proposition 25.59 Homodyne tomography inverts a Radon transform

open in the book · parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:2126 · p. 896

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depends_on definition 17.103 Radon transform ¶
depends_on theorem 17.106 Filtered back-projection in a plane slice ¶
- depends_on theorem 17.32 Fourier inversion ¶
  - depends_on definition 17.29 Fourier transform; the treatise convention ¶
  - depends_on equation 17.42 eq:ft-gaussian ¶
  - proves proof ch:15-fourier-integral-transforms@proof-20 ¶
- depends_on theorem 17.104 Projection-slice theorem ¶
  - depends_on definition 17.103 Radon transform ¶ ↺
  - depends_on definition 17.29 Fourier transform; the treatise convention ¶ ↺
  - proves proof ch:15-fourier-integral-transforms@proof-63 ¶
- proves proof ch:15-fourier-integral-transforms@proof-65 ¶
depends_on theorem 25.57 The Moyal equation, and its classical limit ¶
- depends_on equation 25.51 eq:pq-star ¶
- depends_on equation 25.53 eq:pq-wigner ¶
- depends_on theorem 24.21 Liouville ¶
  - depends_on definition 24.20 Liouville volume ¶
    - depends_on definition 13.102 Wedge product ¶
      - depends_on definition 13.98 $k$-form ¶
      - depends_on equation 13.239 eq:mfd-dual-basis ¶
    - depends_on definition 24.8 Symplectic manifold ¶
      - depends_on definition 13.105 Closed form ¶
      - depends_on definition 13.98 $k$-form ¶ ↺
      - depends_on definition 24.2 Symplectic vector space ¶
    - depends_on theorem 24.12 Darboux ¶
      - depends_on definition 24.8 Symplectic manifold ¶ ↺
      - depends_on theorem A.68 Darboux ¶
      - proves proof ch:07-symplectic-geometry@prooflink-1 ¶
  - depends_on proposition 7.105 Clairaut–Schwarz ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-64 ¶
  - depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
    - depends_on definition 24.14 Hamiltonian vector field ¶
      - depends_on definition 24.8 Symplectic manifold ¶ ↺
      - depends_on equation 22.4 eq:ham-pdot ¶
      - depends_on equation 22.3 eq:ham-qdot ¶
    - depends_on definition 24.8 Symplectic manifold ¶ ↺
    - depends_on proposition 13.128 Cartan's magic formula ¶
      - depends_on definition 13.103 Exterior derivative ¶
      - depends_on equation 13.246 eq:mfd-leibniz-forms ¶
      - depends_on equation 13.245 eq:mfd-nilpotency ¶
      - depends_on proposition 13.127 Component formulas ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-29 ¶
    - proves proof ch:07-symplectic-geometry@proof-5 ¶
  - proves proof ch:07-symplectic-geometry@proof-8 ¶
- proves proof ch:08-poisson-quantum-bridge@proof-22 ¶
proves proof ch:08-poisson-quantum-bridge@proof-23 ¶

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type	direction	node	provenance	where
`depends_on`	→	Radon transform	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:2148
`depends_on`	→	Filtered back-projection in a plane slice	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:2148
`depends_on`	→	The Moyal equation, and its classical limit	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:2148
`proves`	←	ch:08-poisson-quantum-bridge@proof-23	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:2151