theorem 17.9 Dirichlet

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:399 · p. 676

Rests on

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 definition 3.43 Map ¶
  - depends_on equation 7.14 eq:ana-derivh ¶
  - depends_on proposition 7.6 Algebra of limits ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶
    - depends_on proposition 7.3 Triangle inequality ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - proves proof ch:05-real-analysis@proof-1 ¶
    - 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 definition 7.16 Limit ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
      - proves proof ch:05-real-analysis@proof-6 ¶
    - 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 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-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 ¶

Supports

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
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Dirichlet kernel	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:415
`depends_on`	→	Riemann–Lebesgue	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:415
`depends_on`	→	Mean value theorem	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:415
`depends_on`	←	Uniform convergence for $C^{1}$ functions	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:502
`depends_on`	←	Poisson summation	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1858
`proves`	←	ch:15-fourier-integral-transforms@proof-7	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:418