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59d1c80 | 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 | 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
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