Convolution Theorem
Let 
and be arbitrary functions of time 
with Fourier Transforms. Take
 | (1) |
 | (2) |
denotes the inverse Fourier Transform (where thetransform pair is defined to have constants
and 
). Then the Convolution is |  |  | |
 |  | (3) |
Interchange the order of integration,
| |  | |
|  | ||
|  | (4) |
So, applying a Fourier Transform to each side, we have
 | (5) |
 | |  | (6) |
 | | | (7) |
 | |  | (8) |
See also Autocorrelation, Convolution, Fourier Transform, Wiener-Khintchine Theorem
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.
- Multiplicative Character
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