theorem 17.80 Sampling theorem

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2830 · p. 703

Rests on

depends_on definition 17.79 Band-limited function ¶
- depends_on definition 17.29 Fourier transform; the treatise convention ¶
depends_on definition 17.1 Fourier coefficients and Fourier series ¶
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 17.34 Parseval's identity for series ¶
- depends_on lemma 17.2 Orthogonality of the harmonics ¶
  - depends_on definition 17.1 Fourier coefficients and Fourier series ¶ ↺
  - proves proof ch:15-fourier-integral-transforms@proof-1 ¶
- depends_on proposition 17.4 Least squares and Bessel's inequality ¶
  - depends_on definition 17.1 Fourier coefficients and Fourier series ¶ ↺
  - depends_on lemma 17.2 Orthogonality of the harmonics ¶ ↺
  - proves proof ch:15-fourier-integral-transforms@proof-3 ¶
- depends_on theorem 17.22 Fejér ¶
  - depends_on lemma 17.6 Dirichlet kernel ¶
    - depends_on definition 17.1 Fourier coefficients and Fourier series ¶ ↺
    - depends_on lemma 17.2 Orthogonality of the harmonics ¶ ↺
    - proves proof ch:15-fourier-integral-transforms@proof-4 ¶
  - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.4 Convergence ¶
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - proves proof ch:05-real-analysis@proof-10 ¶
  - proves proof ch:15-fourier-integral-transforms@proof-16 ¶
- proves proof ch:15-fourier-integral-transforms@proof-22 ¶
proves proof ch:15-fourier-integral-transforms@proof-50 ¶

Supports

depends_on corollary 17.107 How many projections ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Band-limited function	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2845
`depends_on`	→	Fourier coefficients and Fourier series	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2845
`depends_on`	→	Fourier inversion	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2845
`depends_on`	→	Parseval's identity for series	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2845
`depends_on`	←	How many projections	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3695
`proves`	←	ch:15-fourier-integral-transforms@proof-50	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2848