| import OAI.Geometry.Kahler.BaseDensityFourier |
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| namespace OAI |
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| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open scoped ContDiff ComplexOrder |
| open scoped ContDiff ENNReal |
| open Set Filter Topology |
| open scoped ContDiff |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| noncomputable section |
|
|
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| namespace PinchedHartogs.BaseConstruction |
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| lemma densityCorrection_phase_band {W : Base → ℝ} (hW : Differentiable ℝ W) |
| {ℓ k : ℕ} (hband : PhaseBandwidth W ℓ) (R : ℝ) (f b : ℝ → ℝ) (p : Sphere) : |
| PhaseBandwidth (densityCorrection k R f b W p) (k+ℓ) := by |
| classical |
| intro ξ |
| by_cases hc : Real.exp (-R/k) < ‖bracket (ξ:Base) (p:Base)‖ |
| · have hξ : bracket (ξ:Base) (p:Base) ≠ 0 := norm_pos_iff.mp (lt_trans (Real.exp_pos _) hc) |
| let q : Sphere := ⟨centralPoint p ξ,by simpa only [Metric.mem_sphere,dist_zero_right] using centralPoint_norm p hξ⟩ |
| obtain ⟨s,a,hs,he⟩ := hband q |
| let A : ℤ → ℂ := fun n => (((‖bracket (ξ:Base) (p:Base)‖^2)⁻¹:ℝ) * |
| (bracket (ξ:Base) (p:Base)/(‖bracket (ξ:Base) (p:Base)‖:ℂ))^k * a n * |
| ((f (densityHeight k p ξ) - (n:ℝ)/k*b (densityHeight k p ξ):ℝ):ℂ)) |
| have hh : ∀ z : Circle, densityCorrection k R f b W p ((z:ℂ) • (ξ:Base)) = |
| (phaseSum (s.image (fun n => (k:ℤ)+n)) (fun n => A (n-k)) z).re := by |
| intro z |
| rw [densityCorrection,circle_bracket_norm,ite_eq_left hc] |
| rw [densityCorrectionRaw_phase_expansion k f b hW p ξ hξ s a he z] |
| rw [phaseSum_shift] |
| have hbound : ∀ n ∈ s.image (fun n => (k:ℤ)+n), |n| ≤ ((k+ℓ:ℕ):ℤ) := by |
| intro n hn |
| obtain ⟨m,hm,rfl⟩ := Finset.mem_image.mp hn |
| have ht := abs_add_le (k:ℤ) m |
| have hm' := hs m hm |
| simpa only [Nat.cast_add,abs_of_nonneg (Int.natCast_nonneg k)] using |
| le_trans ht (by simpa using add_le_add_right hm' (k:ℤ)) |
| obtain ⟨b',hb'⟩ := phaseSum_real_band hbound (fun n => A (n-k)) |
| refine ⟨phaseBand (k+ℓ),b',fun n hn => (mem_phaseBand _ _).mp hn,?_⟩ |
| intro z |
| rw [hh,hb'] |
| · refine ⟨∅,fun _ => 0,by simp,?_⟩ |
| intro z |
| simp only [densityCorrection,circle_bracket_norm,ite_eq_right hc,Complex.ofReal_zero,phaseSum,Finset.sum_empty] |
|
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| lemma densityCorrection_orbit_zero {W : Base → ℝ} (hW : Differentiable ℝ W) |
| {ℓ k : ℕ} (hband : PhaseBandwidth W ℓ) (hℓ : ℓ < k) |
| (R : ℝ) (f b : ℝ → ℝ) (p ξ : Sphere) : |
| (∫ z : Circle, densityCorrection k R f b W p ((z:ℂ) • (ξ:Base)) ∂circleMeasure) = 0 := by |
| classical |
| by_cases hc : Real.exp (-R/k) < ‖bracket (ξ:Base) (p:Base)‖ |
| · have hξ : bracket (ξ:Base) (p:Base) ≠ 0 := norm_pos_iff.mp (lt_trans (Real.exp_pos _) hc) |
| let q : Sphere := ⟨centralPoint p ξ,by simpa only [Metric.mem_sphere,dist_zero_right] using centralPoint_norm p hξ⟩ |
| obtain ⟨s,a,hs,he⟩ := hband q |
| let A : ℤ → ℂ := fun n => (((‖bracket (ξ:Base) (p:Base)‖^2)⁻¹:ℝ) * |
| (bracket (ξ:Base) (p:Base)/(‖bracket (ξ:Base) (p:Base)‖:ℂ))^k * a n * |
| ((f (densityHeight k p ξ) - (n:ℝ)/k*b (densityHeight k p ξ):ℝ):ℂ)) |
| have hh : ∀ z : Circle, densityCorrection k R f b W p ((z:ℂ) • (ξ:Base)) = |
| (phaseSum (s.image (fun n => (k:ℤ)+n)) (fun n => A (n-k)) z).re := by |
| intro z |
| rw [densityCorrection,circle_bracket_norm,ite_eq_left hc] |
| rw [densityCorrectionRaw_phase_expansion k f b hW p ξ hξ s a he z,phaseSum_shift] |
| simp_rw [hh] |
| rw [show (∫ z : Circle, (phaseSum (s.image (fun n => (k:ℤ)+n)) (fun n => A (n-k)) z).re ∂circleMeasure) = |
| (∫ z : Circle, phaseSum (s.image (fun n => (k:ℤ)+n)) (fun n => A (n-k)) z ∂circleMeasure).re from |
| integral_re (compact_continuous_integrable (phaseSum_continuous _ _))] |
| have hn : 0 ∉ s.image (fun n => (k:ℤ)+n) := by |
| intro hn |
| obtain ⟨m,hm,he⟩ := Finset.mem_image.mp hn |
| have hm' := (abs_le.mp (hs m hm)).1 |
| omega |
| rw [phaseSum_integral _ hn] |
| rfl |
| · simp only [densityCorrection,circle_bracket_norm,ite_eq_right hc,integral_zero] |
|
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| def densityBandwidth (Q : ℕ) (j : ℕ) : ℕ := ∑ i ∈ Finset.range j, Q^(i+1) |
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| lemma densityBandwidth_succ (Q j : ℕ) : densityBandwidth Q (j+1)=Q^(j+1)+densityBandwidth Q j := by |
| simp [densityBandwidth,Finset.sum_range_succ,Nat.add_comm] |
|
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| lemma densityBandwidth_lt {Q : ℕ} (hQ : 2 ≤ Q) (j : ℕ) : densityBandwidth Q j < Q^(j+1) := by |
| induction j with |
| | zero => simp [densityBandwidth]; omega |
| | succ j ih => |
| rw [densityBandwidth_succ] |
| have hp : 0 < Q^(j+1) := pow_pos (by omega) _ |
| calc |
| Q^(j+1)+densityBandwidth Q j < Q^(j+1)*2 := by omega |
| _ ≤ Q^(j+1)*Q := Nat.mul_le_mul_left _ hQ |
| _ = _ := by simp only [pow_succ] |
|
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| lemma density_bandwidth {Q : ℕ} (hQ : 0 < Q) {R E : ℝ} (hER : E < R) |
| {f b : ℝ → ℝ} (hf : ContDiff ℝ ∞ f) (hb : ContDiff ℝ ∞ b) |
| (htail : ∀ y, E ≤ y → f y=0 ∧ b y=0) (P : ℕ → Finset Sphere) (j : ℕ) : |
| PhaseBandwidth (density Q R f b P j) (densityBandwidth Q j) := by |
| induction j with |
| | zero => exact phaseBandwidth_one |
| | succ j ih => |
| rw [densityBandwidth_succ] |
| apply phaseBandwidth_add (phaseBandwidth_mono ih (Nat.le_add_left _ _)) |
| apply phaseBandwidth_sum |
| intro p hp |
| exact densityCorrection_phase_band ((density_smooth hQ hER hf hb htail P j).differentiable (by simp)) ih R f b p |
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| end PinchedHartogs.BaseConstruction |
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| end |
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| end OAI |
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