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<a name="Chebyshev-Approximations"></a>
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<p>
Next: <a href="Series-Acceleration.html#Series-Acceleration" accesskey="n" rel="next">Series Acceleration</a>, Previous: <a href="Numerical-Differentiation.html#Numerical-Differentiation" accesskey="p" rel="previous">Numerical Differentiation</a>, Up: <a href="index.html#Top" accesskey="u" rel="up">Top</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Chebyshev-Approximations-1"></a>
<h2 class="chapter">30 Chebyshev Approximations</h2>
<a name="index-Chebyshev-series"></a>
<a name="index-fitting_002c-using-Chebyshev-polynomials"></a>
<a name="index-interpolation_002c-using-Chebyshev-polynomials"></a>

<p>This chapter describes routines for computing Chebyshev approximations
to univariate functions.  A Chebyshev approximation is a truncation of
the series <em>f(x) = \sum c_n T_n(x)</em>, where the Chebyshev
polynomials <em>T_n(x) = \cos(n \arccos x)</em> provide an orthogonal
basis of polynomials on the interval <em>[-1,1]</em> with the weight
function <em>1 / \sqrt{1-x^2}</em>.  The first few Chebyshev polynomials are,
<em>T_0(x) = 1</em>, <em>T_1(x) = x</em>, <em>T_2(x) = 2 x^2 - 1</em>.
For further information see Abramowitz &amp; Stegun, Chapter 22. 
</p>
<p>The functions described in this chapter are declared in the header file
<samp>gsl_chebyshev.h</samp>.
</p>
<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top">&bull; <a href="Chebyshev-Definitions.html#Chebyshev-Definitions" accesskey="1">Chebyshev Definitions</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Creation-and-Calculation-of-Chebyshev-Series.html#Creation-and-Calculation-of-Chebyshev-Series" accesskey="2">Creation and Calculation of Chebyshev Series</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Auxiliary-Functions-for-Chebyshev-Series.html#Auxiliary-Functions-for-Chebyshev-Series" accesskey="3">Auxiliary Functions for Chebyshev Series</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Chebyshev-Series-Evaluation.html#Chebyshev-Series-Evaluation" accesskey="4">Chebyshev Series Evaluation</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Derivatives-and-Integrals.html#Derivatives-and-Integrals" accesskey="5">Derivatives and Integrals</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Chebyshev-Approximation-Examples.html#Chebyshev-Approximation-Examples" accesskey="6">Chebyshev Approximation Examples</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top">&bull; <a href="Chebyshev-Approximation-References-and-Further-Reading.html#Chebyshev-Approximation-References-and-Further-Reading" accesskey="7">Chebyshev Approximation References and Further Reading</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
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