Chebyshev Polynomials

Transforming the Interval for Interpolation

    Sometimes it is necessary to take a problem stated on an interval  [a,b]  and reformulate the problem on the interval  [c,d]  where the solution is known.  If the approximation    to    is to be obtained on the interval  [a,b],  then we change the variable so that the problem is reformulated on  [-1,1]:

      or  ,

where  .  The required Chebyshev nodes of    on  [-1,1]  are

    ,  

and the interpolating nodes    on  [a,b]  are obtained using the change of variable    for  .  

Theorem (Lagrange-Chebyshev Approximation).  Assume that    is the Lagrange polynomial that is based on the Chebyshev interpolating nodes    on  [a,b]  mentioned above.  

If    then

          

holds for all  .   

Algorithm ( Lagrange-Chebyshev Approximation ).  The Chebyshev approximation polynomial    of  degree    for  f(x)  over  [-1,1]  can be written as a sum of  :  

    .  

The coefficients    are computed with the formulas  

    ,  
and  

    ,  

for    where   for  .  

Mathematica Subroutine ( Chebyshev Polynomial Approximation ). 

 
The first subroutine "Chebyshev" uses recursion to construct the Chebyshev Approximation Polynomial.  

    The following subroutine "ChebyshevPoly" version uses Mathematica's built-in functions to construct the Chebyshev approximation polynomial.  
In the construction, vector operations are used to assist the computations, since, similar terms occur in the summation for each of the coefficients.
Both give the same results.