4 h v - ∣ ξ ∣^{2} .

Choose nonnegative $f_{0}, g_{0} \in C_{c}^{\infty} (R)$ with $\int f_{0} = \int g_{0} = 1$ , and define

P_{δ} (q, v, ξ) = δ^{- 2} f_{0} (\frac{4 α v - ∣ ξ ∣ ^{2}}{δ}) g_{0} (\frac{( h / β ) q - 4 h v - ∣ ξ ∣ ^{2}}{δ}) .

This is a cone-function regularization in four variables, relative to $d q d v d^{2} ξ$ . It is not (eq:regularization).

For $j = 0, 1$ , put

F_{j} (s, X) = f_{0} (4 α s - ∣ X ∣^{2}) g_{0}^{(j)} (- 4 h s - ∣ X ∣^{2}),

and define

m_{j} = \int F_{j}, s_{j} = \int s F_{j}, q_{j} = \int ∣ X ∣^{2} F_{j},

with integration over $R \times R^{2}$ .

Displayed formulas in this passage

Em01-C@3#eq:scalar-P · line 1415
$P_{δ} (q, v, ξ) = δ^{- 2} f_{0} (\frac{4 α v - ∣ ξ ∣ ^{2}}{δ}) g_{0} (\frac{( h / β ) q - 4 h v - ∣ ξ ∣ ^{2}}{δ}) .$
Em01-C@3#eq:scalar-moments · line 1436
$m_{j} = \int F_{j}, s_{j} = \int s F_{j}, q_{j} = \int ∣ X ∣^{2} F_{j},$
unlabeled · line 1409
$q_{1} = 4 α v - ∣ ξ ∣^{2}, q_{2} = \frac{h}{β} q - 4 h v - ∣ ξ ∣^{2} .$
unlabeled · line 1430
$F_{j} (s, X) = f_{0} (4 α s - ∣ X ∣^{2}) g_{0}^{(j)} (- 4 h s - ∣ X ∣^{2}),$