spherical.inner.products: Inner products of spherical polynomials
Description
This function returns a vector with \(n + 1\) elements containing the inner product
of an order \(k\) spherical polynomial, \(P_k \left( x \right)\),
with itself (i.e. the norm squared) for orders \(k = 0,\;1,\; \ldots ,\;n\).
Usage
spherical.inner.products(n)
Arguments
n
integer value for the highest polynomial order
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 \(n\) orthogonal polynomial
Details
The formula used to compute the inner products of the spherical orthogonal polynomials
is the same as that used for the Legendre orthogonal polynomials.
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.
# NOT RUN {###### generate the inner products vector for the spherical polynomals### of orders 0 to 10.###h <- spherical.inner.products( 10 )
print( h )
# }