| import OAI.Analysis.FiniteFactor.Sampling.Smoothing |
| import Mathlib.Algebra.BigOperators.Ring.Finset |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Finset |
| noncomputable section |
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| / |
| theorem samplingCharacter_product_integrable {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (X Y : Ω → ℝ) |
| (hX : Measurable X) (hY : Measurable Y) (v w : ℤ) : |
| Integrable (fun θ => samplingCharacter v (X θ)*samplingCharacter w (Y θ)) μ := by |
| apply (integrable_const (1 : ℝ)).mono' |
| (((samplingCharacter_continuous v).measurable.comp hX).mul |
| ((samplingCharacter_continuous w).measurable.comp hY)).aestronglyMeasurable |
| exact Filter.Eventually.of_forall (fun θ => by simp [samplingCharacter_norm]) |
|
|
| / |
| joint law; the two input phases need not be independent. -/ |
| theorem samplingFourier_sum_error {Ω α β : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (X Y : Ω → ℝ) |
| (hX : Measurable X) (hY : Measurable Y) |
| (F : Finset α) (G : Finset β) (v : α → ℤ) (w : β → ℤ) |
| (A : α → ℂ) (B : β → ℂ) |
| (hA : ∀ a ∈ F, ‖A a‖ ≤ 1) (hB : ∀ b ∈ G, ‖B b‖ ≤ 1) |
| (ε : ℝ) (hε : 0 ≤ ε) |
| (hFourier : ∀ a ∈ F, ∀ b ∈ G, v a ≠ 0 ∨ w b ≠ 0 → |
| ‖∫ θ, samplingCharacter (v a) (X θ)*samplingCharacter (w b) (Y θ) ∂μ‖ ≤ ε) : |
| ‖(∫ θ, (∑ a ∈ F, A a*samplingCharacter (v a) (X θ)) * |
| (∑ b ∈ G, B b*samplingCharacter (w b) (Y θ)) ∂μ) - |
| (∑ a ∈ F, if v a = 0 then A a else 0) * |
| (∑ b ∈ G, if w b = 0 then B b else 0)‖ ≤ |
| (F.card : ℝ)*(G.card : ℝ)*ε := by |
| classical |
| let τ := fun a b => ∫ θ, samplingCharacter (v a) (X θ)*samplingCharacter (w b) (Y θ) ∂μ |
| let d := fun a b => if v a = 0 ∧ w b = 0 then (1 : ℂ) else 0 |
| have heq (a : α) (b : β) (θ : Ω) : |
| (A a*samplingCharacter (v a) (X θ))*(B b*samplingCharacter (w b) (Y θ)) = |
| (A a*B b)*(samplingCharacter (v a) (X θ)*samplingCharacter (w b) (Y θ)) := by ring |
| have hi (a : α) (b : β) := (samplingCharacter_product_integrable μ X Y hX hY (v a) (w b)).const_mul (A a*B b) |
| have hmain : (∫ θ, (∑ a ∈ F, A a*samplingCharacter (v a) (X θ)) * |
| (∑ b ∈ G, B b*samplingCharacter (w b) (Y θ)) ∂μ) = |
| ∑ a ∈ F, ∑ b ∈ G, A a*B b*τ a b := by |
| simp_rw [sum_mul_sum, heq] |
| rw [integral_finsetSum _ (fun a _ => integrable_finsetSum _ (fun b _ => hi a b))] |
| simp_rw [integral_finsetSum _ (fun b _ => hi _ b), integral_const_mul] |
| rfl |
| have hconstant : (∑ a ∈ F, if v a = 0 then A a else 0) * |
| (∑ b ∈ G, if w b = 0 then B b else 0) = |
| ∑ a ∈ F, ∑ b ∈ G, A a*B b*d a b := by |
| rw [sum_mul_sum] |
| apply sum_congr rfl |
| intro a ha |
| apply sum_congr rfl |
| intro b hb |
| dsimp [d] |
| split_ifs <;> simp_all |
| have hτ (a : α) (ha : a ∈ F) (b : β) (hb : b ∈ G) : ‖τ a b-d a b‖ ≤ ε := by |
| by_cases hz : v a = 0 ∧ w b = 0 |
| · simpa [τ,d,hz.1,hz.2,samplingCharacter_zero] using hε |
| · have hn : v a ≠ 0 ∨ w b ≠ 0 := by tauto |
| simpa [τ,d,hz] using hFourier a ha b hb hn |
| rw [hmain,hconstant, ← sum_sub_distrib] |
| simp_rw [← sum_sub_distrib, ← mul_sub] |
| calc |
| _ ≤ ∑ a ∈ F, ‖∑ b ∈ G, A a*B b*(τ a b-d a b)‖ := norm_sum_le _ _ |
| _ ≤ ∑ a ∈ F, ∑ b ∈ G, ‖A a*B b*(τ a b-d a b)‖ := |
| sum_le_sum fun a ha => norm_sum_le _ _ |
| _ ≤ ∑ _a ∈ F, ∑ _b ∈ G, ε := by |
| apply sum_le_sum |
| intro a ha |
| apply sum_le_sum |
| intro b hb |
| rw [norm_mul,norm_mul] |
| calc |
| _ ≤ 1*1*ε := by gcongr; exact hA a ha; exact hB b hb; exact hτ a ha b hb |
| _ = ε := by ring |
| _ = _ := by simp; ring |
|
|
| end |
| end FiniteFactor |
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| end OAI |
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