| import OAI.Analysis.FiniteFactor.CrossedProduct.AngleLaw |
| import OAI.Analysis.FiniteFactor.PeriodizedUniform |
| import Mathlib.Analysis.SpecialFunctions.Integrals.Basic |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| noncomputable section |
|
|
| / |
| theorem norm_interval_fourier_le (a b w : ℝ) (hw : w ≠ 0) : |
| ‖∫ t in a..b, Complex.exp ((w : ℂ) * Complex.I * t)‖ ≤ 2 / |w| := by |
| rw [integral_exp_mul_complex (mul_ne_zero (Complex.ofReal_ne_zero.mpr hw) Complex.I_ne_zero), |
| norm_div, norm_mul, Complex.norm_real, Complex.norm_I, mul_one, Real.norm_eq_abs] |
| apply div_le_div_of_nonneg_right _ (abs_nonneg w) |
| calc |
| _ ≤ ‖Complex.exp ((w : ℂ) * Complex.I * b)‖ + |
| ‖Complex.exp ((w : ℂ) * Complex.I * a)‖ := norm_sub_le _ _ |
| _ = 2 := by norm_num [Complex.norm_exp, Complex.mul_re, Complex.mul_im] |
|
|
| / |
| theorem sourceAngle_fourier_le (i : SourceCoordinate) (w : ℝ) (hw : w ≠ 0) : |
| ‖∫ t, Complex.exp ((w : ℂ) * Complex.I * t) ∂sourceAngleMarginal i‖ ≤ |
| 2 / (sourceAngleUnit i * |w|) := by |
| have ha := sourceAngleUnit_pos i |
| rw [sourceAngleMarginal, sourceAngleInterval, cond_closed_interval_eq, |
| ProbabilityTheory.cond, integral_smul_measure, Real.volume_Ioc] |
| rw [show 2 * sourceAngleUnit i - sourceAngleUnit i = sourceAngleUnit i by ring] |
| rw [ENNReal.toReal_inv, ENNReal.toReal_ofReal ha.le, norm_smul, |
| Real.norm_eq_abs, abs_of_pos (inv_pos.mpr ha)] |
| rw [← intervalIntegral.integral_of_le (show sourceAngleUnit i ≤ 2*sourceAngleUnit i by linarith)] |
| calc |
| _ ≤ (sourceAngleUnit i)⁻¹ * (2 / |w|) := |
| mul_le_mul_of_nonneg_left (norm_interval_fourier_le _ _ w hw) (inv_nonneg.mpr ha.le) |
| _ = _ := by ring |
|
|
| / |
| theorem fourier_fract (v : ℤ) (t : ℝ) : |
| Complex.exp (2 * Real.pi * Complex.I * ((v : ℂ) * (Int.fract t : ℝ))) = |
| Complex.exp (2 * Real.pi * Complex.I * ((v : ℂ) * (t : ℂ))) := by |
| rw [Int.fract] |
| push_cast |
| have he : 2 * (Real.pi : ℂ) * Complex.I * ((v : ℂ) * ((t : ℂ) - (⌊t⌋ : ℂ))) = |
| 2 * Real.pi * Complex.I * ((v : ℂ) * (t : ℂ)) + |
| ((-(v * ⌊t⌋) : ℤ) : ℂ) * (2 * Real.pi * Complex.I) := by |
| push_cast |
| ring |
| rw [he, Complex.exp_add, Complex.exp_int_mul_two_pi_mul_I, mul_one] |
|
|
| / |
| theorem pair_fourier_fract (v v' : ℤ) (n n' t : ℝ) : |
| Complex.exp (2 * Real.pi * Complex.I * |
| ((v : ℂ) * (Int.fract (n*t) : ℝ) + (v' : ℂ) * (Int.fract (n'*t) : ℝ))) = |
| Complex.exp (((2 * Real.pi * ((v : ℝ)*n + (v' : ℝ)*n') : ℝ) : ℂ) * Complex.I * t) := by |
| rw [mul_add, Complex.exp_add, fourier_fract, fourier_fract, ← Complex.exp_add] |
| congr 1 |
| push_cast |
| ring |
|
|
| theorem sourceAngle_pair_fourier_le (i : SourceCoordinate) (v v' : ℤ) (n n' : ℝ) |
| (hw : (v : ℝ)*n + (v' : ℝ)*n' ≠ 0) : |
| ‖∫ t, Complex.exp (2 * Real.pi * Complex.I * |
| ((v : ℂ) * (Int.fract (n*t) : ℝ) + (v' : ℂ) * (Int.fract (n'*t) : ℝ))) |
| ∂sourceAngleMarginal i‖ ≤ |
| 1 / (Real.pi * sourceAngleUnit i * |(v : ℝ)*n + (v' : ℝ)*n'|) := by |
| simp_rw [pair_fourier_fract] |
| have hp := sourceAngle_fourier_le i (2*Real.pi*((v : ℝ)*n+(v' : ℝ)*n')) |
| (mul_ne_zero (mul_ne_zero (by norm_num) Real.pi_ne_zero) hw) |
| convert hp using 1 |
| rw [abs_mul, abs_mul, abs_of_pos Real.pi_pos] |
| norm_num |
| ring |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
