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f61e547 | 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 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | import OAI.Probability.FKPressure.Coefficients
namespace OAI
noncomputable section
open Finset FormalMultilinearSeries
open scoped Classical Topology NNReal ENNReal
namespace BelowOne.PressureSeries
open MultiaffineDerivative
variable {V : Type*} [Fintype V]
def complexify (p : FormalMultilinearSeries ℝ (V → ℝ) ℝ) :
FormalMultilinearSeries ℂ (V → ℂ) ℂ := fun n =>
∑ τ : Fin n → V, (weight p ⟨n,τ⟩ : ℂ) •
(ContinuousMultilinearMap.mkPiAlgebra ℂ (Fin n) ℂ).compContinuousLinearMap
(fun j => ContinuousLinearMap.proj (τ j))
lemma complexify_apply (p : FormalMultilinearSeries ℝ (V → ℝ) ℝ)
(n : ℕ) (x : Fin n → V → ℂ) :
complexify p n x = ∑ τ : Fin n → V, (weight p ⟨n,τ⟩ : ℂ) * ∏ j, x j (τ j) := by
simp only [complexify, _root_.sum_apply, _root_.smul_apply,
ContinuousMultilinearMap.compContinuousLinearMap_apply,
ContinuousMultilinearMap.mkPiAlgebra_apply, ContinuousLinearMap.proj_apply, smul_eq_mul]
lemma complexify_real (p : FormalMultilinearSeries ℝ (V → ℝ) ℝ)
(n : ℕ) (x : Fin n → V → ℝ) :
complexify p n (fun j i => (x j i : ℂ)) = (p n x : ℂ) := by
rw [complexify_apply, PositiveMultilinear.expansion]
push_cast
apply sum_congr rfl
intro τ hτ
exact mul_comm _ _
lemma complexify_norm_le (p : FormalMultilinearSeries ℝ (V → ℝ) ℝ)
(hp : ∀ a, 0 ≤ weight p a) (n : ℕ) : ‖complexify p n‖ ≤ ‖p n‖ := by
have hpos : 0 ≤ p n (fun _ _ => 1) :=
PositiveMultilinear.nonneg _ (fun τ => hp ⟨n,τ⟩) _ (fun _ _ => zero_le_one)
have hle : ‖complexify p n‖ ≤ p n (fun _ _ => 1) := by
apply (complexify p n).opNorm_le_bound hpos
intro x
rw [complexify_apply, PositiveMultilinear.ones_eq_sum, sum_mul]
refine (norm_sum_le _ _).trans (sum_le_sum ?_)
intro τ hτ
rw [norm_mul, Complex.norm_real, Real.norm_of_nonneg (hp ⟨n,τ⟩), norm_prod]
apply mul_le_mul_of_nonneg_left _ (hp ⟨n,τ⟩)
exact prod_le_prod₀ (fun _ _ => norm_nonneg _) (fun j _ => norm_le_pi_norm (x j) (τ j))
apply hle.trans
rw [← Real.norm_of_nonneg hpos]
apply (p n).le_opNorm _ |>.trans
have hb : ‖(fun _ : V => (1 : ℝ))‖ ≤ 1 :=
(pi_norm_le_iff_of_nonneg zero_le_one).mpr (fun _ => by simp)
have hprod : (∏ _ : Fin n, ‖(fun _ : V => (1 : ℝ))‖) ≤ 1 := by
exact prod_le_one₀ (fun _ _ => norm_nonneg _) (fun _ _ => hb)
simpa only [mul_one] using mul_le_mul_of_nonneg_left hprod (norm_nonneg (p n))
lemma complexify_radius (p : FormalMultilinearSeries ℝ (V → ℝ) ℝ)
(hp : ∀ a, 0 ≤ weight p a) : p.radius ≤ (complexify p).radius := by
unfold FormalMultilinearSeries.radius
apply iSup_le
intro r
apply iSup_le
intro C
apply iSup_le
intro hC
apply (complexify p).le_radius_of_bound C
intro n
exact (mul_le_mul_of_nonneg_right (complexify_norm_le p hp n) (by positivity)).trans (hC n)
lemma norm_lt_of_coordinates {r : ℝ} (hr : 0 < r) {x : V → ℂ}
(hx : ∀ i, ‖x i‖ < r) : ‖x‖ < r := by
exact (pi_norm_lt_iff hr).mpr hx
lemma complexify_hasSum (p : FormalMultilinearSeries ℝ (V → ℝ) ℝ)
(hp : ∀ a, 0 ≤ weight p a) {r : ℝ} (hr : 0 < r)
(hR : ENNReal.ofReal r < p.radius) (x : V → ℂ) (hx : ∀ i, ‖x i‖ < r) :
HasSum (fun m : V → ℕ => (coefficient p m : ℂ) * ∏ i, x i ^ m i) ((complexify p).sum x) := by
have hnorm := norm_lt_of_coordinates hr hx
have he : ‖x‖ₑ < (complexify p).radius := by
exact (show ‖x‖ₑ ≤ ENNReal.ofReal r by
simpa only [ofReal_norm] using ENNReal.ofReal_le_ofReal hnorm.le).trans_lt
(hR.trans_le (complexify_radius p hp))
apply hasSum_coefficients_of_complex_degrees p hp hr.le hR x (fun i => (hx i).le)
exact ((complexify p).summable (mem_eball_zero_iff.mpr he)).hasSum.congr_fun
(fun n => (complexify_apply p n (fun _ => x)).symm)
lemma complexify_sum_real (p : FormalMultilinearSeries ℝ (V → ℝ) ℝ)
(x : V → ℝ) : (complexify p).sum (fun i => (x i : ℂ)) = (p.sum x : ℂ) := by
simp only [FormalMultilinearSeries.sum, complexify_real, Complex.ofReal_tsum]
end BelowOne.PressureSeries
end
end OAI
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