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

			(1)
			(2)

where denotes the inverse Fourier transform (where the transform pair is defined to have constants and ). Then the convolution is

			(3)
			(4)

Interchange the order of integration,

			(5)
			(6)
			(7)
			(8)

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

(9)

The convolution theorem also takes the alternate forms

			(10)
			(11)
			(12)

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, 3rd ed. New York: McGraw-Hill, pp. 108-112, 1999.

CITE THIS AS:

Weisstein, Eric W. "Convolution Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ConvolutionTheorem.html