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8. Prove the following properties of the Fourier convolution: (a) f (x) * g (x) = g (x) * f (x), (b) f * (g * h) = (f * g) * h, (c) f * (ag + bh) = a (f * g) +6(f* h), where a and b are constants,

8. Prove the following properties of the Fourier convolution: (a) f (x) * g (x) = g (x) * f (x), (b) f * (g * h) = (f * g) * h, (c) f * (ag + bh) = a (f * g) +6(f* h), where a and b are constants,

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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8. Prove the following properties of the Fourier convolution: (a) f (x) * g (x) = g (x) * f (x), (b) ƒ * (g * h) = (f * g) * h, (c) f * (ag + bh) = a (ƒ * g) +b(f * h), where a and b are constants, %3D (d) f * 0 = 0 * f = 0, (e) f * 1+ f, || (f) ƒ * /27 8 = f = /27 8 * f, %3D (g) F{f (x) g (x)} = (F * G) (k) = | F(k-)G(£) d£, 2
Transcribed Image Text:8. Prove the following properties of the Fourier convolution: (a) f (x) * g (x) = g (x) * f (x), (b) ƒ * (g * h) = (f * g) * h, (c) f * (ag + bh) = a (ƒ * g) +b(f * h), where a and b are constants, %3D (d) f * 0 = 0 * f = 0, (e) f * 1+ f, || (f) ƒ * /27 8 = f = /27 8 * f, %3D (g) F{f (x) g (x)} = (F * G) (k) = | F(k-)G(£) d£, 2
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