Manuscript III@2 · III.tex · section · 82dad3e5

Fourier compensation error

Sign, inverse, boundary, and kernel audit › Fourier compensation error · lines 3048–3074

The regulated inverse obeys

N_{ρ} Δ = χ_{ρ} (D),

δ_{0} w_{P}_{ρ}^{μ} = - χ_{ρ} (D) ξ^{μ}

and

δ_{ξ} (ϕ + κ w_{P}_{ρ}^{μ} \partial_{μ} ϕ) = κ ((1 - χ_{ρ} (D)) ξ^{μ}) \partial_{μ} ϕ + O (κ^{2}) .

The multiplier need not be a projection. Its error is not the finite-radius boundary error.

For fixed compact smooth test data, this formal compensation error tends to zero. It does not repair the unchanged-product gap, where the scalar is held fixed and the limiting difference is $- κ J$ .

Displayed formulas in this passage

unlabeled · line 3051
$N_{ρ} Δ = χ_{ρ} (D),$
unlabeled · line 3055
$δ_{0} w_{P}_{ρ}^{μ} = - χ_{ρ} (D) ξ^{μ}$
unlabeled · line 3059
$δ_{ξ} (ϕ + κ w_{P}_{ρ}^{μ} \partial_{μ} ϕ) = κ ((1 - χ_{ρ} (D)) ξ^{μ}) \partial_{μ} ϕ + O (κ^{2}) .$