theorem 17.53 Convolution theorem

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2018 · p. 694

Rests on

depends_on definition 17.50 Convolution ¶
depends_on definition 17.29 Fourier transform; the treatise convention ¶
depends_on proposition 17.51 Algebra of convolution ¶
- depends_on definition 17.50 Convolution ¶ ↺
- proves proof ch:15-fourier-integral-transforms@proof-32 ¶
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-34 ¶

Supports

depends_on theorem 17.57 Wiener–Khinchin, finite-energy form ¶
depends_on theorem 10.62 The heat kernel ¶
- depends_on proposition 10.63 Infinite propagation speed ¶

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`	→	Convolution	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2028
`depends_on`	→	Fourier transform; the treatise convention	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2028
`depends_on`	→	Algebra of convolution	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2028
`depends_on`	→	Fourier inversion	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2028
`depends_on`	←	Wiener–Khinchin, finite-energy form	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2138
`depends_on`	←	The heat kernel	declared	parts/02-mathematical-methods/08-pdes.tex:1705
`proves`	←	ch:15-fourier-integral-transforms@proof-34	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2032