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

Predetermined Chebyshev packets

A regular positive-frequency decay regime › Predetermined Chebyshev packets · lines 1216–1248

Hawk02-C@2#eq:packets

Let $U_{n}$ denote Chebyshev polynomials of the second kind:

U_{- 1} = 0, U_{0} = 1, 2 y U_{n} (y) = U_{n + 1} (y) + U_{n - 1} (y) .

Define

ϕ_{n} (ν) = w_{c} (ν - ω_{0}) U_{n} (\frac{ν - ω _{0}}{2 c}), n \geq 0.

Using $U_{n} (cos θ) = sin ((n + 1) θ) / sin θ$ gives

⟨ ϕ_{n}, ϕ_{m} ⟩ = δ_{nm} .

Polynomial density in the weighted space, followed by multiplication by $w_{c}$ , proves completeness in $L^{2} (I_{c}, d ν)$ . The band-edge zeros have measure zero.

Choose

P_{N} = n = 0 \sum^{N - 1} ∣ ϕ_{n} ⟩ ⟨ ϕ_{n} ∣ .

This family is fixed by the Hamiltonian parameters, not by the unknown input or observed outcomes.

Displayed formulas in this passage

Hawk02-C@2#eq:packets · line 1224
$ϕ_{n} (ν) = w_{c} (ν - ω_{0}) U_{n} (\frac{ν - ω _{0}}{2 c}), n \geq 0.$
unlabeled · line 1219
$U_{- 1} = 0, U_{0} = 1, 2 y U_{n} (y) = U_{n + 1} (y) + U_{n - 1} (y) .$
unlabeled · line 1234
$⟨ ϕ_{n}, ϕ_{m} ⟩ = δ_{nm} .$
unlabeled · line 1243
$P_{N} = n = 0 \sum^{N - 1} ∣ ϕ_{n} ⟩ ⟨ ϕ_{n} ∣ .$

Predetermined Chebyshev packets

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