File size: 4,429 Bytes
ef96466 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralBoundedPolynomialAnalytic
import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralOutgoingPolynomialLimit
import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialExteriorJetLimit
namespace OAI
/-! Finite coefficient limits imply norm convergence of the bounded
polynomial jets used in the actual outgoing forcing. -/
open Filter Topology Polynomial
open scoped BoundedContinuousFunction
namespace DefocusingNLS
theorem boundedRadialPolynomial_tendsto (P : ℕ → ℂ[X]) (Q : ℂ[X]) (d : ℕ)
(hdeg : ∀ n, (P n).natDegree ≤ d) (hQ : Q.natDegree ≤ d)
(hP : ∀ k, Tendsto (fun n => (P n).coeff k) atTop (𝓝 (Q.coeff k))) :
Tendsto (fun n => boundedRadialPolynomial (P n)) atTop (𝓝 (boundedRadialPolynomial Q)) := by
have he : (fun n => boundedRadialPolynomial (P n))=
fun n => ∑ k ∈ Finset.range (d+1), (P n).coeff k • boundedRadialPolynomial (X^k) := by
funext n
exact boundedRadialPolynomial_finite (P n) d (hdeg n)
rw [he,boundedRadialPolynomial_finite Q d hQ]
exact tendsto_finsetSum _ (fun k _ => (hP k).smul_const _)
theorem boundedCircularPolynomialComponent_tendsto (P : ℕ → ℂ[X]) (Q : ℂ[X]) (d : ℕ)
(hdeg : ∀ n, (P n).natDegree ≤ d) (hQ : Q.natDegree ≤ d)
(hP : ∀ k, Tendsto (fun n => (P n).coeff k) atTop (𝓝 (Q.coeff k))) :
Tendsto (fun n => boundedCircularPolynomialComponent (P n)) atTop
(𝓝 (boundedCircularPolynomialComponent Q)) := by
have hp := boundedRadialPolynomial_tendsto P Q d hdeg hQ hP
have he := boundedRadialPolynomial_tendsto
(fun n => radialPolynomialEuler (P n)) (radialPolynomialEuler Q) d
(fun n => radialPolynomialEuler_degree (P n) d (hdeg n))
(radialPolynomialEuler_degree Q d hQ)
(fun k => by simpa only [radialPolynomialEuler_coeff] using (hP k).const_mul (-2*(k : ℂ)))
have hlim := (tendsto_iff_norm_sub_tendsto_zero.mp hp).max
(tendsto_iff_norm_sub_tendsto_zero.mp he)
simp only [max_self] at hlim
apply tendsto_iff_norm_sub_tendsto_zero.mpr
apply squeeze_zero (fun _ => norm_nonneg _) _ hlim
intro n
apply (BoundedContinuousFunction.norm_le (le_max_of_le_left (norm_nonneg _))).mpr
intro t
change max ‖boundedRadialPolynomial (P n) t-boundedRadialPolynomial Q t‖
‖boundedRadialPolynomial (radialPolynomialEuler (P n)) t-
boundedRadialPolynomial (radialPolynomialEuler Q) t‖ ≤ _
exact max_le_max
((boundedRadialPolynomial (P n)-boundedRadialPolynomial Q).norm_coe_le_norm t)
((boundedRadialPolynomial (radialPolynomialEuler (P n))-
boundedRadialPolynomial (radialPolynomialEuler Q)).norm_coe_le_norm t)
theorem boundedCircularPolynomialJet_tendsto
(U : ℕ → ℂ[X] × ℂ[X]) (V : ℂ[X] × ℂ[X]) (d : ℕ)
(hdeg : ∀ n, (U n).1.natDegree ≤ d ∧ (U n).2.natDegree ≤ d)
(hV : V.1.natDegree ≤ d ∧ V.2.natDegree ≤ d)
(hU : ∀ k, Tendsto (fun n => (U n).1.coeff k) atTop (𝓝 (V.1.coeff k)) ∧
Tendsto (fun n => (U n).2.coeff k) atTop (𝓝 (V.2.coeff k))) :
Tendsto (fun n => boundedCircularPolynomialJet (U n)) atTop
(𝓝 (boundedCircularPolynomialJet V)) := by
exact (boundedCircularPolynomialComponent_tendsto (fun n => (U n).1) V.1 d
(fun n => (hdeg n).1) hV.1 (fun k => (hU k).1)).prodMk_nhds
(boundedCircularPolynomialComponent_tendsto (fun n => (U n).2) V.2 d
(fun n => (hdeg n).2) hV.2 (fun k => (hU k).2))
theorem spectralOutgoing_bounded_jet_tendsto
(νp νm : ℕ → ℂ) (νp₀ νm₀ η : ℂ)
(hp : Tendsto νp atTop (𝓝 νp₀)) (hm : Tendsto νm atTop (𝓝 νm₀))
(P : ℕ → ℂ[X]) (Q : ℂ[X]) (d : ℕ)
(hdeg : ∀ᶠ n in atTop, (P n).natDegree ≤ d)
(hP : ∀ k, k ≤ d → Tendsto (fun n => (P n).coeff k) atTop (𝓝 (Q.coeff k)))
(hzero : ‖Q.coeff 0‖ < 1) (c : ℂ × ℂ) (j : ℕ) :
Tendsto (fun n => boundedCircularPolynomialJet
(spectralOutgoingPolynomial (νp n) (νm n) η n (P n) c j)) atTop
(𝓝 (boundedCircularPolynomialJet (spectralOutgoingPolynomial νp₀ νm₀ η 1 0 c j))) := by
exact boundedCircularPolynomialJet_tendsto _ _ j
(fun _ => spectralOutgoingPolynomial_degree _ _ _ _ _ _ _)
(spectralOutgoingPolynomial_degree _ _ _ _ _ _ _)
(spectralOutgoingPolynomial_coefficient_limit νp νm νp₀ νm₀ η hp hm P Q d hdeg hP hzero c j)
end DefocusingNLS
end OAI
|