theorem 17.47 Poisson summation

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1841 · p. 692

Rests on

depends_on definition 17.1 Fourier coefficients and Fourier series ¶
depends_on definition 17.29 Fourier transform; the treatise convention ¶
depends_on theorem 17.9 Dirichlet ¶
- 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 ¶
    - depends_on definition 17.1 Fourier coefficients and Fourier series ¶ ↺
    - proves proof ch:15-fourier-integral-transforms@proof-1 ¶
  - proves proof ch:15-fourier-integral-transforms@proof-4 ¶
- 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 ¶
- depends_on theorem 7.35 Mean value theorem ¶
  - depends_on proposition 7.29 Linearity ¶
    - depends_on definition 7.11 Sum of functions ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-13 ¶
  - depends_on theorem 7.34 Rolle ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - proves proof ch:05-real-analysis@proof-18 ¶
  - proves proof ch:05-real-analysis@proof-19 ¶
- proves proof ch:15-fourier-integral-transforms@proof-7 ¶
proves proof ch:15-fourier-integral-transforms@proof-30 ¶

Supports

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
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Fourier coefficients and Fourier series	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1858
`depends_on`	→	Fourier transform; the treatise convention	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1858
`depends_on`	→	Dirichlet	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1858
`depends_on`	←	Sampling periodises the spectrum; aliasing	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2918
`depends_on`	←	Lattice sums and the reciprocal lattice	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1915
`proves`	←	ch:15-fourier-integral-transforms@proof-30	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1861