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Fourier Transform

Suppose f(x) has the Fourier transform F(ω). If a ≠ 0 show that f(ax) has the Fourier Transform 1/|a| F (ω/a).

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From what you say it follows that you know how to do it for a > 0.
Let us now do it for a < 0, and make substitution x = -y/a in the integral that defines Fourier transform:

int_{-infty}^{+infty} f(ax) e^{-i*omega*x} dx = (-1/a) int_{-infty}^{+infty} e^{+i*omega*y/a} dy

= (1/|a|) int_{-infty}^{+infty} e^{-i*omega*y/|a|} dy

= (1/|a|) F(omega/|a|).

Pay attention to all the signs: the limits ...

Solution Summary

A Fourier transform is investigated.