orthopolynom (version 1.0-6.1)

schebyshev.t.inner.products: Inner products of shifted Chebyshev polynomials

Description

This function returns a vector with \(n + 1\) elements containing the inner product of an order \(k\) shifted Chebyshev polynomial of the first kind, \(T_k^* \left( x\right)\), with itself (i.e. the norm squared) for orders \(k = 0,\;1,\; \ldots ,\;n \).

Usage

schebyshev.t.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 is as follows.

\(h_n = \left\langle {T_n^* |T_n^* } \right\rangle = \left\{ {\begin{array}{cc} {\frac{\pi } {2}} & {n \ne 0} \\ \pi & {n = 0} \\ \end{array} } \right.\)

References

Abramowitz, M. and I. A. Stegun, 1968. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., NY.

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.

Examples

###
### generate the inner products vector for the
### shifted T Chebyshev polynomials of orders 0 to 10
###
h <- schebyshev.t.inner.products( 10 )
print( h )