Manuscript Hawk02-C@2 · Hawk02-C.tex · lemma · bb332d66

Lemma: Finite-time packet-tail certificate

A regular positive-frequency decay regime › Predetermined Chebyshev packets · lines 1249–1289 · proof included or linked

Hawk02-C@2#lem:packetbound, Hawk02-C@2#eq:ellbound

For the prescribed packets and $z = 2 c T$ ,

ℓ_{N} (T) \leq min {1 - q (T), η_{N} (T)^{2}}, η_{N} (T) = \frac{z ^{N + 1}}{2 ( N + 1 )!} .

Proof. Writing $x = ν - ω_{0}$ gives

f_{T} (ν) = - i e^{- i ω_{0} T} g (ν) \int_{0}^{T} e^{- i x (T - s)} a_{c} (s) d s .

Replace the exponential by its degree- $(N - 1)$ Taylor polynomial in $x$ . The result belongs to $span {g (ν) x^{k} : 0 \leq k < N} = P_{N} k$ .

For real $y$ ,

e^{- i y} - k = 0 \sum^{N - 1} \frac{( - i y ) ^{k}}{k !} \leq \frac{∣ y ∣ ^{N}}{N !} .

Using $∣ a_{c} (s) ∣ \leq 1$ , $∣ x ∣ \leq 2 c$ , and $∥ g ∥ = c$ , the norm error is at most

\frac{c ( 2 c ) ^{N}}{N !} \int_{0}^{T} (T - s)^{N} d s = \frac{z ^{N + 1}}{2 ( N + 1 )!} .

Orthogonal projection minimizes norm error. The other bound is $ℓ_{N} \leq ∥ f_{T} ∥^{2} = 1 - q (T)$ .

Displayed formulas in this passage

Hawk02-C@2#eq:ellbound · line 1252
$ℓ_{N} (T) \leq min {1 - q (T), η_{N} (T)^{2}}, η_{N} (T) = \frac{z ^{N + 1}}{2 ( N + 1 )!} .$
unlabeled · line 1264
$f_{T} (ν) = - i e^{- i ω_{0} T} g (ν) \int_{0}^{T} e^{- i x (T - s)} a_{c} (s) d s .$
unlabeled · line 1276
$e^{- i y} - k = 0 \sum^{N - 1} \frac{( - i y ) ^{k}}{k !} \leq \frac{∣ y ∣ ^{N}}{N !} .$
unlabeled · line 1283
$\frac{c ( 2 c ) ^{N}}{N !} \int_{0}^{T} (T - s)^{N} d s = \frac{z ^{N + 1}}{2 ( N + 1 )!} .$