View source for Fourier integral

The non-discrete analogue of a [[Fourier series|Fourier series]]. The representation of a function given on a finite interval of the real axis by a Fourier series is very important. An analogous role is played by the representation of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f0410501.png" /> given on the whole axis by a Fourier integral:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f0410502.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f0410503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

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The representation (1) can be constructed formally by assuming the existence of the described integrals. It is valid, for example, for a smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f0410505.png" /> of compact support. There are a great number of tests guaranteeing equation (1) in some sense or other. The substitution of (2) into (1) gives the so-called Fourier integral formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f0410506.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

whose proof reduces to the tests mentioned above. It can be of great advantage to use the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f0410507.png" /> by the simple Fourier integral

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which is obtained from (3) by writing the inner integral as a limit over the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f0410509.png" /> and changing the order of integration. In applied sciences the representation (1) is often interpreted as expansion into harmonics: If

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105010.png" /></td> </tr></table>

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then (1) takes the form:

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and thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105013.png" /> is represented by a superposition of harmonics with frequencies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105014.png" /> which continuously fill the real semi-axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105015.png" />, while the amplitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105016.png" /> and the initial phase <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105017.png" /> depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105018.png" />.

In many cases (in particular for complex-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105019.png" />) it is more convenient to present (1) in exponential form:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105022.png" /> is called the [[Fourier transform|Fourier transform]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105023.png" /> (in applied sciences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105024.png" /> is called the frequency characteristic or the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105025.png" />).

Under the condition that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105026.png" /> is summable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105027.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105028.png" /> is bounded, uniformly continuous on the real axis and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105029.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105030.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105031.png" /> need not be integrable and so the integral (4) need not exist. However, (4) admits a reasonable interpretation if methods of summing integrals are used (here one can consider not only pointwise convergence but also convergence in the mean). For example, the arithmetical means of the truncated Fourier integrals

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105033.png" /></td> </tr></table>

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of a summable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105035.png" /> converge in the mean to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105036.png" /> almost-everywhere as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105037.png" />. With additional restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105038.png" /> one can obtain more specific assertions. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105039.png" /> and has bounded variation in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105040.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105041.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

In applications one often uses the representation

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which is true for an absolutely-integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105043.png" /> that is piecewise smooth in every finite interval, where the integral on the right-hand side is taken to mean the principal value (6). The Fourier integral is also studied when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105044.png" /> is assumed to be locally summable and with some hypotheses imposing restrictions on the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105045.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105046.png" />. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105048.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105049.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

where the limit is understood to be convergence in the mean of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105051.png" /> (however, the limit in (7) also exists in the sense of convergence almost-everywhere). This result acquires a simple form when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105052.png" /> (see [[Plancherel theorem|Plancherel theorem]]).

The theory of multiple Fourier integrals is constructed analogously when one discusses the expansion of a function given on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041050/f04105053.png" />-dimensional space. The concept of the Fourier integral has been extended also to generalized functions.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.C. Titchmarsh,   "Introduction to the theory of Fourier integrals" , Oxford Univ. Press  (1948)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Bochner,   "Lectures on Fourier integrals" , Princeton Univ. Press  (1959)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR></table>