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Chebyshev Polynomials

Figure 3.28:

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 \\

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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See for details.


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