Theorem  (Minimax Property).   Assume that  n  is fixed.  Among all possible choices for  Q(x)  and thus among all possible choices for the distinct nodes    in  [-1,1],  

the polynomial    is the unique choice which has the property

            

Moreover,

            .  

Proof  Chebyshev Polynomials  Chebyshev Polynomials  

Exploration for the theorem.  Construct Q(x) of degree n using the n+1 Chebyshev nodes and compare it to .

Exploration 4.

Construct Q(x) of degree n using the n+1 Chebyshev nodes and compare it to .

Case (i). Using 2 nodes.

Case (ii). Using 3 nodes.

Case (iii). Using 4 nodes.

The symbolic manipulation required to simplify the above polynomial is overwhelming.  However we can simplify the list of coefficients.

(c) John H. Mathews 2004