Relation to trigonometric functions.

    The signal property of Chebyshev polynomials is the trigonometric representation on [-1,1].

Consider the following expansion using the Mathematica command "FunctionExpand."

[Graphics:Images/ChebyshevPolyMod_gr_51.gif]

[Graphics:Images/ChebyshevPolyMod_gr_52.gif]

[Graphics:Images/ChebyshevPolyMod_gr_53.gif]

Exploration 2.

We are interested in the polynomial form of  Cos[n ArcCos[x]],  however we will restrict our analysis to [-1,1].

[Graphics:../Images/ChebyshevPolyMod_gr_54.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_55.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_56.gif]


[Graphics:../Images/ChebyshevPolyMod_gr_57.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_58.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_59.gif]

Here is a list of several expansions.  

[Graphics:../Images/ChebyshevPolyMod_gr_60.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_61.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_62.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_63.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_64.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_65.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_66.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_67.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_68.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_69.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_70.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_71.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_72.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_73.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_74.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_75.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_76.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_77.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_78.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_79.gif]

[Graphics:../Images/ChebyshevPolyMod_gr_80.gif]
[Graphics:../Images/ChebyshevPolyMod_gr_81.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004