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f1f7953 | 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 98 99 | import OAI.Analysis.IrrationalRotation.Sampling.CircleSamplingKernelBridge
namespace OAI
noncomputable section
open MeasureTheory
open scoped BigOperators ENNReal
namespace Invariant233.CircleParameterData
theorem kernel_bounds {m : ℕ} (hm : 0 < m) (z v : ℝ) :
0 ≤ kernel m z v ∧ kernel m z v ≤ (m : ℝ)^2 / kernelWidth m := by
have hmR : (0 : ℝ) < m := by exact_mod_cast hm
have hsq : (0 : ℝ) < (m : ℝ)^2 := sq_pos_of_pos hmR
have hw : 0 < kernelWidth m := Real.rpow_pos_of_pos hmR _
unfold kernel
split_ifs with h
· have hd : 0 < delta z := (one_div_pos.mpr hsq).trans_le h.1
refine ⟨by positivity, ?_⟩
calc
1 / (kernelWidth m * delta z) ≤
1 / (kernelWidth m * (1 / (m : ℝ)^2)) :=
one_div_le_one_div_of_le (mul_pos hw (one_div_pos.mpr hsq))
(mul_le_mul_of_nonneg_left h.1 hw.le)
_ = (m : ℝ)^2 / kernelWidth m := by field_simp
· exact ⟨le_rfl, by positivity⟩
theorem orientedKernel_bounds (endpoint : Bool) {m : ℕ} (hm : 0 < m) (z v : ℝ) :
0 ≤ orientedKernel endpoint m z v ∧
orientedKernel endpoint m z v ≤ (m : ℝ)^2 / kernelWidth m := by
cases endpoint
· exact kernel_bounds hm z v
· exact kernel_bounds hm (z - v) v
theorem samplingAverage_bounds (p : CircleParameterData) (θ : ℝ) (j r : ℕ)
(endpoint : Bool) (hm : 0 < p.m j) (x : Circle) :
0 ≤ p.samplingAverage θ j r endpoint x ∧
p.samplingAverage θ j r endpoint x ≤
((p.m j : ℝ)^2 / kernelWidth (p.m j)) ^ (p.block j).card := by
classical
have hb : 0 ≤ (p.m j : ℝ)^2 / kernelWidth (p.m j) := by
exact div_nonneg (sq_nonneg _) (Real.rpow_nonneg (Nat.cast_nonneg _) _)
have hprod (n : ℕ) :
0 ≤ ∏ i ∈ p.block j, orientedKernel endpoint (p.m j) (p.coordinate i x)
(Int.fract ((n : ℝ) * p.alpha θ i)) := by
exact Finset.prod_nonneg fun i _ => (orientedKernel_bounds endpoint hm _ _).1
have hprod_le (n : ℕ) :
(∏ i ∈ p.block j, orientedKernel endpoint (p.m j) (p.coordinate i x)
(Int.fract ((n : ℝ) * p.alpha θ i))) ≤
((p.m j : ℝ)^2 / kernelWidth (p.m j)) ^ (p.block j).card := by
calc
_ ≤ ∏ _i ∈ p.block j, (p.m j : ℝ)^2 / kernelWidth (p.m j) := by
apply Finset.prod_le_prod₀
· intro i hi
exact (orientedKernel_bounds endpoint hm _ _).1
· intro i hi
exact (orientedKernel_bounds endpoint hm _ _).2
_ = _ := by simp [div_pow]
unfold samplingAverage
refine ⟨mul_nonneg (by positivity) (Finset.sum_nonneg fun n _ => hprod n), ?_⟩
by_cases hs : (p.samples j r).card = 0
· simp only [hs, Nat.cast_zero, div_zero, zero_mul]
exact pow_nonneg hb _
· have hsR : ((p.samples j r).card : ℝ) ≠ 0 := by exact_mod_cast hs
calc
_ ≤ (1 / ((p.samples j r).card : ℝ)) *
∑ _n ∈ p.samples j r,
((p.m j : ℝ)^2 / kernelWidth (p.m j)) ^ (p.block j).card := by
apply mul_le_mul_of_nonneg_left (Finset.sum_le_sum fun n _ => hprod_le n)
positivity
_ = _ := by simp [hsR]
theorem memLp_samplingAverage (p : CircleParameterData) (θ : ℝ) (j r : ℕ)
(endpoint : Bool) (hm : 0 < p.m j) (s : ENNReal) :
MemLp (p.samplingAverage θ j r endpoint) s circleMeasure := by
let : IsFiniteMeasure circleMeasure := by
unfold circleMeasure
infer_instance
apply MemLp.of_bound (p.measurable_samplingAverage θ j r endpoint).aestronglyMeasurable
(((p.m j : ℝ)^2 / kernelWidth (p.m j)) ^ (p.block j).card)
apply Filter.Eventually.of_forall
intro x
have h := p.samplingAverage_bounds θ j r endpoint hm x
simpa only [Real.norm_eq_abs, abs_of_nonneg h.1] using h.2
theorem memLp_samplingError (p : CircleParameterData) (θ : ℝ) (j r : ℕ)
(endpoint : Bool) (hm : 0 < p.m j) (s : ENNReal) :
MemLp (fun x => p.samplingAverage θ j r endpoint x - 1) s circleMeasure := by
let : IsFiniteMeasure circleMeasure := by
unfold circleMeasure
infer_instance
exact (p.memLp_samplingAverage θ j r endpoint hm s).sub (memLp_const (1 : ℝ))
end Invariant233.CircleParameterData
end
end OAI
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