orthopolynom (version 1.0-5)

chebyshev.s.weight: Weight function for the Chebyshev polynomial

Description

This function returns the value of the weight function for the order \(k\) Chebyshev polynomial of the second kind, \(S_k \left( x \right) \).

Usage

chebyshev.s.weight(x)

Arguments

x

the function argument which can be a vector

Value

The value of the weight function.

Details

The function takes on non-zero values in the interval \( \left( -2,2 \right) \). The formula used to compute the weight function is as follows.

\(w\left( x \right) = \sqrt {1 - \frac{{x^2 }}{4}}\)

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.

Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.

Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.

Examples

# NOT RUN {
###
### compute the S Chebyshev weight function for arguments between -2 and 2
###
x <- seq( -2, 2, .01 )
y <- chebyshev.s.weight( x )
plot( x, y )
# }

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