Chebychev Polynomials (of the First Kind)
Chebychev Polynomials (of the First Kind)
In[57]:=
Table[ChebyshevT[k,x],{k,0,6}]
Out[57]=
2 3 2 4
{1, x, -1 + 2 x , -3 x + 4 x , 1 - 8 x + 8 x ,
3 5 2 4 6
5 x - 20 x + 16 x , -1 + 18 x - 48 x + 32 x }
In[58]:=
Plot[Evaluate[Table[ChebyshevT[k,x],{k,0,6}]],{x,0,6}];
The Chebyshev differential equation is
[ Sqrt[(1-x^2)]y' ]' +L 1/Sqrt[(1-x^2)] y=0 on -1<x<1
The eigenvalues are n^2 and the eigenfunctions are the Chebyshev polynomials.
Note that the eigenvalues are real and satisfy L->Infinity.
These polynomials DO solve the differential equation. Check one of them.
In[59]:=
Simplify[D[Sqrt[1-x^2] D[ChebyshevT[2,x],x],x]+2^2 1/Sqrt[1-x^2]ChebyshevT[2,x]]
Out[59]=
0
Check the first seven:
In[60]:=
Table[D[Sqrt[1-x^2] D[ChebyshevT[k,x],x],x]+k^2 1/Sqrt[1-x^2]ChebyshevT[k,x],{k,0,6}]
Out[60]=
2 2
-4 x 2 4 (-1 + 2 x )
{0, 0, ------------ + 4 Sqrt[1 - x ] + -------------,
2 2
Sqrt[1 - x ] Sqrt[1 - x ]
2 3
2 x (-3 + 12 x ) 9 (-3 x + 4 x )
24 x Sqrt[1 - x ] - -------------- + ---------------,
2 2
Sqrt[1 - x ] Sqrt[1 - x ]
3
2 2 x (-16 x + 32 x )
Sqrt[1 - x ] (-16 + 96 x ) - ----------------- +
2
Sqrt[1 - x ]
2 4
16 (1 - 8 x + 8 x )
--------------------,
2
Sqrt[1 - x ]
2 4
2 3 x (5 - 60 x + 80 x )
Sqrt[1 - x ] (-120 x + 320 x ) - --------------------- +
2
Sqrt[1 - x ]
3 5
25 (5 x - 20 x + 16 x )
------------------------,
2
Sqrt[1 - x ]
2 2 4
Sqrt[1 - x ] (36 - 576 x + 960 x ) -
3 5
x (36 x - 192 x + 192 x )
-------------------------- +
2
Sqrt[1 - x ]
2 4 6
36 (-1 + 18 x - 48 x + 32 x )
-------------------------------}
2
Sqrt[1 - x ]
In[61]:=
Simplify[%]
Out[61]=
{0, 0, 0, 0, 0, 0, 0}
By one of the major Sturm-Liouville theorems, the Chebyshev polynomials are orthogonal with respect to the weight 1/Sqrt[1-x^2]. Check one.
In[62]:=
Integrate[1/Sqrt[1-x^2] ChebyshevT[1,x]ChebyshevT[2,x],{x,-1,1}]
Out[62]=
0
Check many at a time:
In[63]:=
Table[Integrate[1/Sqrt[1-x^2] ChebyshevT[k,x]ChebyshevT[m,x],{x,-1,1}],{k,0,3},{m,0,3}];
MatrixForm[%]
Out[63]=
Pi 0 0 0
Pi
--
0 2 0 0
Pi
--
0 0 2 0
Pi
--
0 0 0 2