Fourier transform

From Glossary of Meteorology
(Redirected from Fourier transforms)



Fourier transform

An analytical transformation of a function f(x) obtained (if it exists) by multiplying the function by e-iux and integrating over all x,
ams2001glos-Fe14
where u is the new variable of the transform F(u) and i2 = -1.

If the Fourier transform of a function is known, the function itself may be recovered by use of the inversion formula:
ams2001glos-Fe15
The Fourier transform has the same uses as the Fourier series: For example, the integrand F(u) exp (iux) is a solution of a given linear equation, so that the integral sum of these solutions is the most general solution of the equation. When the variable u is complex, the Fourier transform is equivalent to the Laplace transform.
See also Fourier integral, spectral function.


Copyright 2022 American Meteorological Society (AMS). For permission to reuse any portion of this work, please contact permissions@ametsoc.org. Any use of material in this work that is determined to be “fair use” under Section 107 of the U.S. Copyright Act (17 U.S. Code § 107) or that satisfies the conditions specified in Section 108 of the U.S.Copyright Act (17 USC § 108) does not require AMS’s permission. Republication, systematic reproduction, posting in electronic form, such as on a website or in a searchable database, or other uses of this material, except as exempted by the above statement, require written permission or a license from AMS. Additional details are provided in the AMS Copyright Policy statement.