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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 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralPolynomialAnalytic
import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralPolynomialTailData
namespace OAI
/-! Finite holomorphic coefficient families define holomorphic bounded exterior data. -/
open Polynomial
open scoped BoundedContinuousFunction
namespace DefocusingNLS
theorem boundedRadialPolynomial_finite (P : ℂ[X]) (d : ℕ) (hd : P.natDegree ≤ d) :
boundedRadialPolynomial P=
∑ k ∈ Finset.range (d+1), P.coeff k • boundedRadialPolynomial (X^k) := by
apply BoundedContinuousFunction.ext
intro t
change P.eval (radialExteriorClampedVariable t : ℂ)=
(BoundedContinuousFunction.evalCLM ℂ t)
(∑ k ∈ Finset.range (d+1), P.coeff k • boundedRadialPolynomial (X^k))
simp only [map_sum,map_smul,BoundedContinuousFunction.evalCLM_apply,
boundedRadialPolynomial_apply,smul_eq_mul,eval_pow,eval_X]
exact Polynomial.eval_eq_sum_range' (Nat.lt_succ_of_le hd) _
theorem boundedRadialPolynomial_analyticAt (P : ℂ → ℂ[X]) (d : ℕ) (z : ℂ)
(hd : ∀ lam, (P lam).natDegree ≤ d)
(hP : ∀ k, AnalyticAt ℂ (fun lam => (P lam).coeff k) z) :
AnalyticAt ℂ (fun lam => boundedRadialPolynomial (P lam)) z := by
have he : (fun lam => boundedRadialPolynomial (P lam))=
fun lam => ∑ k ∈ Finset.range (d+1), (P lam).coeff k • boundedRadialPolynomial (X^k) := by
funext lam
exact boundedRadialPolynomial_finite (P lam) d (hd lam)
rw [he]
exact (Finset.range (d+1)).analyticAt_fun_sum (fun k _ => (hP k).smul analyticAt_const)
theorem polynomial_eval_analyticAt (P : ℂ → ℂ[X]) (d : ℕ) (z x : ℂ)
(hd : ∀ lam, (P lam).natDegree ≤ d)
(hP : ∀ k, AnalyticAt ℂ (fun lam => (P lam).coeff k) z) :
AnalyticAt ℂ (fun lam => (P lam).eval x) z := by
have he : (fun lam => (P lam).eval x)=
fun lam => ∑ k ∈ Finset.range (d+1), (P lam).coeff k*x^k := by
funext lam
exact Polynomial.eval_eq_sum_range' (Nat.lt_succ_of_le (hd lam)) x
rw [he]
exact (Finset.range (d+1)).analyticAt_fun_sum (fun k _ => (hP k).mul analyticAt_const)
theorem spectralPolynomialJet_analyticAt (νp νm : ℂ → ℂ) (η : ℂ)
(m : ℕ) (P : ℂ[X]) (c : ℂ × ℂ) (j : ℕ) (z : ℂ) (t : ℝ)
(hp : AnalyticAt ℂ νp z) (hm : AnalyticAt ℂ νm z) :
AnalyticAt ℂ (fun lam => circularPolynomialJet
(spectralOutgoingPolynomial (νp lam) (νm lam) η m P c j) t) z := by
let U := fun lam => spectralOutgoingPolynomial (νp lam) (νm lam) η m P c j
have hd (lam : ℂ) := spectralOutgoingPolynomial_degree (νp lam) (νm lam) η m P c j
have hc (k : ℕ) := spectralOutgoingPolynomial_coeff_analytic νp νm η m P c z hp hm j k
have hEuler (Q : ℂ[X]) (hQ : Q.natDegree ≤ j) : (radialPolynomialEuler Q).natDegree ≤ j := by
apply natDegree_le_iff_coeff_eq_zero.mpr
intro k hk
rw [radialPolynomialEuler_coeff,coeff_eq_zero_of_natDegree_lt (hQ.trans_lt hk),mul_zero]
have h1 := polynomial_eval_analyticAt (fun lam => (U lam).1) j z
(Real.exp (-2*t) : ℂ) (fun lam => (hd lam).1) (fun k => (hc k).1)
have h2 := polynomial_eval_analyticAt (fun lam => (U lam).2) j z
(Real.exp (-2*t) : ℂ) (fun lam => (hd lam).2) (fun k => (hc k).2)
have h1' := polynomial_eval_analyticAt (fun lam => radialPolynomialEuler (U lam).1) j z
(Real.exp (-2*t) : ℂ) (fun lam => hEuler _ (hd lam).1) (fun k => by
simp only [radialPolynomialEuler_coeff]
exact analyticAt_const.mul (hc k).1)
have h2' := polynomial_eval_analyticAt (fun lam => radialPolynomialEuler (U lam).2) j z
(Real.exp (-2*t) : ℂ) (fun lam => hEuler _ (hd lam).2) (fun k => by
simp only [radialPolynomialEuler_coeff]
exact analyticAt_const.mul (hc k).2)
exact (h1.prod h1').prod (h2.prod h2')
end DefocusingNLS
end OAI
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