definition 17.40 Transform and derivative of a distribution

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1569 · p. 689

ground object -- no derivation owed

Rests on

depends_on definition 17.38 Schwartz space and tempered distributions ¶
depends_on proposition 17.39 The transform preserves $\mathcal{S}$ ¶
- depends_on definition 17.38 Schwartz space and tempered distributions ¶ ↺
- depends_on proposition 17.30 Elementary properties ¶
  - depends_on definition 17.29 Fourier transform; the treatise convention ¶
  - depends_on equation 7.28 eq:ana-parts ¶
  - depends_on lemma 17.7 Riemann–Lebesgue ¶
    - 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 ¶
  - proves proof ch:15-fourier-integral-transforms@proof-19 ¶
- 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 ¶
- proves proof ch:15-fourier-integral-transforms@proof-25 ¶

Supports

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 definition 28.25 Complex susceptibility ¶
  - depends_on proposition 17.61 Mean-square response of a damped resonator ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Schwartz space and tempered distributions	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1581
`depends_on`	→	The transform preserves $\mathcal{S}$	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1581
`depends_on`	←	Green's function	declared	parts/02-mathematical-methods/08-pdes.tex:1159
`depends_on`	←	The identities physics uses	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1611