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    Fourier Transform

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    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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    Solution Preview

    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 ...

    Solution Summary

    A Fourier transform is investigated and solved in the solution.