The Fourier transform is a generalization of the Complex Fourier Series in the limit as
. Replace the discrete with the continuous while letting . Then change the sum to
an Integral, and the equations become

(1) | |||

(2) |

Here,

(3) |

(4) |

(5) | |||

(6) |

In general, the Fourier transform pair may be defined using two arbitrary constants and as

(7) | |||

(8) |

The

Since any function can be split up into Even and Odd portions and
,

(9) |

(10) |

A function has a forward and inverse Fourier transform such that

(11) |

- 1. exists.
- 2. Any discontinuities are finite.
- 3. The function has bounded variation. A Sufficient weaker condition is fulfillment of the Lipschitz Condition.

The Fourier transform is linear, since if and have Fourier Transforms and , then

(12) |

(13) |

The Fourier transform is also symmetric since implies .

Let denote the Convolution, then the transforms of convolutions of functions have particularly
nice transforms,

(14) | |||

(15) | |||

(16) | |||

(17) |

The first of these is derived as follows:

(18) |

where .

There is also a somewhat surprising and extremely important relationship between the Autocorrelation and the Fourier
transform known as the Wiener-Khintchine Theorem. Let
, and denote the Complex
Conjugate of , then the Fourier Transform of the Absolute Square of is given by

(19) |

The Fourier transform of a Derivative of a function is simply related to the transform of the
function itself. Consider

(20) |

(21) |

(22) |

(23) |

(24) |

(25) |

(26) |

(27) |

The important Modulation Theorem of Fourier transforms allows to be expressed in terms of as follows,

(28) |

Since the Derivative of the Fourier Transform is given by

(29) |

(30) |

(31) |

(32) |

(33) |

If has the Fourier Transform , then the Fourier transform has the shift property

(34) |

so has the Fourier Transform

(35) |

If has a Fourier Transform , then the Fourier transform obeys a similarity theorem.

(36) |

The ``equivalent width'' of a Fourier transform is

(37) |

(38) |

Any operation on which leaves its Area unchanged leaves unchanged, since

(39) |

In 2-D, the Fourier transform becomes

(40) |

(41) |

(42) | |||

(43) |

**References**

Arfken, G. ``Development of the Fourier Integral,'' ``Fourier Transforms--Inversion Theorem,'' and
``Fourier Transform of Derivatives.'' §15.2-15.4 in *Mathematical Methods for Physicists, 3rd ed.*
Orlando, FL: Academic Press, pp. 794-810, 1985.

Blackman, R. B. and Tukey, J. W. *The Measurement of Power Spectra, From the Point of View of Communications Engineering.*
New York: Dover, 1959.

Bracewell, R. *The Fourier Transform and Its Applications.* New York: McGraw-Hill, 1965.

Brigham, E. O. *The Fast Fourier Transform and Applications.* Englewood Cliffs, NJ: Prentice Hall, 1988.

James, J. F. *A Student's Guide to Fourier Transforms with Applications in Physics and Engineering.*
New York: Cambridge University Press, 1995.

Körner, T. W. *Fourier Analysis.* Cambridge, England: Cambridge University Press, 1988.

Morrison, N. *Introduction to Fourier Analysis.* New York: Wiley, 1994.

Morse, P. M. and Feshbach, H. ``Fourier Transforms.'' §4.8 in
*Methods of Theoretical Physics, Part I.* New York:
McGraw-Hill, pp. 453-471, 1953.

Papoulis, A. *The Fourier Integral and Its Applications.* New York: McGraw-Hill, 1962.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T.
*Numerical Recipes in C: The Art of Scientific Computing.* Cambridge, England: Cambridge University Press, 1989.

Sansone, G. ``The Fourier Transform.'' §2.13 in *Orthogonal Functions, rev. English ed.*
New York: Dover, pp. 158-168, 1991.

Sneddon, I. N. *Fourier Transforms.* New York: Dover, 1995.

Sogge, C. D. *Fourier Integrals in Classical Analysis.* New York: Cambridge University Press, 1993.

Spiegel, M. R. *Theory and Problems of Fourier Analysis with Applications to Boundary Value Problems.*
New York: McGraw-Hill, 1974.

Strichartz, R. *Fourier Transforms and Distribution Theory.* Boca Raton, FL: CRC Press, 1993.

Titchmarsh, E. C. *Introduction to the Theory of Fourier Integrals, 3rd ed.* Oxford, England: Clarendon Press, 1948.

Tolstov, G. P. *Fourier Series.* New York: Dover, 1976.

Walker, J. S. *Fast Fourier Transforms, 2nd ed.* Boca Raton, FL: CRC Press, 1996.

© 1996-9

1999-05-26