proposition 10.47 Bilinear expansion

open in the book · parts/02-mathematical-methods/08-pdes.tex:1263 · p. 353

Rests on

depends_on definition 10.42 Green's function ¶
- depends_on definition 17.40 Transform and derivative of a distribution ¶
  - depends_on definition 17.38 Schwartz space and tempered distributions ¶
  - depends_on proposition 17.39 The transform preserves $\mathcal{S}$ ¶
    - depends_on definition 17.38 Schwartz space and tempered distributions ¶ ↺
    - depends_on proposition 17.30 Elementary properties ¶
      - depends_on definition 17.29 Fourier transform; the treatise convention ¶
      - depends_on equation 7.28 eq:ana-parts ¶
      - depends_on lemma 17.7 Riemann–Lebesgue ¶
      - proves proof ch:15-fourier-integral-transforms@proof-19 ¶
    - 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 ¶
    - proves proof ch:15-fourier-integral-transforms@proof-25 ¶
- depends_on definition 10.4 The second-order operator ¶
  - depends_on definition 10.3 Principal part and principal symbol ¶
    - depends_on definition 10.2 Linear, semilinear, quasilinear ¶
      - depends_on definition 10.1 Partial differential equation; order ¶
    - depends_on equation 10.2 eq:pde-multi-index ¶
  - depends_on proposition 7.105 Clairaut–Schwarz ¶
    - 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 theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-64 ¶
depends_on remark 9.53 rem:slt-parseval-completeness ¶
proves proof ch:08-pdes@proof-16 ¶

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type	direction	node	provenance	where
`depends_on`	→	Green's function	declared	parts/02-mathematical-methods/08-pdes.tex:1273
`depends_on`	→	rem:slt-parseval-completeness	declared	parts/02-mathematical-methods/08-pdes.tex:1273
`proves`	←	ch:08-pdes@proof-16	declared	parts/02-mathematical-methods/08-pdes.tex:1276