Fourier transform: Difference between revisions

Latest revision as of 14:15, 20 February 2012

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,

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:

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.

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-<div class="definition"><div class="short_definition">An analytical transformation of a function ''f''(''x'') obtained (if it exists) by multiplying  the function by ''e''<sup>&minus;''iux''</sup> and integrating over all ''x'',  <div class="display-formula"><blockquote>[[File:ams2001glos-Fe14.gif|link=|center|ams2001glos-Fe14]]</blockquote></div> where ''u'' is the new [[variable]] of the transform ''F''(''u'') and ''i''<sup>2</sup> = &minus;1.</div><br/> <div class="paragraph">If the Fourier transform of a function is known, the function itself may be recovered by use of  the inversion formula:  <div class="display-formula"><blockquote>[[File:ams2001glos-Fe15.gif|link=|center|ams2001glos-Fe15]]</blockquote></div> 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]]. <br/>''See also'' [[Fourier integral]], [[spectral function]].</div><br/> </div>
+<div class="definition"><div class="short_definition">An analytical transformation of a function ''f''(''x'') obtained (if it exists) by multiplying  the function by ''e''<sup>-''iux''</sup> and integrating over all ''x'',  <div class="display-formula"><blockquote>[[File:ams2001glos-Fe14.gif|link=|center|ams2001glos-Fe14]]</blockquote></div> where ''u'' is the new [[variable]] of the transform ''F''(''u'') and ''i''<sup>2</sup> = -1.</div><br/> <div class="paragraph">If the Fourier transform of a function is known, the function itself may be recovered by use of  the inversion formula:  <div class="display-formula"><blockquote>[[File:ams2001glos-Fe15.gif|link=|center|ams2001glos-Fe15]]</blockquote></div> 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]]. <br/>''See also'' [[Fourier integral]], [[spectral function]].</div><br/> </div>
 </div>