orthopolynom (version 1.0-6.1)

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)

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

Arguments

n

integer value for the highest polynomial order

Author

Frederick Novomestky

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.

See Also

legendre.inner.products

Examples

###
### generate the inner products vector for the spherical polynomals
### of orders 0 to 10.
###
h <- spherical.inner.products( 10 )
print( h )