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<table width="100%" summary="page for chebyshev.s.inner.products"><tr><td>chebyshev.s.inner.products</td><td style="text-align: right;">R Documentation</td></tr></table>

<h2> Inner products of Chebyshev polynomials </h2>

<h3>Description</h3>

<p>This function returns a vector with <i>n + 1</i> elements containing the inner product of 
an order <i>k</i> Chebyshev polynomial of the second kind, <i>S_k &le;ft( x\right)</i>, 
with itself (i.e. the norm squared) for orders <i>k = 0,\;1,\; &hellip; ,\;n </i>.
</p>


<h3>Usage</h3>

<pre>
chebyshev.s.inner.products(n)
</pre>


<h3>Arguments</h3>

<table summary="R argblock">
<tr valign="top"><td><code>n</code></td>
<td>
<p> integer value for the highest polynomial order </p>
</td></tr>
</table>


<h3>Details</h3>

<p>The formula used to compute the inner products is as follows.
</p>
<p><i>h_n  = &le;ft\langle {S_n |S_n } \right\rangle  = &pi; </i>.
</p>


<h3>Value</h3>

<p>A vector with <i>n + 1</i> elements
</p>
<table summary="R valueblock">
<tr valign="top"><td><code>1 </code></td>
<td>
<p>inner product of order 0 orthogonal polynomial</p>
</td></tr>
<tr valign="top"><td><code>2 </code></td>
<td>
<p>inner product of order 1 orthogonal polynomial</p>
</td></tr>
</table>
<p>...
</p>
<table summary="R valueblock">
<tr valign="top"><td><code>n+1 </code></td>
<td>
<p>inner product of order <i>n</i> orthogonal polynomial</p>
</td></tr>
</table>


<h3>Author(s)</h3>

<p> Frederick Novomestky <a href="mailto:fnovomes@poly.edu">fnovomes@poly.edu</a> </p>


<h3>References</h3>

<p>Abramowitz, M. and I. A. Stegun, 1968. <em>Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables</em>, Dover Publications, Inc., New York.
</p>
<p>Courant, R., and D. Hilbert, 1989. <em>Methods of Mathematical Physics</em>,
John Wiley, New York, NY.
</p>
<p>Szego, G., 1939. <em>Orthogonal Polynomials</em>, 23, American Mathematical Society
Colloquium Publications, Providence, RI.
</p>


<h3>Examples</h3>

<pre>
###
### generate the inner products vector for the 
### S Chebyshev polynomials of orders 0 to 10
###
h &lt;- chebyshev.s.inner.products( 10 )
print( h )
</pre>


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