Manuscript En01-C@1 · En01-C.tex · proposition · 00a2161c

Proposition: Finite-width Gaussian bound

Finite packet and finite-window corrections › A physical two-mode diagnostic · lines 1190–1237 · proof included or linked

En01-C@1#prop:width

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Suppose the filtered Gaussian kernel on these supports has the form

τ (x, y) = n (x) n (y) e^{- a (x - y)^{2}} e^{i (φ (x) - φ (y))}, a \geq 0,

where, after the common compensation,

∣ φ (x) - φ (y) ∣ \leq θ < π /2

on the union of the packet supports. Assume $P_{\pm} > 0$ . Then

- lo g V_{mode} - a d^{2} \leq 4 a d ℓ + 4 a ℓ^{2} - lo g (cos θ) .

Proof. Put $A_{\pm} = \int u_{\pm} (x) n (x) d x > 0$ . Within either support, $∣ x - y ∣ \leq 2 ℓ$ . Across the supports,

max (d - 2 ℓ, 0) \leq ∣ x - y ∣ \leq d + 2 ℓ .

Taking real parts for a lower bound and absolute values for an upper bound yields

cos θ e^{- 4 a ℓ^{2}} A_{\pm}^{2} \leq P_{\pm} \leq A_{\pm}^{2},

cos θ e^{- a (d + 2 ℓ)^{2}} A_{+} A_{-} \leq ∣ B ∣ \leq e^{- a m a x (d - 2 ℓ, 0)^{2}} A_{+} A_{-} .

Divide by the bounds on $P_{+} P_{-}$ and take logarithms. For $d \geq 2 ℓ$ the lower logarithmic endpoint differs from $a d^{2}$ by at most $4 a d ℓ - lo g (cos θ)$ . For $d < 2 ℓ$ , the displayed larger bound remains valid. The upper endpoint contributes $4 a d ℓ + 4 a ℓ^{2}$ .

Displayed formulas in this passage

unlabeled · line 1193
$τ (x, y) = n (x) n (y) e^{- a (x - y)^{2}} e^{i (φ (x) - φ (y))}, a \geq 0,$
unlabeled · line 1200
$∣ φ (x) - φ (y) ∣ \leq θ < π /2$
unlabeled · line 1206
$- lo g V_{mode} - a d^{2} \leq 4 a d ℓ + 4 a ℓ^{2} - lo g (cos θ) .$
unlabeled · line 1218
$max (d - 2 ℓ, 0) \leq ∣ x - y ∣ \leq d + 2 ℓ .$
unlabeled · line 1223
$cos θ e^{- 4 a ℓ^{2}} A_{\pm}^{2} \leq P_{\pm} \leq A_{\pm}^{2},$
unlabeled · line 1227
$cos θ e^{- a (d + 2 ℓ)^{2}} A_{+} A_{-} \leq ∣ B ∣ \leq e^{- a m a x (d - 2 ℓ, 0)^{2}} A_{+} A_{-} .$

Proposition: Finite-width Gaussian bound

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