R: Chebyshev Polynomials

chebCoeff {pracma} | R Documentation

## Chebyshev Polynomials

### Description

Chebyshev Coefficients for Chebyshev polynomials of the first kind.

### Usage

    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).

### Note

See the “Chebfun Project” <http://www.maths.ox.ac.uk/chebfun/> by Nick Trefethen.

### References

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

### 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

[Package pracma version 2.0.4 Index]