Chebyshev Polynomials

Chebyshev Coefficients for Chebyshev polynomials of the first kind.

chebCoeff(fun, a, b, n)

Arguments

fun

function to be approximated.

a, b

endpoints of the interval.

n

an integer >= 0.

Details

For a function fun on on the interval [a, b] determines the coefficients of the Chebyshev polynomials up to degree n that will approximate the function (in L2 norm).

Value

Vector of coefficients for the Chebyshev polynomials, from low to high degrees (see the example).

References

Weisstein, Eric W. “Chebyshev Polynomial of the First Kind." From MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html

Note

See the “Chebfun Project” <https://www.chebfun.org/> by Nick Trefethen.

See also

chebPoly, chebApprox

Examples

##  Chebyshev coefficients for x^2 + 1
n <- 4
f2 <- function(x) x^2 + 1
cC <- chebCoeff(f2, -1, 1, n)  #  3.0   0  0.5   0   0
cC[1] <- cC[1]/2               # correcting the absolute Chebyshev term
                               # i.e.  1.5*T_0 + 0.5*T_2
cP <- chebPoly(n)              # summing up the polynomial coefficients
p <- cC %*% cP                 #  0 0 1 0 1