Let and be arbitrary functions of time with Fourier Transforms. Take

(1) |

(2) |

(3) |

Interchange the order of integration,

(4) |

So, applying a Fourier Transform to each side, we have

(5) |

(6) | |||

(7) | |||

(8) |

**References**

Arfken, G. ``Convolution Theorem.'' §15.5 in *Mathematical Methods for Physicists, 3rd ed.*
Orlando, FL: Academic Press, pp. 810-814, 1985.

Bracewell, R. ``Convolution Theorem.'' *The Fourier Transform and Its Applications.* New York: McGraw-Hill, pp. 108-112, 1965.

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1999-05-25