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

  $$\begin{small}\begin{array}{l} T_0(x)=1 T_1(x)=x T_2(x)=2x^2... ... T_{10}(x)=512x^{10}-1280x^8+1120x^6-400x^4+50x^2-1 \end{array} \end{small}$$ (6.1)

  
  

• Note that the coefficient of $x^n$ in $T_n(x)$ is always $2^{n-1}$.
• In Fig. 6.1, we plot the first four polynomials of Eqn.6.1.

Figure 6.1: Plot of the first four polynomials of the Chebyshev polynomials.

• The members of this series of polynomials can be generated from the two-term recursion formula
  $$\begin{array}{l} T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)\\ T_0(x)=1    \&   T_1(x)=x\\ \end{array}$$

• They form an orthogonal set,
  $$\int_{-1}^1\frac{T_n(x)T_m(x)}{\sqrt{1-x^2}}dx=\left\lbrace ... ... & n=m=0 \\ \pi /2, & n=m\neq 0 \\ \end{array}\right\rbrace$$

• The Chebyshev polynomials are also terms of a Fourier series, because
  $$T_n(x) = cos (n \theta)$$
  where $\theta =arc(cos x)$. Observe that
  $$\begin{array}{lcl} n=0; & cos 0=1 \rightarrow & T_0=1\\ n=1... ...os \theta = cos(arc(cos x))=x \rightarrow& T_1=x\\ \end{array}$$

• 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 6.1 shows.

• MATLAB has no commands for these polynomials but this M-file will compute them:
  

  
  
  if symop does not exist, http://siber.cankaya.edu.tr/ozdogan/NumericalComputations/mfiles/chapter4/symop.m download.

• 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 of this upper bound.
• Example m-file: Comparison of Lagrangian interpolation polynomials for equidistant and non-equidistant (Chebyshev) sample points for the function $f(x) = \frac{1}{1+x^2}$ (http://siber.cankaya.edu.tr/ozdogan/NumericalComputations/mfiles/chapter4/lagrange_chebyshev.mlagrange_chebyshev.m )