Manuscript Em02-E@1 · Em02-E-v1.tex · section · f3795050

Linear and quadratic functionals

Classical covariance and the quadratic kernel bridge › Linear and quadratic functionals · lines 889–929

Em02-E@1#eq:PB, Em02-E@1#eq:Llocal

Fix a real compact test density

f = f_{0} (t, x) d t d^{3} x .

For an affine perturbation $η = (h, φ)$ , set $k = ψ = 0$ in the coefficient formulas and define

L_{f} (η) = \int f n_{1} [η], Q_{f} (η) = \frac{1}{2} \int f n_{2} [η] .

Then

O_{f} (Γ_{0} + λ η) = O_{f} (Γ_{0}) + λ L_{f} (η) + λ^{2} Q_{f} (η) (mod λ^{3}) .

Define the linear differential operations

P η = n_{1} [η], B^{i} η = \frac{1}{2} \partial_{i} tr Q_{1} [η] - \partial_{j} Q_{1}^{j i} [η] .

Set

L_{loc} [η] = \frac{1}{2} N_{2} + u_{1} N ˙_{1} + \frac{1}{2} N ˙_{0} u_{2} + \frac{1}{2} N ¨_{0} u_{1}^{2}, k = ψ = 0.

This is a local quadratic expression. Every coefficient in it is fixed by the preceding formulas.

Displayed formulas in this passage

Em02-E@1#eq:PB · line 911
$P η = n_{1} [η], B^{i} η = \frac{1}{2} \partial_{i} tr Q_{1} [η] - \partial_{j} Q_{1}^{j i} [η] .$
Em02-E@1#eq:Llocal · line 919
$L_{loc} [η] = \frac{1}{2} N_{2} + u_{1} N ˙_{1} + \frac{1}{2} N ˙_{0} u_{2} + \frac{1}{2} N ¨_{0} u_{1}^{2}, k = ψ = 0.$
unlabeled · line 892
$f = f_{0} (t, x) d t d^{3} x .$
unlabeled · line 897
$L_{f} (η) = \int f n_{1} [η], Q_{f} (η) = \frac{1}{2} \int f n_{2} [η] .$
unlabeled · line 903
$O_{f} (Γ_{0} + λ η) = O_{f} (Γ_{0}) + λ L_{f} (η) + λ^{2} Q_{f} (η) (mod λ^{3}) .$