| gegenbauer.inner.products {orthopolynom} | R Documentation |
Inner products of Gegenbauer polynomials
Description
This function returns a vector with n + 1 elements containing the inner product of
an order k Gegenbauer polynomial, C_k^{\left( \alpha \right)} \left( x \right),
with itself (i.e. the norm squared) for orders k = 0,\;1,\; \ldots ,\;n .
Usage
gegenbauer.inner.products(n,alpha)
Arguments
n |
integer value for the highest polynomial order |
alpha |
numeric value for the polynomial parameter |
Details
The formula used to compute the inner products is as follows.
h_n = \left\langle {C_n^{\left( \alpha \right)} |C_n^{\left( \alpha \right)} } \right\rangle = \left\{ {\begin{array}{cc}
{\frac{{\pi \;2^{1 - 2\,\alpha } \,\Gamma \left( {n + 2\,\alpha } \right)}}
{{n!\;\left( {n + \alpha } \right)\,\left[ {\Gamma \left( \alpha \right)} \right]^2 }}} & {\alpha \ne 0} \\
{\frac{{2\;\pi }}
{{n^2 }}} & {\alpha = 0} \\
\end{array} } \right..
Value
A vector with n + 1 elements
1 |
inner product of order 0 orthogonal polynomial |
2 |
inner product of order 1 orthogonal polynomial |
...
n+1 |
inner product of order |
Author(s)
Frederick Novomestky fnovomes@poly.edu
References
Abramowitz, M. and I. A. Stegun, 1968. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York.
Courant, R., and D. Hilbert, 1989. Methods of Mathematical Physics, John Wiley, New York, NY.
Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.
See Also
Examples
###
### generate the inner products vector for the
### Gegenbauer polynomials of orders 0 to 10
### the polynomial parameter is 1.0
###
h <- gegenbauer.inner.products( 10, 1 )
print( h )