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

Profile moments and leading estimates

Weighted convolution and leading master estimates › Profile moments and leading estimates · lines 919–948

Em02-D@4#eq:moments, Em02-D@4#eq:translation

The density of $P^{0} - n \cdot P$ under $p$ is

P_{*} (t) = \frac{4 π N _{p}}{2} \int_{0}^{\infty} q^{2} b_{0} (t^{2} /2 + q^{2}) d q .

Set

H_{*} = P_{*} * P_{*}, ν = H_{*} (0) > 0, κ = H_{*}^{''} (0) = - \int_{R} ∣ P_{*}^{'} (t) ∣^{2} d t < 0.

The density $H_{*}$ is even, so $H_{*}^{'} (0) = H_{*}^{'''} (0) = 0$ .

Product convolution gives the exact translation formula

J_{δ} (S; w) = \int ρ (P, Q) J (S - δ (P + Q); w (\cdot + δ P)) d^{4} P d^{4} Q .

Displayed formulas in this passage

Em02-D@4#eq:moments · line 928
$H_{*} = P_{*} * P_{*}, ν = H_{*} (0) > 0, κ = H_{*}^{''} (0) = - \int_{R} ∣ P_{*}^{'} (t) ∣^{2} d t < 0.$
Em02-D@4#eq:translation · line 940
$J_{δ} (S; w) = \int ρ (P, Q) J (S - δ (P + Q); w (\cdot + δ P)) d^{4} P d^{4} Q .$
unlabeled · line 922
$P_{*} (t) = \frac{4 π N _{p}}{2} \int_{0}^{\infty} q^{2} b_{0} (t^{2} /2 + q^{2}) d q .$