Example 5.  What is the maximum over the interval  [0, 1]  for each of the quantities, where    is the Chebyshev polynomial.
5 (a).  Find  .  
5 (b).  Find  .  
5 (c).  Find  .  
5 (d).  Find  .  
5 (e).  Find  .  
5 (f).  Compare the result with the error bounds for the Lagrange polynomial based on equally spaced points.

Solution 5.

5 (a-e).  The error bounds for the Chebyshev polynomials are:

5 (f).  The error bounds for the Lagrange polynomial based on equally spaced abscissas were

      

The error bound for the Chebyshev polynomial is smaller than the error bound for the Lagrange polynomial using equally spaced abscissas.

(c) John H. Mathews 2004