lemma 17.7 Riemann–Lebesgue

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:334 · p. 675

Rests on

depends_on definition 7.39 Darboux sums and the definite integral ¶
- depends_on axiom 7.1 Completeness of $\R$ ¶
proves proof ch:15-fourier-integral-transforms@proof-5 ¶

Supports

depends_on corollary 17.8 Localisation ¶
depends_on proposition 17.10 Smoothness and coefficient decay ¶
- depends_on corollary 17.11 Uniform convergence for $C^{1}$ functions ¶
depends_on proposition 17.30 Elementary properties ¶
- depends_on example 17.31 The three standard pairs ¶
- 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 proposition 17.90 Hankel transform as the axially symmetric Fourier transform ¶
      - depends_on proposition 17.52 Linear translation-invariant systems ¶
- 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 17.9 Dirichlet ¶
- depends_on corollary 17.11 Uniform convergence for $C^{1}$ functions ¶ ↺
- depends_on theorem 17.47 Poisson summation ¶
  - depends_on corollary 17.81 Sampling periodises the spectrum; aliasing ¶
  - depends_on corollary 17.49 Lattice sums and the reciprocal lattice ¶
    - depends_on definition 17.25 Multiple Fourier series ¶
      - depends_on proposition 17.26 The lattice harmonics are an orthonormal basis ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
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Edges

type	direction	node	provenance	where
`depends_on`	→	Darboux sums and the definite integral	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:341
`depends_on`	←	Localisation	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:376
`depends_on`	←	Smoothness and coefficient decay	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:472
`depends_on`	←	Elementary properties	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1226
`depends_on`	←	Dirichlet	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:415
`proves`	←	ch:15-fourier-integral-transforms@proof-5	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:344