theorem 17.34 Parseval's identity for series

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1368 · p. 687

Rests on

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 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶
    - 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 definition A.36 Cut ¶
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-32 ¶
    - depends_on corollary A.46 Monotone convergence ¶
      - depends_on theorem A.40 Least-upper-bound property ¶ ↺
      - proves proof app:A-long-proofs@proof-33 ¶
    - 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 ¶

Supports

depends_on proposition 28.48 Modal energy ¶
depends_on theorem 17.80 Sampling theorem ¶
- depends_on corollary 17.107 How many projections ¶

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`	→	Orthogonality of the harmonics	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1381
`depends_on`	→	Least squares and Bessel's inequality	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1381
`depends_on`	→	Fejér	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1381
`depends_on`	←	Modal energy	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:1719
`depends_on`	←	Sampling theorem	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2845
`proves`	←	ch:15-fourier-integral-transforms@proof-22	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1384