Chebyshev Polynomials and Chebyshev Series

• If we want to represent a known function as a polynomial, one way to do it is with a Taylor series. Given a function, $f(x)$, we write
  
  $$P_n(x)=a_0 + a_1(x - a) + a_2(x - a)^2 + a_3(x - a)^3 +\ldots+ a_n(x - a)^n + \ldots$$
  
  
  where $a_i=f^{(i)} (a)/i!$ (we remember that $f^{(0)}$ is just $f(a)$). Unless $f(x)$ is itself a polynomial, the series may have an infinite number of terms. Terminating the series incurs an error, the truncation error. The error after the $(x - a)^n$ term,
  
  $$Error=\frac{(x-a)^{n+1}}{(n+1)!}f^{(n+1)}(\xi), \xi~in~[a,x]$$
  
  

• A problem with using the Taylor series to get polynomial approximations to a transcendental function is that the error grows rapidly as $x$-values depart from $x = a$.
• For $f(x)=e^x$, the Taylor series is easy to write because the derivatives are so simple: $f^{(a)}=e^a$ for all orders and we have, for $a =0$ (which is then called a Maclaurin series)
  
  $$e^x\approx 1+1(x-0)+1/2(x - 0)^2 + 1/6(x - 0)^3$$
  
  

• if we use only terms through $x^3$; the error term shows that the error of this will grow about proportional to $x^4$ as $x$-values depart from zero. There is a way to deal with this rapid growth of the errors, and that is to write the polynomial approximation to $f(x)$ in terms of Chebyshev polynomials.