| import OAI.Analysis.IrrationalRotation.Sampling.CircleSamplingKernelBridge |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory |
| open scoped BigOperators ENNReal |
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| namespace Invariant233.CircleParameterData |
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| 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⟩ |
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| 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 |
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| 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] |
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| 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 |
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| 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)) |
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| end Invariant233.CircleParameterData |
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| end |
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| end OAI |
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