Chebyshev Polynomials

Figure 3.28:
\includegraphics[width=\twidth]{eps/first-even-chebs-c}

The $ n$th Chebyshev polynomial may be defined by

$\displaystyle T_n(x) = \left\{\begin{array}{ll}
\cos[n\cos^{-1}(x)], & \vert x\vert\le1 \\ [5pt]
\cosh[n\cosh^{-1}(x)], & \vert x\vert>1 \\
\end{array}\right..
$

The first three even-order cases are plotted in Fig.3.29. (We will only need the even orders for making Chebyshev windows.) Clearly, $ T_0(x)=1$ and $ T_1(x)=x$. Using the double-angle trig formula $ \cos(2\theta)=2\cos^2(\theta)-1$, it can be verified that

$\displaystyle T_n(x) = 2x T_{n-1}(x) - T_{n-2}(x)
$

for $ n\ge 2$. The following properties of the Chebyshev polynomials are well known:
  • $ T_n(x)$ is an $ n$th-order polynomial in $ x$.
  • $ T_n(x)$ is an even function when $ n$ is an even integer, and odd when $ n$ is odd.
  • $ T_n(x)$ has $ n$ zeros in the open interval $ (-1,1)$, and $ n+1$ extrema in the closed interval $ [-1,1]$.
  • $ T_n(x)>1$ for $ x>1$.


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Dolph-Chebyshev Window Definition
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