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Difference between revisions of "Fourier coefficients"

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The coefficients
 
The coefficients
 
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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/f041010/f0410101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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c_i = \frac{\int_X f \phi_i dx}{\int_X \phi_i^2 dx} \ \ \text{or}\ \ c_i = \frac{\int_X f \bar{\phi_i}dx}{\int_X \phi_i \bar{\phi_i} dx}  \tag{*}
 
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in the expansion of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041010/f0410102.png" /> defined on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041010/f0410103.png" /> with respect to an orthogonal system of real-valued (complex-valued) functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041010/f0410104.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041010/f0410105.png" /> is an orthogonal system in a Hilbert (pre-Hilbert) space, then, given an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041010/f0410106.png" /> of this space, the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041010/f0410107.png" /> are also called the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041010/f0410108.png" /> with respect to the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041010/f0410109.png" />. J. Fourier first investigated trigonometric series with coefficients defined by (*).
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in the expansion of a function $f$ defined on a space $X$ with respect to an orthogonal system of real-valued (complex-valued) functions on $X$. If $\{ \phi_i \}$ is an orthogonal system in a Hilbert (pre-Hilbert) space, then, given an element $f$ of this space, the numbers $c_i = (f,\phi_i)/(\phi_i,\phi_i)$ are also called the Fourier coefficients of $f$ with respect to the system $\{ \phi_i \}$. J. Fourier first investigated trigonometric series with coefficients defined by (*).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR></table>

Latest revision as of 20:16, 13 October 2014

The coefficients $$ c_i = \frac{\int_X f \phi_i dx}{\int_X \phi_i^2 dx} \ \ \text{or}\ \ c_i = \frac{\int_X f \bar{\phi_i}dx}{\int_X \phi_i \bar{\phi_i} dx} \tag{*} $$ in the expansion of a function $f$ defined on a space $X$ with respect to an orthogonal system of real-valued (complex-valued) functions on $X$. If $\{ \phi_i \}$ is an orthogonal system in a Hilbert (pre-Hilbert) space, then, given an element $f$ of this space, the numbers $c_i = (f,\phi_i)/(\phi_i,\phi_i)$ are also called the Fourier coefficients of $f$ with respect to the system $\{ \phi_i \}$. J. Fourier first investigated trigonometric series with coefficients defined by (*).

References

[1] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
How to Cite This Entry:
Fourier coefficients. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier_coefficients&oldid=17816
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article