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)
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 is as follows.
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.
# NOT RUN {###### generate the inner products vector for the### shifted T Chebyshev polynomials of orders 0 to 10###h <- schebyshev.t.inner.products( 10 )
print( h )
# }