proposition 17.52 Linear translation-invariant systems

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1979 · p. 694

Rests on

depends_on definition 17.50 Convolution ¶
depends_on definition 17.38 Schwartz space and tempered distributions ¶
depends_on proposition 17.41 The identities physics uses ¶
- depends_on definition 17.40 Transform and derivative of a distribution ¶
  - 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 ¶
      - 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 ¶
- depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
  - depends_on definition 7.41 Antiderivative ¶
    - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-20 ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.9 Real function ¶
  - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - proves proof ch:05-real-analysis@proof-23 ¶
    - depends_on theorem 7.24 Extreme value theorem ¶ ↺
    - proves proof ch:05-real-analysis@proof-24 ¶
  - proves proof ch:05-real-analysis@proof-25 ¶
- depends_on theorem 17.32 Fourier inversion ¶ ↺
- proves proof ch:15-fourier-integral-transforms@proof-26 ¶
proves proof ch:15-fourier-integral-transforms@proof-33 ¶

Supports

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`	→	Convolution	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1992
`depends_on`	→	Schwartz space and tempered distributions	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1992
`depends_on`	→	The identities physics uses	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1992
`depends_on`	←	Complex susceptibility	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:767
`depends_on`	←	Mean-square response of a damped resonator	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2267
`proves`	←	ch:15-fourier-integral-transforms@proof-33	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1995