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Convolution of Fourier transforms : Associativity Law

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The convolution of two functions f and g is defined to be the new function f * g.

whenever the integral converges.
(a)Is it true that f*g = g*f? Why?
(b) If F(k) = (i//) f°° exp (?ikx) f(x) dx and G(k) = (i//) f°° exp (?ikx) g(x) dx are the Fourier transforms of f(x) and g(x), FIND the Fourier transform of h(x) = f *g (x).
(c) Is it true that (f * g) * h (x) = f * (g * h) (x)? If yes, can this Associativity Law be validated with the help of (b)'?


Solution Summary

A convolution of Fourier transfors is investigated. The solution is detailed and well presented.