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67aee1d | 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 | import OAI.Analysis.FiniteFactor.Sampling.Smoothing
import Mathlib.Algebra.BigOperators.Ring.Finset
namespace OAI
namespace FiniteFactor
open MeasureTheory Finset
noncomputable section
/-- Bounded characters are integrable under every probability law. -/
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])
/-- Finite Fourier polynomials require only Fourier bounds on the actual
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
end OAI
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