Manuscript VII-H@1 · VII-H.tex · section · d20d2a9f

Explicit product tests

A delayed source/readout realization › Explicit product tests · lines 1172–1208

VII-H@1#eq:delta-choice, VII-H@1#eq:experimental-tests

Put $q = t_{0} - a > 0$ and

p_{2} (F_{*}) = ∣ α ∣ \leq 2 max z sup ∣ \partial^{α} F_{*} (z) ∣.

Choose a positive number $δ$ satisfying

0 < δ < min {a, \frac{q}{4}, \frac{t _{0}}{32 a p _{2} ( F _{*} ) ( 2/ q + 3 )}} .

Let

η (u) = \frac{ρ ( u )}{\int _{- 1} ^{1} ρ ( v ) d v}, y_{0} = (2 δ, 0, 0, 0),

and define

h_{δ} (y) = δ^{- 4} α = 0 \prod^{3} η (\frac{y ^{α} - y _{0} ^{α}}{δ}), f_{δ} (x) = F_{*} (x - y_{0}) .

The source test is nonnegative and has integral one. The source lies in $δ < y^{0} < 3 δ$ , while the readout lies at $x^{0} > 2 δ + q > 3 δ$ . They are strictly time ordered and disjoint.

Displayed formulas in this passage

VII-H@1#eq:delta-choice · line 1179
$0 < δ < min {a, \frac{q}{4}, \frac{t _{0}}{32 a p _{2} ( F _{*} ) ( 2/ q + 3 )}} .$
VII-H@1#eq:experimental-tests · line 1194
$h_{δ} (y) = δ^{- 4} α = 0 \prod^{3} η (\frac{y ^{α} - y _{0} ^{α}}{δ}), f_{δ} (x) = F_{*} (x - y_{0}) .$
unlabeled · line 1175
$p_{2} (F_{*}) = ∣ α ∣ \leq 2 max z sup ∣ \partial^{α} F_{*} (z) ∣.$
unlabeled · line 1189
$η (u) = \frac{ρ ( u )}{\int _{- 1} ^{1} ρ ( v ) d v}, y_{0} = (2 δ, 0, 0, 0),$

Explicit product tests

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