Chebyshev Polynomials

• A Maclaurin series can be thought of as representing $f(x)$ as a weighted sum of polynomials. The kind of polynomials that are used are just the successive powers of $x$: $1, x, x^2,x^3,\ldots$. Chebyshev polynomials are not as simple; the first 11 of these are
  

  $$\begin{array}{l} T_0(x)=1 \\ T_1(x)=x \\ T_2(x)=2x^2-1 \\ ... ...T_10(x)=512x^{10}-1280x^8+1120x^6-400x^4+50x^2-1\\ \end{array}$$ (1)

  
  
  The members of this series of polynomials can be generated from the two-term recursion formula
  
  $$T_{n+1}(x)=2xT_n(x)-T_{n-1}(x),~T_0(x)=1,~T_1(x)=x$$
  
  

• Note that the coefficient of $x^n$ in $T_n(x)$ is always $2^{n-1}$. In Fig. 1, we plot the first four polynomials of Eqn.1.
• They form an orthogonal set,
  
  $$\int_{-1}^1\frac{T_n(x)T_m(x)}{\sqrt{1-x^2}}dx=\left\{ \begi... ...\neq m \\ \pi, & n=m=0 \\ \pi /2, & n=m\neq 0 \\ \end{array}$$
  
  

• The Chebyshev polynomials are also terms of a Fourier series, because
  
  $$T_n(x) = cos n \theta$$
  
  
  where $\theta =arccos x$. Observe that $cos 0=1, cos \theta = cos(arccos x)=x$.

  Figure 1: Plot of the first four polynomials of the Chebyshev polynomials.

  
• Because of the relation $T_n(x) = cos(n\theta )$, the Chebyshev polynomials will have a succession of maxima and minima of alternating signs, as Figure 1 shows.
• All polynomials of degree $n$ that have a coefficient of one on $x^n$, the polynomial
  
  $$\frac{1}{2^{n-1}}T_n(x)$$
  
  
  has a smaller upper bound to its magnitude in the interval [-1, 1]. This is important because we will be able to write power function approximations to functions whose maximum errors are given in terms ofthis upper bound.
• MATLAB has no commands for these polynomials but this M-file will compute them:

  function T=Tch(n)
  if n==0
  	disp('1')
  elseif n==1
  	disp('x')
  else
  	t0='1';
  	t1='x';
  	for i=2:n
  		T=symop('2*x','*',t1,'-',t0);
  		t0=t1;
  		t1=T;
  	end 
  end
  >>Tch(5)
  >>collect(ans)
  ans= 16*x^5-20*x^3+5*x