OAI/NumberTheory/Ostmann/Construction/PhaseGiantTest.lean

GitHub source file

15 declarations, 0 contain sorry, 0 declared as axioms

The Fourier-phase giant and its real physical inverse

  1. def OAI.Ostmann.complexUnitPhase line 11
  2. theorem OAI.Ostmann.complexUnitPhase_zero line 13
  3. theorem OAI.Ostmann.complexUnitPhase_norm_le line 16
  4. theorem OAI.Ostmann.complexUnitPhase_conj line 22
  5. theorem OAI.Ostmann.complexUnitPhase_mul_conj line 26
  6. def OAI.Ostmann.densityFourierInverse line 35
  7. def OAI.Ostmann.phaseGiantPhysical line 38
  8. theorem OAI.Ostmann.additiveFourier_reflect line 42
  9. theorem OAI.Ostmann.densityFourierInverse_real line 50
  10. theorem OAI.Ostmann.phaseGiantPhysical_real line 63
  11. theorem OAI.Ostmann.densityFourierInverse_energy line 70
  12. theorem OAI.Ostmann.phaseGiantPhysical_energy line 75
  13. theorem OAI.Ostmann.phaseGiantPhysical_bound line 84
  14. theorem OAI.Ostmann.densityFourier_pairing line 93

    Source documentation

    The unitary bilinear identity, with the inverse sign explicit.

    theorem densityFourier_pairing {p : ℕ} [NeZero p]
    (f g : ZMod p → ℂ) :
    (∑ x, f x * densityFourierInverse g x) =
      ∑ b, densityFourier f (-b) * g b := by
…

    Mentions: OAI.Ostmann.densityFourierInverse, OAI.Ostmann.densityFourier

  15. theorem OAI.Ostmann.phaseGiantPhysical_pairing line 108

    Source documentation

    Flattening preserves the exact positive Fourier pairing used in (7.2).

    theorem phaseGiantPhysical_pairing {p : ℕ} [NeZero p]
    (f : ZMod p → ℂ)
    (hf : ∀ b, densityFourier f (-b) = conj (densityFourier f b)) :
    (∑ x, f x * phaseGiantPhysical (densityFourier f) x) =
      ((∑ b, ‖densityFourier f b‖ : ℝ) : ℂ) := by
…

    Mentions: OAI.Ostmann.densityFourier, OAI.Ostmann.phaseGiantPhysical