Manuscript V-B@1 · V-B-v1.tex · section · 3fadc1ad

Raw and harmonic criteria

Admissibility on the declared solution spaces › Raw and harmonic criteria · lines 689–726

V-B@1#eq:zetadata

Fix a time $t_{-} < 0$ . On

S_{ret}^{-} = {h \in S : D h = 0 for t \leq t_{-}},

let $w_{R} = G_{1, R} D h$ . For $h \in S$ , let $w_{0}$ solve $□ w_{0} = D h$ with zero data at $0$ . Define

s_{R} h = h - K w_{R}, s_{0} h = h - K w_{0},

B_{R} (h) = B_{f} (s_{R} h, ϕ), B_{0} (h) = B_{f} (s_{0} h, ϕ) .

The fixed $t_{-}$ ensures that the finite-time laboratory interpretation has a common lower time bound.

Put

ζ = E_{1} H, A_{H} = G_{1, A} H, J_{sec} = J_{raw} - D^{*} A_{H} .

Then

K^{*} J_{sec} = 0.

If $ζ = 0$ , $A_{H}$ and $J_{sec}$ are compact.

The propagated defect has explicit Cauchy data:

ζ (0) ζ ˙ (0) = b_{0} - \int_{0}^{b} S_{w} (s) γ (s) d s, = a + \int_{0}^{b} cos (s ∣ D ∣) γ (s) d s .

These follow by convolving (eq:H) with $S_{w}$ .

Displayed formulas in this passage

V-B@1#eq:zetadata · line 720
$ζ (0) ζ ˙ (0) = b_{0} - \int_{0}^{b} S_{w} (s) γ (s) d s, = a + \int_{0}^{b} cos (s ∣ D ∣) γ (s) d s .$
unlabeled · line 692
$S_{ret}^{-} = {h \in S : D h = 0 for t \leq t_{-}},$
unlabeled · line 698
$s_{R} h = h - K w_{R}, s_{0} h = h - K w_{0},$
unlabeled · line 701
$B_{R} (h) = B_{f} (s_{R} h, ϕ), B_{0} (h) = B_{f} (s_{0} h, ϕ) .$
unlabeled · line 709
$ζ = E_{1} H, A_{H} = G_{1, A} H, J_{sec} = J_{raw} - D^{*} A_{H} .$
unlabeled · line 714
$K^{*} J_{sec} = 0.$