| import OAI.Geometry.Kahler.BaseLogApproximation | |
| namespace OAI | |
| universe uKahler12650_1 | |
| 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 ENNReal Pointwise | |
| open scoped ContDiff | |
| open Set Filter Topology MeasureTheory | |
| noncomputable section | |
| open Set Filter Topology MeasureTheory | |
| namespace PinchedHartogs.BaseConstruction | |
| lemma logNormalized_unit_mul (ε : ℝ) {c : ℂ} (hc : ‖c‖=1) (v : ℂ) : | |
| logNormalized ε (c*v)=c*logNormalized ε v := by | |
| simp only [logNormalized,norm_mul,hc,one_mul,mul_div_assoc] | |
| lemma logMean_unit_mul (ε : ℝ) {c : ℂ} (hc : ‖c‖=1) (v : ℂ) : | |
| logMean ε (c*v)=logMean ε v := by simp only [logMean,norm_mul,hc,one_mul] | |
| lemma logNormalized_peak_phase (ε : ℝ) (c : ℂ) (P : Finset Sphere) (k n : ℕ) : | |
| ∀ z : Circle, ∀ ξ : Sphere, | |
| (logNormalized ε (c*peakPolynomial P k (phaseAction z ξ)))^n = | |
| phaseChar (k*n) z*(logNormalized ε (c*peakPolynomial P k ξ))^n := by | |
| intro z ξ | |
| rw [phaseAction_coe,peakPolynomial_homogeneous] | |
| have he : c*((z:ℂ)^k*peakPolynomial P k ξ)=(z:ℂ)^k*(c*peakPolynomial P k ξ) := by ring | |
| rw [he,logNormalized_unit_mul ε (by rw [norm_pow,z.norm_coe,one_pow] : ‖(z:ℂ)^k‖=1),mul_pow,← pow_mul] | |
| simp only [← Nat.cast_mul,phaseChar,zpow_natCast,Circle.coe_pow] | |
| lemma logMean_peak_phase (ε : ℝ) (c : ℂ) (P : Finset Sphere) (k : ℕ) : | |
| ∀ z : Circle, ∀ ξ : Sphere, | |
| (logMean ε (c*peakPolynomial P k (phaseAction z ξ)):ℂ) = | |
| phaseChar (0:ℕ) z*(logMean ε (c*peakPolynomial P k ξ):ℂ) := by | |
| intro z ξ | |
| rw [phaseAction_coe,peakPolynomial_homogeneous] | |
| have he : c*((z:ℂ)^k*peakPolynomial P k ξ)=(z:ℂ)^k*(c*peakPolynomial P k ξ) := by ring | |
| rw [he,logMean_unit_mul ε (by rw [norm_pow,z.norm_coe,one_pow] : ‖(z:ℂ)^k‖=1)] | |
| simp | |
| lemma probability_integral_error {X : Type uKahler12650_1} [MeasurableSpace X] {μ : Measure X} | |
| [IsProbabilityMeasure μ] {f g : X → ℝ} (hf : Integrable f μ) (hg : Integrable g μ) | |
| {E : ℝ} (h : ∀ x, |f x-g x| ≤ E) : |(∫ x,f x ∂μ)-(∫ x,g x ∂μ)| ≤ E := by | |
| rw [← integral_sub hf hg,← Real.norm_eq_abs] | |
| calc | |
| _ ≤ ∫ x, ‖f x-g x‖ ∂μ := norm_integral_le_integral_norm _ | |
| _ ≤ ∫ x, E ∂μ := integral_mono (hf.sub hg).norm (integrable_const E) | |
| (fun x => by simpa only [Real.norm_eq_abs] using h x) | |
| _ = E := by simp | |
| end PinchedHartogs.BaseConstruction | |
| end | |
| end OAI | |