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  [Graphics:Images/ChebyshevPolyMod_gr_110.gif]  in  [-1,1],  

the polynomial  
[Graphics:Images/ChebyshevPolyMod_gr_111.gif]  is the unique choice which has the property

            
[Graphics:Images/ChebyshevPolyMod_gr_112.gif]

Moreover,

            
[Graphics:Images/ChebyshevPolyMod_gr_113.gif].  

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 [Graphics:Images/ChebyshevPolyMod_gr_114.gif].

Exploration 4.

Construct Q(x) of degree n using the n+1 Chebyshev nodes and compare it to [Graphics:../Images/ChebyshevPolyMod_gr_115.gif].

 

Case (i). Using 2 nodes.

[Graphics:../Images/ChebyshevPolyMod_gr_116.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_117.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_118.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_119.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_120.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_121.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_122.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_123.gif]


Case (ii). Using 3 nodes.

[Graphics:../Images/ChebyshevPolyMod_gr_124.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_125.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_126.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_127.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_128.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_129.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_130.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_131.gif]


Case (iii). Using 4 nodes.

[Graphics:../Images/ChebyshevPolyMod_gr_132.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_133.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_134.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_135.gif]

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

[Graphics:../Images/ChebyshevPolyMod_gr_136.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_137.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_138.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_139.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_140.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_141.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_142.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_143.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_144.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_145.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004