openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralBoundedPolynomialLimit.lean
| 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 | |