Difference between revisions of "Fourier coefficients"
From Encyclopedia of Mathematics
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The coefficients | The coefficients | ||
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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{*} | |
| − | + | $$ | |
| − | in the expansion of a function | + | in the expansion of a function $f$ defined on a space $X$ with respect to an orthogonal system of real-valued (complex-valued) functions on $X$. If $\{ \phi_i \}$ is an orthogonal system in a Hilbert (pre-Hilbert) space, then, given an element $f$ of this space, the numbers $c_i = (f,\phi_i)/(\phi_i,\phi_i)$ are also called the Fourier coefficients of $f$ with respect to the system $\{ \phi_i \}$. J. Fourier first investigated trigonometric series with coefficients defined by (*). |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)</TD></TR></table> | ||
Latest revision as of 20:16, 13 October 2014
The coefficients $$ 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
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