theorem 17.32 Fourier inversion

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1302 · p. 686

Rests on

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 ¶

Supports

depends_on corollary 17.81 Sampling periodises the spectrum; aliasing ¶
depends_on corollary 17.33 The transform is injective ¶
depends_on lemma A.584 Fourier injectivity in $n$ variables ¶
- depends_on theorem A.587 The Gaussian average is a non-zero projector that absorbs the Weyl operators ¶
  - depends_on lemma A.590 The Gram matrix is universal ¶
    - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶
  - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
    - depends_on lemma A.590 The Gram matrix is universal ¶ ↺
    - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
  - depends_on proposition A.592 The vacuum of the Schrödinger system ¶
depends_on proposition 17.41 The identities physics uses ¶
- depends_on proposition 17.90 Hankel transform as the axially symmetric Fourier transform ¶
- depends_on proposition 17.52 Linear translation-invariant systems ¶
  - depends_on definition 28.25 Complex susceptibility ¶
  - depends_on proposition 17.61 Mean-square response of a damped resonator ¶
depends_on proposition 17.42 Sokhotski–Plemelj ¶
- depends_on proposition 105.22 The two-state effective generator ¶
  - depends_on proposition 105.23 Symmetry constraints on the mass matrix ¶
- depends_on theorem 17.78 Causality implies dispersion relations ¶
  - depends_on remark 33.10 Fibre anelasticity, and why it biases one method and not the other ¶
  - depends_on remark 28.26 Causality is visible in the susceptibility ¶
  - depends_on remark 28.62 Neither velocity is a speed limit ¶
depends_on proposition 17.39 The transform preserves $\mathcal{S}$ ¶
- depends_on definition 17.40 Transform and derivative of a distribution ¶
  - depends_on definition 10.42 Green's function ¶
    - depends_on definition 10.50 Retarded and advanced Green's functions ¶
    - depends_on example 10.49 The method of images ¶
    - depends_on proposition 28.24 Impulse response of the damped oscillator ¶
    - depends_on proposition 10.47 Bilinear expansion ¶
    - depends_on proposition 10.43 Representation of the solution ¶
    - depends_on theorem 10.45 Symmetry of the Green's function ¶
  - depends_on proposition 17.41 The identities physics uses ¶ ↺
depends_on theorem 17.69 Bromwich inversion integral ¶
- depends_on corollary 17.71 Residue evaluation and causality ¶
  - depends_on corollary 17.76 Heaviside's expansion theorem ¶
  - depends_on proposition 17.77 Poles and stability ¶
depends_on theorem 17.53 Convolution theorem ¶
- depends_on theorem 17.57 Wiener–Khinchin, finite-energy form ¶
- depends_on theorem 10.62 The heat kernel ¶
  - depends_on proposition 10.63 Infinite propagation speed ¶
depends_on theorem 17.106 Filtered back-projection in a plane slice ¶
- depends_on proposition 25.59 Homodyne tomography inverts a Radon transform ¶
depends_on theorem 17.94 Mellin inversion ¶
- depends_on proposition 17.96 Poles give asymptotics ¶
- depends_on theorem 17.100 Moments diagonalise a convolution evolution ¶
  - depends_on corollary 17.101 Coupled channels ¶
depends_on theorem 17.36 Plancherel ¶
- depends_on theorem 17.43 Bandwidth theorem ¶
  - depends_on corollary 17.44 Duration and bandwidth in SI ¶
  - depends_on remark 17.46 The quantum reading, in SI ¶
depends_on theorem 17.105 Radon inversion in three-dimensional space ¶
… 3 more

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Fourier transform; the treatise convention	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1309
`depends_on`	→	eq:ft-gaussian	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1309
`depends_on`	←	Sampling periodises the spectrum; aliasing	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2918
`depends_on`	←	The transform is injective	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1357
`depends_on`	←	Fourier injectivity in $n$ variables	declared	appendices/A-long-proofs.tex:27974
`depends_on`	←	The identities physics uses	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1611
`depends_on`	←	Sokhotski–Plemelj	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1661
`depends_on`	←	The transform preserves $\mathcal{S}$	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1544
`depends_on`	←	Bromwich inversion integral	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2502
`depends_on`	←	Convolution theorem	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2028
`depends_on`	←	Filtered back-projection in a plane slice	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3656
`depends_on`	←	Mellin inversion	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3258
`depends_on`	←	Plancherel	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1447
`depends_on`	←	Radon inversion in three-dimensional space	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3611
`depends_on`	←	Sampling theorem	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2845
`depends_on`	←	Wiener–Khinchin	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2187
`depends_on`	←	Properties of the Wigner function	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1902
`proves`	←	ch:15-fourier-integral-transforms@proof-20	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1312