lemma 17.14 Uniformly bounded sine sums

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:586 · p. 678

Rests on

depends_on proposition 8.4 Euler's formula ¶
- depends_on equation 8.2 eq:cpx-exp-series ¶
- depends_on proposition 7.47 Comparison; absolute convergence ¶
  - depends_on corollary A.46 Monotone convergence ¶
    - depends_on theorem A.40 Least-upper-bound property ¶
      - depends_on definition A.36 Cut ¶
      - depends_on definition A.38 Order on $\R$ ¶
      - proves proof app:A-long-proofs@proof-29 ¶
    - proves proof app:A-long-proofs@proof-33 ¶
  - depends_on definition 7.45 Series ¶
    - depends_on definition 7.2 Absolute value ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
  - depends_on theorem 7.8 Cauchy criterion ¶
    - depends_on corollary A.47 Cauchy completeness ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on theorem A.40 Least-upper-bound property ¶ ↺
      - proves proof app:A-long-proofs@proof-34 ¶
    - depends_on definition 7.4 Convergence ¶ ↺
    - proves proof ch:05-real-analysis@proof-5 ¶
  - proves proof ch:05-real-analysis@proof-28 ¶
- proves proof ch:06-complex-analysis@proof-2 ¶
proves proof ch:15-fourier-integral-transforms@proof-12 ¶

Supports

depends_on lemma 17.15 A polynomial whose partial sum spikes at the origin ¶
- depends_on proposition 17.16 An explicit continuous function with a divergent Fourier series ¶

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`	→	Euler's formula	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:593
`depends_on`	←	A polynomial whose partial sum spikes at the origin	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:658
`proves`	←	ch:15-fourier-integral-transforms@proof-12	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:596