Chebyshev Series

• By rearranging the Chebyshev polynomials, we can express powers of $x$ in terms of them:

  $$\begin{array}{l} 1=T_0 x = T_1 x^2 = \frac{1}{2}(T_0 + T_... ... x^9=\frac{1}{256}(126T_1 + 84T_3+36T_5+9T_7+T_9) \end{array}$$ (6.2)

  
  

• By substituting these identities into an infinite Taylor series and collecting terms in $T_i(x)$, we create a Chebyshev series. For example, we can get the first four terms of a Chebyshev series by starting with the Maclaurin expansion for $e^x$. Such a series converges more rapidly than does a Taylor series on $[-1, 1]$;
  $$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\ldots$$
  Replacing terms by Eqn. 6.2, but omitting polynomials beyond $T_3(x)$ because we want only four terms (The number of terms that are employed determines the accuracy of the computed values), we have;
  $$e^x=1.2661T_0+1.1302T_1+0.2715T_2+0.0443T_3+\ldots$$
  To compare the Chebyshev expansion with the Maclaurin series, we convert back to powers of $x$, using Eqn. 6.1:

  $$e^x=0.9946+0.9973x+0.5430x^2+0.1772x^3+\ldots$$ (6.3)

  
  

• Table 6.2 and Figure 6.2 compare the error of the Chebyshev expansion, Eqn. 6.3, with the Maclaurin series $(1+x+ O.5x^2 +0.1667x^3)$, using terms through $x^3$ in each case.
  • The errors can be considered to be distributed more or less uniformly throughout the interval.
  • In contrast to this, the Maclaurin expansion, which gives very small errors near the origin, allows the error to bunch up at the ends of the interval.
  
  

  Table 6.2: Comparison of Chebyshev series for $e^x$ with Maclaurin series.

  
  
  

  Figure 6.2: Comparison of the error of Chebyshev series for $e^x$ with the error of Maclaurin series.